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Markets, Games, and Strategic Behavior:

Recipes for Interactive Learning

Charles A. HoltUniversity of Virginia Comments

welcome: [email protected]

August 2005

his !oo" will !e #u!lished !y Addison $esley in 200%& and anyre'uest to use it #rior to #u!lication should !e cleared with their editor& Adrienne ()Am!rosio& Ac'uisitions *ditor& *conomics&

Addison+$esley& %,-+/+-50& Adrienne.()Am!rosio@A$1.com

Charles A. HoltUniversity of Virginia

Co#yright& All 3ights 3eserved All

rights reserved.

,



Markets, Games, and Strategic Behavior Charles A. Holt

4ar"ets& ames and 6trategic 7ehavior

Charles A. Holt Co#yright& All 3ights 3eserved

a!le of Contents

Preface................................................................................................................ 7

Part I. Basic Concepts: Decisions, Game heor!, and Market "#$i%i&ri$m ''

Chapter 1. Introduction ................................................................................... 13

Chapter . ! Pit Market................................................................................... "

Chapter 3. Some Simp#e Games$ Competition, Coordination, and Guessing.. 3%

Chapter &. 'isk and (ecision Making............................................................. "1

Chapter ". 'andomi)ed Strategies................................................................... *3

Part II. Individ$a% Decision "(periments......................................................... )*

Chapter *. Pro+a+i#it Matching..................................................................... 77

Chapter 7. -otter Choice !noma#ies.............................................................. 7

Chapter . IS/ 0In Search of 2 ..................................................................... %7

Part III. Game heor! "(periments: reas$res and Int$itive

Contradictions .........................................................................................................

.................... '+)

Chapter %. Mu#tiStage Games in 45tensive 6orm......................................... 1%

Chapter 1. Genera#i)ed Matching Pennies.................................................. 11%

Chapter 11. 8he 8rave#er9s (i#emma............................................................ 131

Chapter 1. Coordination Games................................................................... 1&"

Part I. Market "(periments.......................................................................... '*)

Chapter 13. Monopo# and Cournot Markets................................................ 1"%

Chapter 1&. :ertica# Market 'e#ationships................................................... 171

2



Chapter 1". Market Institutions and Po;er ................................................... 17%

Chapter 1*. Co##usion and Price Competition .............................................. 1%"

Chapter 17. Product <ua#it, !smmetric Information, and Market 6ai#ure 3 Chapter 1. ! -imit /rder !sset Market....................................................... %

Part . -$ctions................................................................................................ /

Chapter 1%. Private :a#ue !uctions .............................................................. "

Chapter . 8he 8akeover Game................................................................... 3%

Chapter 1. 8he =inner9s Curse................................................................... &"

Chapter . Mu#ti>nit !uctions ................................................................... ""

Part I. Bargaining and 0airness.................................................................... 1)

Chapter 3. >#timatum Bargaining............................................................... *%

Chapter &. 8rust, 'eciprocit, and Principa#!gent Games........................ 1

Part II. P$&%ic Choice .................................................................................... 2)

Chapter ". :oting 0in progress2................................................................... %

Chapter *. :o#untar Contri+utions............................................................ %1

Chapter 7. 8he :o#unteer9s (i#emma.......................................................... 3"

Chapter . 45terna#ities, Congestion, and Common Poo# 'esources ......... 31"

Chapter %. 'ent Seeking............................................................................... 37

Part III. Information and Learning ............................................................. //*

Chapter 3. Baes9 'u#e................................................................................ 337

Chapter 31. Information Cascades................................................................ 3"3

Chapter 3. Statistica# (iscrimination.......................................................... 3*1

-ppendices: Instr$ctions for C%ass "(periments........................................... /)/

Pit Market Instructions 0Chapter 2 ............................................................... 373

Game Instructions 0Chapter 32....................................................................... 37* -otter Choice Instructions 0Chapter &2......................................................... 37

BS Game Instructions 0Chapter "2.................................................................. 3

Binar Prediction Game 0Chapter *2 ............................................................. 3




-otter Choice Instructions 0Chapter 72......................................................... 3&

Search Instructions 0Chapter 2...................................................................... 3*

Instructions$ 8rave#er9s (i#emma 0Chapter 112.............................................. 3% Instructions$ Coordination Game 0Chapter 12 ............................................. 3%1

Instructions$ Market Game 0Chapter 132...................................................... 3%3

Instructions$ Price?<ua#it Market 0Chapter 172.......................................... 3%"

Private :a#ue !uction 0Chapter 1%2 ............................................................... 3%%

Instructions for 8akeover Game 0Chapter 2................................................ &1

Common :a#ue !uction 0Chapter 12............................................................. &3

Mu#ti>nit !uction 0Chapter 2.................................................................... &"

>#timatum Bargaining 0Chapter 32 .............................................................. &

P#aor@eep Game 0Chapter *2 .................................................................. &%

:o#unteer9s (i#emma Game 0Chapter 72...................................................... &11

-o++ing Game Instructions 0Chapter %2..................................................... &13

Baes9 'u#e Instructions 0Chapter 32 ........................................................... &1"

References.......................................................................................................... 3'4

/



8reface

*conomics is en9oying a resurgence of interest in !ehavioral

considerations& i.e. in the study of how #eo#le actually ma"e decisions.

1a!oratory e#eriments are increasingly used to study !ehavior in mar"ets&

games& and other strategic situations. he rising interest in e#eriments is

reflected in the 2002 ;o!el 8ri<e& which went to an e#erimental economist and

an e#erimental #sychologist. $hole new su!+disci#lines are arising in the

literature =e.g.& !ehavioral game theory& !ehavioral law and economics& !ehavioral

finance& neuroeconomics>& and e#eriments #rovide "ey em#irical guide#osts for

develo#ments in these areas.

his !oo" is designed to com!ine a !ehavioral a##roach with active

classroom learning eercises. his !oo" is a collection of cha#ters& each

containing a single game and the associated analysis of the results. he games set

u# sim#le economic situations& e.g. a mar"et or auction& which highlight several

related economic ideas. *ach cha#ter #rovides the reading for a #articular class&

in a one+a+day a##roach. he reading can serve either as a su##lement to other

material& or =#refera!ly> it can !e assigned in con9unction with an in+class

e#eriment in which students #lay the game with each other. (oing the

e#eriments +efore the assigned reading enhances the teaching value of these

e#eriments. 4any of the games can !e run in class !y hand& with dice or #laying

cards. he a##endi contains instructions for hand+run games in many cases.

?or those with student com#uter access& all games are availa!le online at theVeconla! site:

htt#:veconla!.econ.virginia.eduadmin.htm =for instructor setu#>&htt#:veconla!.econ.virginia.edulogin.htm =for #artici#ant login>.

hese we!+!ased games can !e set u# and run from any standard !rowser

=nternet *#lorer or ;etsca#e> that is connected to the internet& without loading

any software. ?or instructions& see the lin" on the admin #age a!ove. he

we!!ased #rograms have fully integrated instructions that automatically conform

to the features selected !y the instructor in the setu#. $e!+!ased games are

'uic"er to administer& and the instructor data dis#lays can #rovide records of decisions& earnings& round+!y+round data averages& and in some cases& theoretical

calculations. hese dis#lays can !e #rinted or #ro9ected for #ost+e#eriment

discussions. here is an etensive menu of setu# o#tions for each game that lets

5




the instructor select #arameters& e.g. the num!ers of !uyers& sellers& decision

rounds& fied #ayments& #ayoffs& etc. ?or eam#le& the #rivate+value auction

setu# menu allows one to choose the range of randomly determined #rivate

values& the num!er of !idders& the num!er of rounds& the #ricing rule =first+#rice

or second+#rice& and winner+#ays or all+#ay >. have even taught classes where

students design their own e#eriments and run them on the others in the class&

with a formal =8ower+8oint> #resentation of results in the following class.

he first several cha#ters contain eam#les of mar"ets with !uyers and

sellers& sim#le two+#erson games& and individual lottery choice decisions. hese

initial cha#ters raise a few methodological issues& such as whether and when

financial incentives are needed for research andor teaching e#eriments. n

addition& some !asic notions of decision ma"ing and e'uili!rium are introduced.

he focus in these cha#ters is on the !asicsB discussion of anomalies and

alternative theories is deferred until later. After these cha#ters have !een covered&there is a lot of flei!ility in terms of the order of coverage of the remaining

cha#ters& which are divided into grou#s: individual decisions& games& auctions&

mar"ets& !argaining& #u!lic choice& and asymmetric information. t is #ossi!le to

#ic" and choose& !ased on the level and su!9ect matter of the course.

his !oo" #rovides an organi<ing device for courses in game theory&

to#ics in microeconomics& and introductory e#erimental economics. ?or an

u##er+level or graduate course in e#erimental economics& (avis and Holt s

=,> 45perimenta# 4conomics has the advantage of !eing organi<ed around the

main classes of e#eriments& with #resentations of the associated theory and

methodological conce#ts. his !oo"& in contrast& #resents a se'uence of #articular

games& one #er cha#ter& with a carefully measured amount of theory andmethodology that is mainly #resented in the contet of s#ecific eam#les. hese

cha#ters are relatively self+contained& which ma"es it #ossi!le to choose a

selection tailored for a #articular course& e.g. #u!lic economics. he !oo" could

also !e integrated into courses in microeconomics& managerial economics& or

strategy at the 4.7.A. level. 4any of the e#erimental designs may !e of interest

to non+economists& e.g. students of #olitical science& anthro#ology& and

#sychology& as well as anyone interested in !ehavioral finance or !ehavioral law

and economics.

4athematical arguments are sim#le& since e#eriments are ty#ically !ased on

#arametric cases that distinguish alternative theories. he mathematical

calculations are sometimes illustrated with s#readsheet #rograms that are

constructed in a ste#+!y+ste# #rocess. hen a #rocess of co#ying !loc"s of cells

results in iterative calculations that converge to e'uili!rium solutions. Calculus is

avoided& ece#t in a cou#le of #laces& e.g.& in the cha#ters on auctions. hese

%



o#tional sections are #receded !y discrete eam#les that #rovide the intuition

!ehind the more general results.

have tried to "ee# the tet sim#le. here are no footnotes. 3eferences to other

#a#ers are very limited& and most are confined to an etensions and further reading

section at the end of each cha#ter. he !oo" contains no etensive surveys of related

literature on research e#eriments. 6uch surveys can !e found in (avis and Holt

=,>& Dagel and 3oth s =,5> Aand+ook of 45perimenta# 4conomics& and Hey s

=,/> 45perimenta# 4conomics& which are #itched at a level a##ro#riate for advanced

undergraduates& graduate students& and researchers in the field. ?or more discussion of

methodology& see ?riedman and 6under s =,/> 45perimenta# Methods. he

references in all of these !oo"s are somewhat out of date& since the num!er of

#u!lished economics #a#ers using la!oratory methods has almost dou!led since ,5&

when the last of these !oo"s was #u!lished. his is an increase of a!out ,00

#u!lications& and there are many more un#u!lished wor"ing #a#ers =see ?igure ,., inCha#ter ,>. hese new #u!lications are listed and categori<ed !y "eyword on the

!i!liogra#hy of e#erimental economics and social science that is availa!le on+line at

htt#:www.#eo#le.virginia.eduEcah2"y2".htm

A friend once as"ed me who my intellectual hero was& and without

hesitation mentioned Vernon 6mith =who was then at Ari<ona and is currently at

eorge 4ason>. he wor" that he has done with his colleagues and coauthors has

always served as an ins#iration for me. His 2002 ;o!el 8ri<e in *conomics

=together with (anny Dahneman> is richly deserved& and the effects of his wor"

#ervade many #arts of this !oo".

would es#ecially li"e to than" my coauthor 1isa Anderson and her

$illiam and 4ary students for many hel#ful suggestions on this !oo" and on thesoftware that it utili<es. 4uch of what "now a!out these to#ics is due to 9oint

research #ro9ects with Faco! oeree =UVA + University van Amsterdam>& and with

others: 1isa Anderson =$illiam and 4ary>& 6imon Anderson =Virginia>& 6heryl

7all =Virginia ech>& Fordi 7randts =7arcelona>& 4onica Ca#ra =$ashington and

1ee>& Catherine *c"el =Virginia ech>& Fean *nsminger =Caltech>& 3oland ?ryer

=ChicagoHarvard>& 3osario ome< =4alaga>& 6usan 1aury =eorgia 6tate>&

(avid 3eiley =Ari<ona>& om 8alfrey =Cal ech>& Al 3oth =Harvard>& 3oger

6herman =Houston>& and 3ic" $ilson =3ice>. also received useful suggestions

from Fuan Camilo Cardenas =Universidad de los Andes& Colum!ia>& Fames Co

=Ari<ona>& ;ic" ?eltovich =Houston>& (an ?riedman =U.C. 6anta Cru<>& 8hil

rossman =6t. Cloud>& 6hachar Dariv =7er"eley>& Howard 1eatherman

=4aryland>& Fames 4ur#hy =4assachusetts>& and im 6almon =?lorida 6tate>.

Annie alman suggested the relevance of the film =a## Street to the material in

Cha#ter 20. =6he is an actress who #layed the role of 4ichael (ouglas s secretary

-




in the film.> n addition& was fortunate to have an unusually talented and

enthusiastic grou# of Virginia graduate and undergraduate students who read #arts

of the manuscri#t: *mily 7ec"& AF 7ostian& Feanna Com#osti& Dari *lasson& *rin

olu!& Datie Fohnson& 6helley Fohnson& Durt 4itman& Angela 4oore& 4ai 8ham&

1oren 8itt& and Uliana 8o#ova. wo other students& Foe 4onaco and A.F. 7ostian&

set u# the 1inu servers and #rocedures for running the #rograms on a networ" of

hand+held& wireless #oc"et 8Cs with color& touchsensitive screens. magine a

game theory class& outside on the University of Virginia lawn& with students

com#eting in an auction via wireless #oc"et 8C sG



8art . 7asic Conce#ts: (ecisions& ame heory&

and 4ar"et *'uili!rium

he net several cha#ters #rovide an introduction to the three main ty#es of

e#eriments: individual decisions& games& and mar"ets. *ach cha#ter introduces

some "ey conce#ts& such as e#ected value& ris" aversion& ;ash e'uili!rium& and

mar"et efficiency. Dnowledge of these conce#ts ma"es it #ossi!le to #ic" and

choose among the remaining cha#ters !ased on the goals and level of the course.

All cha#ters in this section should !e com#leted !efore #roceeding to the sections

that follow.

he first #art of Cha#ter , #rovides an introduction to the use of e#erimental

games for teaching and research& which is followed !y an overview of the to#icscovered in the !oo". he final two sections of this cha#ter contain a very !rief

discussion of methodology and a history of the develo#ment of e#erimental

economics. hese final two sections are o#tional for students in most courses

=other than e#erimental economics>.

Cha#ter 2 introduces students to a #it mar"et that corres#onds to the

trading of futures contracts in a trading #it& with free intermingling of #ros#ective

!uyers and sellers. he cha#ter descri!es a design that illustrates the role of

#rices in achieving an efficient set of trades. he !est #rocedure is to run a class

#it mar"et e#eriment +efore discussing the cha#ter material. nstructions for this

are #rovided in the a##endi& and the instructor is free to use #laying cards to

design a setu# that will com#lement or su##lement the lessons that can !e derived

from the mar"et design #resented in the cha#ter. he hand+run e#eriment is a

recommended way to stimulate a discussion of how mar"ets wor". However&

some with com#uter access may #refer to set u# the e#eriment using the (ou!le

Auction #rogram& which is listed as (A on the Veconla! instructor admin menu

under 4ar"ets.

Cha#ter #resents some sim#le games of com#etition& coordination and

guessing that serve to introduce the notion of a ;ash e'uili!rium. he 8risoner s

(ilemma and Coordination games can !e administered !y giving students #laying

cards and using the instructions #rovided in the a##endi& or these games can !e

run from the 4atri ame #rogram =4> on the Veconla! menu. he uessingame& which is easily run !y hand or with the #rogram & can !e used to focus

a discussion of how !ehavior may converge to a ;ash e'uili!rium.

he fourth cha#ter shifts attention to individual decisions in situations

with random elements that affect money #ayoffs. his #ermits a discussion of




e#ected money value for someone who is neutral towards ris". ;on+neutral

attitudes =ris" aversion or ris" see"ing> may also arise in some situations.

he notions of decision ma"ing in the #resence of random elements are used in

Cha#ter 5 to analy<e sim#le matri games in which the e'uili!rium may involve

randomi<ation. $e consider games of chic"en& !attle+of+sees& and matching

#ennies. he techni'ues used to find e'uili!ria in these sim#le games will !e

a##lied in more com#le situations encountered later& such as the choice of #rice

when there is danger of !eing undercut slightly !y com#etitors.

,0



Cha#ter ,. ntroduction

I. 5rigins

1i"e other scientists& economists o!serve naturally occurring data #atterns and

then try to construct e#lanations. hen the resulting theories are evaluated in

terms of factors li"e #lausi!ility& generality& and #redictive success. As with other

sciences& it is often difficult to sort out cause and effect when many factors are

changing at the same time. hus& there may !e several reasona!le theories that are

roughly consistent with the same o!servations. As Deynes noted& without a

la!oratory to control for etraneous factors& economists often test their theories !y

gauging reactions of colleagues. n such an environment& theories may gain

su##ort on the !asis of mathematical elegance& #ersuasion& and focal events ineconomic history li"e the reat (e#ression. heories may fall from fashion& !ut

the a!sence of shar# em#irical tests leaves an unsettling clutter of #lausi!le

alternatives. ?or eam#le& economists are fond of using the word e'uili!rium

#receded !y a 9uicy ad9ective =e.g. #ro#er& #erfect& divine& or universally divine>.

his clutter is often not a##arent in refined tet!oo" #resentations.

he develo#ment of so#histicated econometric methods has added an

im#ortant disci#line to the #rocess of devising and evaluating theoretical models.

;evertheless& any statistical analysis of naturally occurring economic data is

ty#ically !ased on a host of auiliary assum#tions. *conomics has only recently

moved in the direction of !ecoming a la!oratory science in the sense that "eytheories and #olicy recommendations are sus#ect if they cannot #rovide intended

results in controlled e#eriments. his !oo" #rovides an introduction to the study

of economic !ehavior& organi<ed around games that can !e #layed in class.

he first classroom mar"et games were conducted in a Harvard class !y

*dward Cham!erlin =,/>. He had #ro#osed a new theory of mono#olistic

com#etition& and he used e#eriments to highlight failures of the standard model

of #erfect com#etition. 6tudents were given !uyer and seller roles and

instructions a!out how trades could !e arranged. ?or eam#le& a seller would !e

given a card with a cost in dollars. f the seller were to find a !uyer who agreed to

#ay a #rice a!ove this cost& the seller would earn the difference. 6imilarly& a

!uyer would !e given a card with a resale value& and the !uyer could earn thedifference if a #urchase could !e arranged at a #rice !elow this resale value.

(ifferent sellers may !e given cards with different cost num!ers& and li"ewise&

!uyers may receive different values. hese values and costs are the "ey elements

that any theory of mar"et #rice determination would use to derive #redictions& as

,,




e#lained in Cha#ter 2. $ithout going into detail& it should !e clear that it is

#ossi!le to set u# a la!oratory mar"et& and to #rovide strong financial incentives

!y using large value and cost num!ers and !y #aying su!9ects their earnings in

cash.

Cham!erlin s classroom mar"ets #roduced some inefficiencies& which he

attri!uted to the tendency for !uyers and sellers in a mar"et to !rea" off and

negotiate in small grou#s. Vernon 6mith& who attended Cham!erlin s class& later

!egan running classroom mar"ets with an enforced central clearinghouse for all

offers to !uy and sell. his trading institution is called a dou!le auction since

sellers as" #rices tend to decline at the same time as !uyers !id #rices rise. A

trade occurs when the !id+as" s#read closes and someone acce#ts another s offer

to !uy or sell. 6mith o!served efficient com#etitive outcomes& even with as few

as %+,0 traders. his result was significant& since the classical large num!ers

assum#tions were not realistic a##roimations for most mar"et settings. 6mith searly wor" on the dou!le auction mar"et figured #rominently in his 2002 ;o!el

8ri<e in *conomics.

hese classroom mar"ets can !e 'uite useful& !ecause students learn what

an economic situation is li"e from the inside !efore standard #resentations of the

relevant economic theory. A classroom game =followed !y structured 'uestion+

and+answer discussion> can let students discover the relevant economic #rinci#le

for themselves& which enhances the credi!ility of seemingly a!stract economic

models. *ach of the cha#ters that follow will !e !uilt around a single game that

can !e run in class& using a we!+!ased software suite andor sim#le #ro#s li"e dice

and #laying cards.

II. 5vervie6

Individua# (ecisions

A mar"et ty#ically involves a relatively com#le set of interactions

!etween multi#le !uyers and sellers over time. 6ometimes it is useful to study

"ey as#ects of !ehavior of individuals in isolation. ?or eam#le& stoc" mar"ets

involve ma9or ris"s of gains and losses& !ut such ris"s de#end on antici#ated

!ehavior of others in the mar"et. t is straightforward to set u# a sim#le decision

e#eriment !y giving a #erson a choice !etween gam!les or lotteries& e.g. !etween

a sure ,0 and a coin fli# that yields 25 in the event of heads. 4any of the earlier

cha#ters #ertain to such individual decisions& where #ayoffs may de#end on

random events !ut do not de#end on the decisions made !y others. 6ome of the

to#ics in these early cha#ters& li"e e#ected values and ris" aversion& are useful in

,2



the analysis of interactive situations in su!se'uent cha#ters. n addition& the

cha#ters in #art of the !oo" are focused on a series of s#ecific decision ma"ing

situations& e.g. #rediction& search& and information #rocessing.

Game 8heor

he sim#lest strategic interactions among economic agents have !een

modeled as games. n a matching+#ennies game& for eam#le& each #layer

chooses heads or tails with the #rior "nowledge that one will win a sum of money

when the coins match& and the other will win when the coins do not match. *ach

#erson s o#timal decision in such a situation de#ends on what the other #layer is

e#ected to do. he systematic study of such situations !egan with Fohn von

;eumann and Iscar 4orgensern s =,//> 8heor of Games and 4conomic

Behavior . hey asserted that standard economic theory of com#etitive mar"ets

did not a##ly to the !ilateral and small+grou# interactions that ma"e u# asignificant #art of economic activity. heir solution was incom#lete& ece#t for

the case of <ero+sum games in which one #erson s loss is another s gain. $hile

the <ero+sum assum#tion may a##ly to some etremely com#etitive situations&

li"e s#orts contests or matching #ennies games& it does not a##ly to situations

where all #layers might #refer some outcomes to others. n #articular& economists

and mathematicians at the 3A;( Cor#oration in 6anta 4onica& California were

trying to a##ly game+theoretic reasoning to military tactics at the dawn of the

Cold $ar. n many nuclear scenarios it is easy to imagine that the winner may !e

much worse off than would !e the case in the a!sence of war& which results in a

non+<ero+sum game. At a!out this time& a young graduate student at 8rinceton

entered von ;eumann s office with a notion of e'uili!rium that a##lies to a wideclass of games& including the s#ecial case of those that satisfy the <ero+sum

#ro#erty. Fohn ;ash s notion of e'uili!rium and the half+#age #roof that it

generally eists were recogni<ed !y the ;o!el 8ri<e committee a!out 50 years

later. $ith the ;ash e'uili!rium as its "eystone& game theory has recently

achieved the central role that von ;eumann and 4orgenstern envisioned. ndeed&

with the ece#tion of su##ly and demand& the ;ash e'uili!rium is #ro!a!ly used

as often today as any other construct in economics.

ntuitively s#ea"ing& a ;ash e'uili!rium is a set of strategies& one for each

#layer& with the #ro#erty that no!ody would wish to deviate from their #lanned

action given the strategies !eing used !y the other #layers. Consider a

coordination game in which two sellers must decide inde#endently whether to sell

in a !lac" mar"et& or in an alternative = red > mar"et. here is not enough room for

two entrants in either mar"etB they !oth earn <ero for any redred or !lac"!lac"

outcome. $ith a red!lac" outcome& the entrant in the !lac" mar"et earns ,0

,




while the entrant in the red mar"et only earns 5. n this game& the congested

outcomes =redred or !lac"!lac"> that yield <ero #ayoffs are not ;ash e'uili!ria&

since each #erson would wish to deviate from their decision given the other s

decision to enter the same mar"et. 7y this time you should have reali<ed that the

;ash e'uili!rium is a red!lac" outcome& since each #erson earns a #ositive

amount =5 or ,0>& so neither would want to switch to the other s mar"et and end

u# earning 0.

his analysis does not settle the issue of which #erson gets to enter the

more #rofita!le !lac" mar"et& and many develo#ments in game theory are focused

on the issue of how to #redict which of several ;ash e'uili!ria will have more

#redictive #ower in a #articular game. 6uch games are easy to set u# as

e#eriments !y letting each #erson #lay a red card =Hearts or (iamonds> or a

!lac" card =Clu!s or 6#ades>. $hen eisting theory ma"es no #rediction&

o!served !ehavioral tendencies in such e#eriments can #rovide im#ortantguide#osts for the develo#ment of new theories. ndeed& the mathematicians and

economists at the 3A;( Cor#oration !egan running game e#eriments at a!out

the same time as Cham!erlin s classroom mar"et e#eriments. he game theory

cha#ters in #art #ertain to sim#le games& with a careful consideration of

conditions under which !ehavior does or does not conform to e'uili!rium

#redictions.

Markets

he cha#ters in #art V cover several ways that economists have modeled

mar"et interactions& with firms choosing #rices& #roduction 'uantities& or entry

decisions. 6ome mar"ets have distinct grou#s of !uyers and sellers& and othersmore closely resem!le stoc" mar"ets in which #urchase and resale is common.

4ar"et e#eriments can !e used to assess the antitrust im#lications of mergers&

contracts& and other mar"et conditions. Ine goal of such e#eriments is to assess

factors that increase the etent to which mar"ets achieve all #ossi!le gains from

trade& i.e. the efficienc of the mar"et. 4ar"ets with enough #rice flei!ility and

good information a!out going #rices tend to !e highly efficient.

!uctions

he advent of we!+!ased communications has greatly e#anded the

#ossi!ilities for setting u# auctions that connect large num!ers of geogra#hicallydis#ersed !uyers and sellers. ?or eam#le& several economists were recently

involved in designing and running an auction where!y farmers made offers to

sus#end irrigation on designated tracts of land in southwest eorgia during the

200, growing season. 7ids were collected from sites and were ran"ed and

,/



#rocessed !y a we!+!ased #rogram. he interim results were viewed online

hundreds of miles away !y eorgia *nvironmental 8rotection (ivision officials

who had to decide when to terminate !idding =Cummings& Holt& and 1aury&

forthcoming 200/>. $e!+!ased auctions have also !een used with great success

in the sale of communications !andwith in the U.6. and in *uro#e.

here are numerous decisions that must !e made in setting u# an auction& for

eam#le& whether to allow !id revisions during the auction. *#eriments can !e used

to test!ed alternative sets of rules. n the eorgia rrigation 3eduction Auction& for

eam#le& economists at eorgia 6tate !egan running e#eriments as soon as the law

#assed. 6tate officials o!served some of these e#eriments !efore drafting the actual

auction #rocedures. A large+scale dry run with over a hundred !idders at 5 southwest

eorgia locations #receded the actual auction& which involved a!out 200 farmers and

5 million dollars in irrigation reduction #ayments. he auction had much of the loo"

and feel of an e#eriment& with the reading of instructions& a round+!y+roundcollection of !ids& and we!+!ased !id collection and #rocessing. he cha#ters in #art

V #ertain to auctions& including a we!+!ased irrigation reduction auction.

Bargaining

*conomic decisions in small+grou# settings often raise issues of fairness

since earnings may !e ine'uita!le. n a sim#le ultimatum !argaining game& one

#erson #ro#oses a way to divide a fied amount of money& say ,00& and the other

may either agree to the division& which is im#lemented& or may re9ect the division&

which results in <ero earnings for !oth. Attitudes a!out ine'uity are more difficult

to model than the sim#ler selfish money+see"ing motives that may dominate in

im#ersonal mar"et situations. n this case& research e#eriments can #rovideinsights in areas where theory is silent or less develo#ed. 4any to#ics in 1aw and

*conomics involve !argaining =e.g. !an"ru#tcy& #re+trial settlements>& and

e#eriments can #rovide useful insights. *#eriments such as these also have

!een used !y anthro#ologists =*nsminger& 200,> to study attitudes a!out fairness

in #rimitive societies in Africa and 6outh America. he cha#ters in #art V

#ertain to !argaining and other classes of games where fairness& e'uity& and other

inter#ersonal factors seem to matter.

Pu+#ic Choice

nefficiencies can occur when some costs and values are not reflected in

#rices. ?or eam#le& it may !e difficult to set u# a mar"et that allows #u!lic

goods li"e national defense to !e #rovided !y decentrali<ed contri!utions.

Another source of #otential inefficiency occurs when one #erson s activity has a

negative im#act on others well !eing& as is the case with #ollution or the over+use

,5




of a freely availa!le& shared resource. ;on+#rice allocation mechanisms often

involve the dedication of real resources to lo!!ying. ;o individual contestant

would want the value of their effort to eceed the value of the o!9ect !eing

sought& !ut the aggregate lo!!ying costs for a num!er of individuals may !e large

relative to the value of the #ri<e. he #u!lic choice cha#ters in #art V #ertain to

such situationsB to#ics include voluntary contri!utions and the use of a

common#ool resource.

Information

4ar"ets may also fail to generate efficient results when #rices do not

convey #rivate information. $ith limited information& individuals may rely on

signals li"e educational credentials or ethnic !ac"ground. nformational

asymmetries can #roduce interesting #atterns of conformity that may have large

effects on stoc" #rices& hiring #atterns& etc. *#eriments are #articularly useful inthese cases& since the effect of informational dis#arities is to #roduce many ;ash

e'uili!ria. he information cha#ters in #art V include e#eriments on signaling

and discrimination.

III. - Brief Comment on Methodo%og!

his section and the net are o#tional for students who are #rimarily

interested in microeconomics and game theory& as o##osed to research in

e#erimental economics.

8reatmentsn an e#eriment& a treatment is a com#letely s#ecified set of #rocedures&

which includes instructions& incentives& rules of #lay& etc. Fust as scientific

instruments need to !e cali!rated& it is useful to cali!rate economics e#eriments&

which ty#ically involves esta!lishing a !aseline treatment for com#arisons. ?or

eam#le& su##ose that individuals are given sums of money that can either !e

invested in an individual or a grou# account& where investments in the grou#

account have a lower return to the individual& !ut a higher return to all grou#

mem!ers. f the ty#ical #attern of !ehavior is to invest half in each account& then

this might !e attri!uted either to the #articular investment return functions used in

the e#eriment or to going fifty+fifty in an unfamiliar situation. n this case& a #air

of treatments with differing returns to the individual account may !e used.

6u##ose that the investment rate for the individual account is fifty #ercent in one

treatment& which could !e attri!uted to confusion or uncertainty& as indicated

a!ove. he im#ortance of the relevant economic incentives could !e esta!lished

,%



if this investment rate falls& say to ten #ercent& when the return for investing in the

individual account is reduced.

;et& consider a mar"et eam#le. High #rices may !e attri!uted to small

num!ers of sellers or to the way in which !uyers are constrained from re'uesting

#rivate discounts. hese issues could !e investigated !y changing the num!er of

sellers& holding !uyers sho##ing rules constant& or !y changing !uyers sho##ing

o##ortunities while holding the num!er of sellers constant. 4any economics

e#eriments involve a 22 design with treatments in each of the four cells: low

num!ers with !uyer sho##ingB low num!ers with no sho##ingB high num!ers with

no sho##ingB high num!ers with sho##ing.

8reatment Structure$ Bet;eenSu+ectsD :ersus =ithinSu+ectsD (esigns

*ach of the following cha#ters is !ased on a single e#eriment. n order

to #reserve time for discussion& the e#eriments ty#ically involve a #air of treatments. Ine issue is whether half of the #eo#le are assigned to each treatment&

which is called a !etween+su!9ects design. he alternative is to let each #erson

ma"e decisions in !oth treatments& which #uts more #eo#le into each treatment

!ut affords less time to com#lete multi#le rounds of decision ma"ing in each

treatment. his is called a within+su!9ects design& since each #erson s !ehavior in

one treatment is com#ared with the same #erson s !ehavior in the other treatment.

*ach method has its advantages. f !ehavior is slow to converge or if many

o!servations are re'uired to measure what is !eing investigated& then the !etween+

su!9ects design may !e #referred since it will generate the most o!servations in

the limited time availa!le. 4oreover& su!9ects are only e#osed to a single

treatment& which avoids se'uence effects. ?or eam#le& a mar"et e#eriment thatlets sellers discuss #rices may result in some successful collusive arrangements&

with high #rices that may carry over even if communication is not allowed in a

second treatment. hese se'uence effects may !e avoided with a !etween+

su!9ects design. he advantage of a within+su!9ects design is that individual

differences are controlled for !y letting each #erson serve as their own control.

?or eam#le& su##ose that you have a grou# of eight adults& and you want to

determine whether they run as fast wearing !lue 9eans as with running shorts. n

any grou# of eight adults& running s#eeds may vary !y factors of 2 or &

de#ending on weight& age& health& etc. n this case it would !e desira!le to time

running s#eeds for each #erson under !oth conditions unless the distance were so

great that fatigue would cause ma9or se'uence effects. n general& a

withinsu!9ects design is more a##ealing if there is high !ehavioral varia!ility

across individuals.

,-




Incentives

*conomics e#eriments ty#ically involve monetary decisions li"e #rices&

costly efforts& etc. 4ost economists are sus#icious of results of e#eriments done

with hy#othetical incentives& and therefore real cash #ayments are generally used

in research e#eriments. As we shall see& sometimes incentives matter a lot and

sometimes not much at all. ?or eam#le& #eo#le have !een shown to !e more

generous in offers to others when such offers are hy#othetical than when

generosity has a real cost. ?or #ur#oses of teaching& it is ty#ically not #ossi!le or

even desira!le to use monetary #ayments. ;evertheless& the results of class

e#eriments can #rovide a useful learning e#erience as long as the effects of

#ayments in research e#eriments are #rovided when im#ortant incentive effects

have !een documented. herefore& the #resentations in the cha#ters that follow

will !e !ased on a miture of class and research e#eriments. $hen the term

e#eriment or research e#eriment is used& this will mean that all earnings were atreasona!le levels and were #aid in cash. he term classroom e#eriment

indicates that #ayoffs were !asically hy#othetical. However& for non+mar"et

classroom e#eriments discussed in this !oo"& would #ic" one #erson at random

e5 post and #ay a small fraction of earnings& usually several dollars. his

#rocedure is not generally necessary& !ut it was followed here to reduce

une#ected differences !etween classroom and research data.

'ep#ication

Ine of the main advantages of e#erimental analysis is the a!ility to re#eat the

same setu# numerous times in order to determine average tendencies that are

relatively insensitive to individual or grou# effects. 3e#lication re'uires thatinstructions and #rocedures !e carefully documented. ?or eam#le& it is essential

that instructions to su!9ects !e written as a scri#t that is followed in eactly the

same manner with each cohort that is !rought to the la!oratory. Having a set of

written instructions hel#s ensure that !iased terminology is avoided& and it

#ermits other researchers to re#licate the re#orted results. he general rule is that

enough detail should !e re#orted so that someone else could redo the e#eriment

in a manner that the original author=s> would acce#t as !eing valid& even if the

results turned out to !e different. ?or eam#le& if the e#erimenters #rovide a

num!er of eam#les of how #rices determine #ayoffs in a mar"et e#eriment& and

if these eam#les are not contained in the written instructions& the different resultsin a re#lication may !e due to differences in the way the #ro!lem is #resented to

the su!9ects.

Contro#

,



A second main advantage of e#erimentation is the a!ility to control the factors

that may !e affecting o!served !ehavior& so that etraneous factors are held

constant =controlled> as the treatment varia!le changes. he most common cause

of loss of control is changing more than one factor at the same time& so that it is

difficult to determine the cause of any o!served change in !ehavior. ?or eam#le&

a well+"nown study of lottery choice com#ared the tendency of su!9ects to choose

a sure amount of money with a lottery that may yield higher or lower #ayoffs. n

one treatment& the #ayoffs were in the <ero to ,0 range& and in another treatment

the #ayoffs were in the thousands of dollars. hen the author concluded that there

were no incentive effects& since there was no o!served difference in the tendency

to choose the safe #ayoff. he trou!le with this conclusion is that the high+#ayoff

treatment was conducted under hy#othetical conditions& so two factors were !eing

changed: the scale of the #ayoffs& and whether or not they were real or

hy#othetical. his conclusion turns out to !e 'uestiona!le& given results of e#eriments that change one factor at a time. ?or eam#le& Holt and 1aury

=200,> re#ort that scaling u# hy#othetical choices has little effect on the tendency

to select the safer o#tion in their e#eriments. hus& choice #atterns with low real

#ayoffs do loo" li"e choice #atterns with !oth low and high hy#othetical #ayoffs.

However& scaling u# the #ayoffs in the real =nonhy#othetical> treatments to

hundreds of dollars causes a shar# increase in the tendency to choose the safer

o#tion.

Control is also lost when #rocedures ma"e it difficult to determine the incentives

that #artici#ants actually faced in an e#eriment. f #eo#le are trading #hysical

o!9ects li"e university sweatshirts& differences in individual valuations ma"e it

hard to reconstruct the nature of demand in a mar"et e#eriment. here are& of course& situations where non+monetary rewards are desira!le& such as e#eriments

designed to test whether ownershi# of a #hysical o!9ect ma"es it more desired

= the endowment effect >. hus control should always !e 9udged in the contet of

the #ur#ose of the e#eriment.

6ata# 4rrors

8rofessional economists often loo" to e#erimental #a#ers for data #atterns that

su##ort eisting theories or that suggest desira!le #ro#erties of new theories.

herefore& the researcher needs to !e a!le to distinguish !etween results that are

re#lica!le from those that are artifacts of im#ro#er #rocedures. *ven students in

e#erimental sciences should !e sensitive to #rocedural matters so that they can

evaluate others results critically. *#eriments also #rovide a rich source of to#ics

for term #a#ers& senior dissertations& etc. 6tudents and others who are new to

e#erimental methods in economics should !e warned that there are some fatal

,




errors that can render useless the results of economics e#eriments. As the a!ove

discussion indicates& these include: ina##ro#riate incentives& nonstandari<ed

instructions and #rocedures& loss of control due to the failure to #rovide a

cali!rated !aseline treatment or the change in more than one design factor at the

same time.

A more detailed treatment of methodology can !e found in Cha#ters , and of

(avis and Holt =,> 45perimenta# 4conomics and ?riedman and 6under =,/>

45perimenta# Methods.

I. - Brief 7istor! of "(perimenta% "conomics

?igure ,., !elow shows the trends in #u!lished #a#ers in e#erimental

economics. he first #a#ers !y Cham!erlin and some of the game theorists at

3A;( were written in the late ,/0 s and early ,50 s. n addition& some of theearly interest in e#erimental methods was generated !y the wor" of ?oura"er and

6iegel =,%2& ,%>. 6iegel was a #sychologist with high methodological

standardsB some of his wor" on #ro!a!ility matching will !e discussed in a later

cha#ter. n the late ,50 s& some !usiness school faculty at #laces li"e

Carnegie4ellon !ecame interested in !usiness games& !oth for teaching and

research. And Vernon 6mith s early mar"et e#eriments were #u!lished in ,%2.

*ven so& there were less than ,0 #u!lications #er year !efore ,%5& and less than

0 #er year !efore ,-5. 6ome of the interesting wor" during this #eriod was

!eing done !y 3einhard 6elten and other ermans& and there was an international

conference on e#erimental economics held in ermany in ,-. (uring the late

,-0 s& Vernon 6mith was a visitor at Caltech& where he !egan wor"ing withCharles 8lott& who had studied at Virginia under Fames 7uchanan and was

interested in #u!lic choice issues. 8lott s =,-> first voting e#eriments

stimulated wor" on voting and agendas !y #olitical scientists in the early ,0 s.

Ither interesting wor" included the 7attalio& reen& and Dagel =,,>

e#eriments with rats and #igeons& and Al 3oth s early !argaining e#eriments&

e.g. 3oth and 4alouf =,->. here were still fewer than 50 #u!lications #er year

in the area !efore ,5. At this time& my former thesis advisor and colleague& *d

8rescott& told me that *#erimental economics was dead end in the ,%0 s and it

will !e dead end in the ,0 s.

n the ,0 s& Vernon 6mith and his colleagues and students at Ari<ona

esta!lished the first large la!oratory and !egan the #rocess of develo#ing

com#uteri<ed interfaces for e#eriments. n #articular& Arlington $illiams wrote

the first #osted+offer #rogram. After a series of conferences in ucson& the

*conomic 6cience Association was founded in ,%& and the su!se'uent

20



0255075

100125150175200225250275

Number of Publications

#residents constitute a #artial list of "ey contri!utors =6mith& 8lott& 7attalio&

Hoffman& Holt& ?orsythe& 8alfrey& Co& 6chotter& Camerer& and ?ehr>.

,/,5,5,%,%,-,-,,,,

0ig$re '.'. 8$m&ers of P$&%ished Papers in "(perimenta% "conomics

Vernon 6mith and 4ar" saac !egin editing 'esearch In 45perimenta#

4conomics& a series of collected #a#ers a##earing a!out every other year since

,-. he first com#rehensive !oo"s in this area were #u!lished in the ,0 s&

e.g. (avis and Holt s =,> and Hey =,/>. he ,5 Aand+ook of

45perimenta# 4conomics& edited !y Dagel and 3oth& contains survey #a#ers on

"ey to#ics li"e auctions& !argaining& #u!lic goods& etc& and these are still good

sources for reference. he first s#ecialty 9ournal& 45perimenta# 4conomics& was

started in ,. he strong interest among *uro#eans is indicated !y the fact that

one of the founding coeditors is Arthur 6chram from Amsterdam. he *conomic

6cience Association now has an annual international meeting and additional

regional meetings in the U.6. and *uro#e. hese develo#ments have resulted in

over a hundred #u!lications in this area every year since ,0& with highs of over

200 #er year since ,.hese num!ers indicate the growing acce#tance of e#erimental methods

in economics. 6ince the total num!er of #u!lications in all areas of economics

has increased& it is natural to consider #a#ers #u!lished in the to# 9ournals& where

the total article count is roughly constant. n the !merican 4conomic 'evie;& for

2,




eam#le& there was one e#erimental economics article in the ,50 s& articles in

the ,%0 s& ,, in the ,-0 s& /0 in the ,0 s& and % articles in the ,0 s.

6imilarly& the num!er of #u!lications in 4conometrica went from 2 in the ,-0 s

to ,2 in the ,0 s to 2- in the ,0 s. A searcha!le !i!liogra#hy of over /&000

#a#ers in e#erimental economics and related social sciences can !e found in the

author s E@ Bi+#iograph of 45perimenta# 4conomics$

htt#:www.#eo#le.virginia.eduEcah2"y2".htm =which also contains H41

reading lists for s#ecific to#ics>.

here are lots of eciting develo#ments in this field in recent years.

*conomics e#eriments are !eing integrated into introductory courses and the

wor"!oo"s of some ma9or tets. heorists are loo"ing at la!oratory results for

a##lications and tests of their ideas& and #olicy ma"ers are increasingly willing to

loo" at how #ro#osed mechanisms #erform in controlled tests. *#erimental

methods have !een used to design large auctions =e.g. the ?CC auction for !andwidth and the eorgia rrigation Auction> and systems for matching #eo#le

with 9o!s =e.g. medical residents and hos#itals>. wo of the reci#ients of the ,/

;o!el 8ri<e in *conomics& ;ash and 6elten& were game theorists who had run

their own e#eriments. 4ost significantly& the 2002 ;o!el 8ri<e in *conomics

was awarded to an e#erimental economist =6mith> and to an e#erimental

#sychologist =Dahneman> who is widely cited in the economics literature.

*conomics is well on its way to !ecoming an e#erimental science.

22



Cha#ter 2. A 8it 4ar"et

$hen !uyers and sellers can communicate freely and o#enly as in a trading #it&

the #rice and 'uantity outcomes can !e #redicta!le and efficient. (eviations tend

to !e relatively small and may !e due to informational im#erfections. he

discussion should !e #receded !y a class e#eriment& either using #laying cards

and the instructions #rovided in the A##endi& or using the (ou!le Auction

#rogram& which is listed under the 4ar"ets menu on the Veconla! site.

I. - Simp%e "(amp%e

Cham!erlin =,/> set u# the first mar"et e#eriment !y letting students with

!uyer or seller roles negotiate trading #rices. he #ur#ose was to illustratesystematic deviations from the standard theory of #erfect com#etition. ronically&

this e#eriment is most useful today in terms of what factors it suggests are

needed to #romote efficient& com#etitive mar"et outcomes.

*ach seller was given a card with a dollar amount or cost written on it. ?or

eam#le& one seller may have a cost of 2& and another may have a cost of .

he seller earns the difference !etween the sale #rice and the cost& so the low+cost

seller would !e searching for a #rice a!ove 2 and the high+cost seller would !e

searching for a #rice a!ove . he cost is not incurred unless a sale is made& i.e.

the #roduct is made to order. he seller would not want to sell !elow cost& since

the resulting loss is worse than the <ero earnings from no sale.6imilarly& each !uyer was given a card with a dollar amount or value written on

it. A !uyer with a value of ,0& for eam#le& would earn the difference if a #rice

!elow this amount could !e negotiated. A !uyer with a lower value& say /&

would refuse #rices a!ove that level& since a #urchase a!ove value would result in

negative earnings.

he mar"et is com#osed of grou#s of !uyers and sellers who can negotiate trades

with each other& either !ilaterally or in larger grou#s. ?or eam#le& su##ose that

the mar"et structure is shown in a!le 2.,. here are four !uyers with values of

,0& and four !uyers with values of /. 6imilarly& there are four sellers with costs

of 2& and four sellers with costs of .

n addition to the structural elements of the mar"et =num!ers of !uyers andsellers& and their values and costs>& we must consider the nature of the mar"et

#rice negotiations. A market institution is a full s#ecification of the rules of trade.

?or eam#le& one might let sellers #ost catalogue #rices and then let !uyers

contact sellers if they wish to #urchase at a #osted #rice& with discounts not

2




#ermitted. his institution& "nown as a #osted+offer auction& is sometimes used in

la!oratory studies of retail mar"ets. he asymmetry& with one side #osting and

the other res#onding& is common when there are many #eo#le on the

a!le 2.,. A 4ar"et *am#le

Values Costs

7uyer , ,0 6eller 2

7uyer 2 ,0 6eller ,0 2

7uyer ,0 6eller ,, 2

7uyer / ,0 6eller ,2 2

7uyer 5 / 6eller ,

7uyer % / 6eller ,/

7uyer - / 6eller ,5

7uyer / 6eller ,%

res#onding side and few on the #osting side. 8osting on the thin side may

conserve on information costs& and agents on the thin side may have the #ower to

im#ose #rices on a ta"e+it+or+leave+it !asis. n contrast& Cham!erlin used an

institution that was symmetric and less structuredB he let !uyers and sellers mi

together and negotiate !ilaterally or in small grou#s& much as traders of futures

contracts interact in a trading #it. 6ometimes he announced transactions #rices as

they occurred& much as the mar"et officials watching from a #ul#it over the

trading #it would "ey in contract #rices that are #osted electronically and flashed

to other mar"ets around the world. At other times& Cham!erlin did not announce

#rices as they occurred& which may have resulted in more decentrali<ed trading

negotiations.

II. - C%assroom "(periment

?igure 2., shows the results of a classroom #it mar"et e#eriment using

the setu# in a!le 2.,. 8artici#ants were University of Virginia *ducation 6chool

students who were interested in new a##roaches to teaching economics at the

secondary school level. $hen a !uyer and a seller agreed on a #rice& they came

together to the recording des"& where the #rice was chec"ed to ensure that it was

no lower than the seller s cost and no higher than the !uyer s value& as re'uired !ythe instructions given in the a##endi of this cha#ter. =he #rohi!ition of trading

at a loss was used since the earnings in this classroom e#eriment were only

hy#othetical.> *ach dot in the figure corres#onds to a trade. ;otice that the

2/



$0

$1

$2

$3

$4

$5

$6

$7

$8

$9

$10

0 2 3 41

'uantity of trades !egins at 5& goes u# to %& and then declines to four. he #rices

are varia!le in the first two #eriods and then sta!ili<e in the 5+- range.

Consider the 'uestion of why the #rices converged to the 5+- range& with a

transactions 'uantity of four units #er #eriod. ;otice that the 'uantity could have

!een as high as eight. ?or eam#le& su##ose that the four high+value =,0> !uyers

negotiated with high+cost => sellers and agreed on #rices of . 6imilarly&

su##ose that the negotiated #rices were for sales from the low+cost =2> sellers

to the low+value =/> !uyers. n this scenario& all of the sellers units sell& the

'uantity is eight& and each #erson earns , for the #eriod. 8rices& however& are

'uite varia!le. hese #atterns are not o!served in the #rice se'uence of trades

shown in ?igure 2.,. here is high varia!ility initially& as some of the high+cost

units are sold at high #rices& and some of the low+value units are #urchased at low

#rices. 7y the third #eriod& #rices have converged to the 5+- range that

#revents high+cost sellers from selling at a #rofit and low+value !uyers from !uying at a #rofit. he result is that only the four low+cost units are sold to the

four high+value !uyers.

8eriod

0ig$re .'. Contract Price Se#$ence

he intuitive reason for the low #rice dis#ersion in later #eriods is fairly o!vious.

At a #rice of & all eight sellers would !e willing =and #erha#s eager> to sell& !utonly the four high+value !uyers would !e willing to !uy& and #erha#s not so eager

given the low !uyer earnings at that high #rice. his creates a com#etitive

situation in which sellers may try to lower #rices to get a sale& causing a #rice

decline. Conversely& su##ose that #rices !egan in the range. At these low

25




#rices& all eight !uyers would !e willing to !uy& !ut only the four low+cost sellers

would !e willing to sell. his gives sellers the #ower to raise #rice without losing

sales.

?inally& consider what ha##ens with #rices in the intermediate range& from

/ to . At a #rice of %& for eam#le& each low cost seller earns / =J % + 2>&

and each high+value !uyer earns / =J ,0 + %>. ogether& the eight #eo#le who

ma"e trades earn a total of 2. hese earnings are much higher than , #er

#erson that was earned with #rice dis#ersion& for a total of , times ,% #eo#le J

,%. n this eam#le& the effect of reduced #rice dis#ersion is to reduce 'uantity

!y half and to dou!le total earnings. he reduced #rice dis#ersion !enefits !uyers

and sellers as a grou#& since total earnings go u#& !ut the ecluded high+cost

sellers and low+value !uyers are worse off. ;evertheless& economic efficiency

has increased !y these eclusions. he sellers have high costs !ecause the

o##ortunity costs of the resources they use are high& i.e. the value of the resourcesthey would em#loy =#erha#s inefficiently> is higher in alternative uses. 4oreover&

the !uyers with low values are not willing to #ay an amount that covers the

o##ortunity cost of the etra #roduction needed to serve them.

Ine way to measure the efficiency of a mar"et is to com#are the actual

earnings of all #artici#ants with the maimum #ossi!le earnings. t is a sim#le

calculation to verify that 2 is the highest total earnings level that can !e

achieved !y any com!ination of trades in this mar"et. he efficiency was ,00K

for the final two #eriods shown in ?igure 2.,& since the four units that traded in

each of those #eriods involved low costs and high values. f a fifth unit had

traded& as ha##ened in the first #eriod& the aggregate earnings must go down since

the high cost => for the fifth unit eceed the low value =/> for this unit. husthe earnings total goes down !y the difference /& reducing earnings from the

maimum of 2 to 2. n this case& the outcome with #rice dis#ersion and five

units traded only has an efficiency level of 22 J -.5 #ercent. he trading of a

sith unit in the second #eriod reduced efficiency even more& to -5 #ercent.

he o#eration of this mar"et can !e illustrated with the standard su##ly

and demand gra#h. ?irst consider a seller with a cost of 2. his seller would !e

unwilling to su##ly any units at #rices !elow this cost& and would offer the entire

ca#acity =, unit> at higher #rices. hus the seller s individual su##ly function has

a ste# at 2. he total 'uantity su##lied !y all sellers in the mar"et is <ero for

#rices !elow 2& !ut mar"et su##ly 9um#s to four units at slightly higher #rices as

the four low+cost sellers offer their units. he su##ly function has another ste# at

when the four high+cost sellers offer their units at #rices slightly a!ove this

high+cost ste#. his resulting mar"et su##ly function& with ste#s at each of the

two cost levels& is shown !y the solid line in ?igure 2.2. he mar"et demand

2%



$0$2

$4

$6

$8

$10

$12

$14

6 7 8 90 1 2 3 4 5

Supplyeman!

function is constructed analogously. At #rices a!ove ,0& no !uyer is willing to

#urchase& !ut the 'uantity demanded 9um#s to four units at #rices slightly !elow

this value. he demand function has another ste# at /& as shown !y the dashed

line in ?igure 2.2. (emand is vertical at #rices !elow /& since all !uyers will

wish to #urchase at any lower #rice. he su##ly and demand functions overla# at

a 'uantity of / in the range of #rices from / to . At any #rice in this region&

the 'uantity su##lied e'uals the 'uantity demanded. At lower #rices& there is

ecess demand& which would tend to drive #rices u#. At #rices a!ove the region

of overla#& there is ecess su##ly that would tend to drive #rices !ac" down.

he fact that the maimum aggregate earnings are 2 for this design can !e seen

directly from ?igure 2.2. 6u##ose that the #rice is % for all trades. he value of

the first unit on the left is ,0& and since the !uyer #ays %& the sur#lus value is

the difference& or /. he sur#lus on the second& third& and fourth units is also /&

so the consumers sur#lus is the sum of the sur#luses on individual consumersunits& or ,%. ;otice that consumers sur#lus is the area under the

Luantity

0ig$re .. - Simp%e Market Design

demand curve and a!ove the #rice #aid. 6ince sellers earn the difference !etween

#rice and cost& the #roducers sur#lus is the area =also ,%> a!ove the su##ly curve

and !elow the #rice. hus the total sur#lus is the sum of these areas& whiche'uals the area under demand and a!ove su##ly =to the left of the intersection>.

his total sur#lus area is four times the vertical distance =,0 + 2>& or 2. he

total sur#lus is actually inde#endent of the #articular #rice at which the units

trade. ?or eam#le& a higher #rice would reduce consumers sur#lus and increase

2-



$0

$1

$2

$3

$4

$5

$6

$7

$8$9

$10

$0

$1

$2

$3

$4

$5

$6

$7

$8

$9

$10


#roducers sur#lus& !ut the total area would remain fied at 2. Adding a fifth

unit would reduce sur#lus since the cost => is greater than the value =/>.

hese ty#es of sur#lus calculations do not de#end on the #articular form

of the demand and su##ly functions in ?igure 2.2. ndividual sur#lus amounts on

each unit are the difference !etween the value of the unit =the height of demand>

and the cost of the unit =the height of su##ly>. hus the area !etween demand and

su##ly to the left of the intersection re#resents the maimum #ossi!le total

sur#lus& even if demand and su##ly have more ste#s than the eam#le in ?igure

2.2.

?igure 2. shows the results of a classroom e#eriment with !uyers and

sellers& using a design with more ste#s and an asymmetric structure. ;otice that

demand is relatively flat on the left side& and therefore that the com#etitive #rice

range =from - to > is relatively high. ?or #rices in the com#etitive range& the

consumers sur#lus will !e much smaller than the #roducers sur#lus. he rightside shows the results of two #eriods of #it mar"et trading& with the #rices #lotted

in the order of trade. 8rices start at a!out 5& in the middle of the range !etween

the lowest cost and the highest value. he #rices seem to !e converging to the

com#etitive range from !elow& which could !e due to !uyer resistance to

increasingly une'ual earnings. n !oth #eriods& all seven of the higher value =,0

and > units were #urchased& and all seven of the lower cost =2 and -> units

were sold. As !efore& #rices stayed in a range needed to eclude the high+cost and

low+value units& and efficiency was ,00K in !oth #eriods.

0 , 2 / 5 % - ,0 + ' / 3 * 1 ) 2 4'+''''/'3'*'1')

Luantity 8rice 6eries for wo rading 8eriods

0ig$re ./. Demand, S$pp%!, and ransactions Prices for a Pit Market

2



he asymmetric structure in ?igure 2. was used to ensure that #rices did

not start in the e'uili!rium range& in order to illustrate some ty#ical features of

#rice ad9ustments. he first units that traded in each #eriod were the ones with

,0 values and 2 costs on the left side of the su##ly and demand figure. After

these initial transactions in the 5+- range& the remaining traders were closer to

the com#etitive margin. hese marginal !uyers were those with values of and

/. he marginal sellers were those with costs of - and . Clearly these units

will have to sell for #rices a!ove -& which forces the #rices closer to the

com#etitive #rediction at the end of the #eriod. $hen sellers who sold early in

the #eriod see these higher #rices& they may hold out for higher #rices in the net

#eriod. 6imilarly& !uyers will come to e#ect #rices to rise later in the #eriod& so

they will scram!le to !uy early& which will tend to drive #rices u# earlier in each

su!se'uent #eriod. o summari<e& the convergence #rocess is influenced !y thetendency for the highly #rofita!le units =on the left side of demand and su##ly> to

trade early& leaving traders at the margin where #rice negotiations tend to !e near

com#etitive levels. 8rice dis#ersion is narrowed in su!se'uent #eriods as traders

come to e#ect the higher #rices at the end of the #eriod.

III. Cham&er%in9s Res$%ts and ernon Smith9s Reaction

he negotiations in the class e#eriments discussed a!ove too" #lace in a central

area that served as the trading #it& although some #eo#le tended to !rea" off in

#airs to finali<e deals. *ven so& the #artici#ants could hear the #u!lic offers !eing

made !y others& a #rocess which should reduce #rice dis#ersion. A highvalue !uyer& who is !etter off #aying u# to ,0 instead of not trading& may not !e

willing to #ay such high #rices when some sellers are ma"ing lower offers.

6imilarly& a low+cost seller will !e less willing to acce#t a low #rice when other

!uyers are o!served to #ay more. n this manner& good mar"et information a!out

the going #rices will tend to reduce #rice dis#ersion. A high dis#ersion is needed

for high+cost sellers and low+value !uyers to !e a!le to find trading #artners& so

less dis#ersion will tend to eclude these etra+marginal traders.

Cham!erlin did re#ort some tendency for the mar"ets to yield too many

trades relative to com#etitive #redictions& which he attri!uted to the dis#ersion

that can result from small grou# negotiations. n order to evaluate this con9ecture&

he too" the value and cost cards from his e#eriment and used them to simu#ate a

decentra#i)ed trading #rocess. hese simulations were not la!oratory e#eriments

with student tradersB they were mechanical& the way one would do com#uter

simulations today.

2




?or grou#s of si<e two& Cham!erlin would shuffle the cost cards and the

value cards& and then he would match one cost with one value and ma"e a trade at

an intermediate #rice if the value eceeded the cost. Cards for trades that were

not made were returned to the dec" to !e reshuffled and re+matched. t is useful

to see how this random matching would wor" in a sim#le eam#le with only four

!uyers and four sellers& as shown in a!le 2.2. he random #airing #rocess would

result in only two trades if the two low+cost units were matched with the two

high+value units. t would result in four trades when !oth low+cost units were

matched with the low+value units& and high+cost units are matched with highvalue

units. he intermediate cases would result in three trades& for eam#le when the

valuecost com!inations are: '+;& '+;2& 3;& and /. he three

valuecost #airs that result in a trade are shown in !old. o summari<e& random

matches in this eam#le #roduce a 'uantity of trades of either two& three& or four&

and some simulations should convince you that on average there will !e threeunits traded.

a!le 2.2. An *ight+rader *am#le

Values Costs

7uyer , ,0 6eller 5 2

7uyer 2 ,0 6eller % 2

7uyer / 6eller -

7uyer / / 6eller

?or grou#s larger than two traders& Cham!erlin would shuffle and deal the cards

into grou#s and would calculate the com#etitive e'uili!rium 'uantity for each

grou#& as determined !y the intersection of su##ly and demand for that grou#

alone. n the four+!uyerfour+seller eam#le in a!le 2.2& there is only one grou#

of si<e /& i.e. all four traders& and the e'uili!rium for all four is the com#etitive

'uantity of 2 units. his illustrates Cham!erlin s general finding that 'uantity

tended to decrease with larger grou# si<e in his simulations.

;otice the relationshi# !etween the use of simulations and la!oratory

e#eriments with human #artici#ants. he e#eriment #rovided the em#irical

regularity =ecess 'uantity> that motivated a theoretical model =com#etitive

e'uili!rium for su!grou#s>& and the simulation confirmed that the same regularity

would !e #roduced !y this model. n general& com#uter simulations can !e used

to derive #ro#erties of models that are too com#le to solve analytically& which is

0



often the case for models of out+of+e'uili!rium !ehavior and dynamic ad9ustment.

he methodological order can& of course& !e reversed& with com#uter simulations

!eing used to derive #redictions that can !e tested with la!oratory e#eriments

using human #artici#ants.

he simulation analysis given a!ove suggests that mar"et efficiency will !e

higher when #rice information is centrali<ed& so that all traders "now the going

levels of !id& as"& and transactions #rices. Vernon 6mith& who attended some of

Cham!erlin s e#eriments when he was a student at Harvard& used this intuition to

design a trading institution that #romoted efficiency. After some classroom

e#eriments of his own& he !egan using a dou!le auction in which !uyers made

!ids& sellers made offers =or as"s >& and all could see the highest outstanding !id

and the lowest outstanding as". 7uyers could raise the current !est !id at any

time& and sellers could undercut the current !est as" at any time. n this manner&

the !idas" s#read would ty#ically diminish until someone closed a contract !yacce#ting the terms from the other side of the mar"et& i.e. until a seller acce#ted a

!uyer s !id or a !uyer acce#ted a seller s as". his is called a dou+#e auction&

since it involves !oth !uyers =!idding u#& as in an auction for anti'ues> and sellers

=!idding down& as would occur if contractors undercut each others #rices>. A trade

occurs when these #rocesses meet& i.e. when a !uyer acce#ts a seller s as" or

when a seller acce#ts a !uyer s !id.

a!le 2. shows a ty#ical se'uence of !ids and as"s in a dou!le auction. 7uyer 2

!ids .00& and seller 5 offers to sell for .00. 6eller % comes in with an as"

#rice of -.50& and !uyer , !ids /.00 and then /.50. At this #oint seller - offers

to sell at %.00& which !uyer 2 acce#ts.

a!le 2.. A 8rice ;egotiation 6e'uence

Bid !sk

7uyer 2 .00

.00 6eller 5

-.50 6eller %

7uyer , /.00

7uyer , /.50

%.00 6eller -

7uyer 2 Acce#ts %.00

n a dou!le auction& there is always good #u!lic information a!out the

!idas" s#read and a!out #ast contract #rices. his information creates a

,




largegrou# setting that tends to diminish #rice varia!ility and increase efficiency.

6mith =,%2& ,%/> re#orted that this dou!le auction resulted in efficiency

measures of over ninety #ercent& even with relatively small num!ers of traders

=/%> and with no information a!out others values and costs. he dou!le auction

tends to !e somewhat more centrali<ed than a #it mar"et& which does not close off

the #ossi!ility of !ilateral negotiations that are not o!served !y others. (ou!le

auction trading more closely resem!les trading for securities on the ;ew Mor"

6toc" *change& where the s#ecialist collects !ids and as"s& and all trades come

across the tic"er ta#e.

7esides introducing a centrali<ed #osting of all !ids& as"s& and trading

#rices& 6mith introduced a second "ey feature into his early mar"et e#eriments:

the re#etition of trading in successive mar"et #eriods or trading days. he effects

of re#etition are a##arent in ?igure 2./& which was done with the V econla! we!+

!ased interface in a classroom setting. he su##ly and demand curves for the firsttwo #eriods are shown in !old on the left sideB the values and costs that determine

the locations of the ste#s were randomly generated. here were four trading

#eriods& which are delineated !y the vertical lines that se#arate the dots that show

the transactions #rice se'uence. he #rices converge to near com#etitive levels in

the second #eriod. he demand and su##ly shift in #eriod =higher red lines on

the left> causes a 'uic" rise in #rices& and convergence is o!served !y the end of

#eriod /.

2



0ig$re .3. Demand, S$pp%!, and ransactions Prices for a Do$&%e -$ction 6ith a

Str$ct$ra% Change after the Second Period

?igure 2.5 shows the results of a dou!le auction mar"et conducted under

research conditions with a much more etreme configuration of values and costs

that was used to #rovide a stress test of the convergence #ro#erties of a dou!le

auction. he four !uyers !egin with four units each& for a total of ,% units& all

with values of %.0. he four sellers have a total of ,, units& all with costs of

5.-0. n addition& each #erson received a commission of 5 cents for each unit

that they !ought or sold. he left side of the figure shows the resulting su##ly

function& which is vertical at a 'uantity of ,, where it crosses demand. he

intersection is at a #rice of %.0& and at lower #rices there is ecess demand. he

#eriod+!y+#eriod #rice se'uences are shown on the right side of the figure. ;otice

that first+#eriod #rices start in the middle of the range !etween values and costs&

and then rise late in the #eriod. 8rices in #eriod 2 start higher and rise again. his

u#ward trend continues until #rices reach the com#etitive level of a!out %.0 in

#eriod /. At this #oint& !uyers are only earning #ennies =e.g. the commission> on

each transaction. A second treatment& not shown& !egan in #eriod % with the total

num!er of !uyer units reduced to ,, and the num!er of seller units increased to

,%. 7uyers& who were already earning very little on each unit& were disa##ointed

at having fewer units. 7ut #rices started to fall immediatelyB the very first trading

#rice in #eriod % was only %.%5. 8rices declined steadily& reaching the new

com#etitive #rice of 5.-0 !y #eriod ,0. his session illustrates the strongconvergence #ro#erties of the dou!le auction& where ecess demand or su##ly

#ressures can #ush #rices to create severe earnings ine'uities that are unli"ely to

arise in !ilateral !argaining situations.




8rice 6eries for ?our rading 8eriods Luantity

0ig$re .*. Demand, S$pp%!, and ransactions Prices for a Do$&%e -$ction

I. "(tensions

he sim#le e#eriment #resented in this cha#ter can !e varied in numerous useful

directions. ?or eam#le& an u#ward shift in demand& accom#lished !y increasing

!uyers values& should raise #rices. An increase in the num!er of sellers would

tend to shift su##ly outward& which has the effect of lowering #rices. he

im#osition of a , #er+unit ta on !uyers would =in theory> shift demand down !y,. o see this& note that if one s value is ,0& !ut a ta of , must !e #aid u#on

#urchase& then the net value is only . All demand ste#s would shift down !y ,

in this manner. Alternatively& a , ta #er unit im#osed on sellers would shift

su##ly u# !y ,& since the ta is analogous to a , increase in cost. 6ome

etensions will !e considered in the cha#ter on dou!le auctions that is in the

mar"et e#eriments section of this !oo".

$e have discussed several alternative mar"et institutions here.

Cham!erlin s pit market corres#onds most closely to trading of futures contracts

in a trading #it& whereas 6mith s dou+#e auction is more li"e the trading of

securities on the ;ew Mor" 6toc" *change. A postedoffer auction =where

sellers set #rices on a ta"e+it+or+leave+it !asis> is more li"e a retail mar"et withmany !uyers and few sellers. hese different trading institutions have different

#ro#erties and a##lications& and outcomes need not match com#etitive

#redictions. ;evertheless& it is useful to consider outcomes in terms of mar"et

efficiency& measured as the #ercentage of maimum earnings achieved !y the

/



trades that are made. n considering alternative designs for ways to auction off

some licenses& for eam#le& one might want to consider the efficiency of the

allocations along with the amounts of revenue generated for the seller=s>. he

theoretical #redictions discussed in this cha#ter are derived from an analysis of

su##ly and demand. n the com#etitive model& all !uyers and sellers ta"e #rice as

given. his model may not #rovide good #redictions when some traders #erceive

themselves as !eing #rice ma"ers& with #ower to #ush #rices in their favor. A

single+seller mono#olist may !e a!le to raise #rices a!ove com#etitive levels& and

a small grou# of sellers with enough of a concentration of mar"et ca#acity may !e

a!le to raise #rices as well. he eercise of this ty#e of mar"et #ower is discussed

in the cha#ter on #osted+offer auctions& in which sellers #ost #rices on a ta"e+it+or+

leave+it !asis. he eercise of mar"et #ower !y #ricesetting sellers is a strategic

decision& and the relevant theoretical models are those of game theor& which is

the analysis of interrelated strategic decisions. he word interrelated is criticalhere& since the amount that one firm may wish to raise #rice de#ends on how

much others are e#ected to raise #rices. $e turn to a sim#le introduction of

game theory in the net cha#ter. he discussion will !e in terms of sim#le games

with two #layers and two decisions& !ut the same #rinci#les will !e a##lied later

in the !oo" to the analysis of more com#le games with many sellers and many

#ossi!le #rice levels.

<$estions:

,. 6u##ose that all of the num!ered (iamonds and 6#ades from a dec" of cards

=ecluding Ace& Ding& Lueen& and Fac"> are used to set u# a mar"et. he

(iamonds determine (emand& e.g. a ,0 re#resents a !uyer with a redem#tionvalue of ,0. 6imilarly& the 6#ades determine 6u##ly. here are !uyers and

sellers& each with a single card.

a> ra#h su##ly and demand and derive the com#etitive #rice and 'uantity

#redictions.

!> $hat is the #redicted effect of removing the & /& and 5 of 6#adesN

2. $hen #eo#le are as"ed to come u# with alternative theories a!out why #rices in a

#it mar"et sta!ili<e at a #articular level o!served in class& they sometimes suggest

ta"ing the average of all !uyers values and sellers costs.

a> (evise and gra#h two e#erimental designs =list all !uyers values and all sellers

costs> that have the same com#etitive #rice #rediction !ut different averages of values and costs.

!> (evise and gra#h two e#erimental designs that have the same average of all

values and costs& !ut which have different com#etitive #rice #redictions.

5




. ;et consider the following #rice+determination theory& which was suggested in a

recent e#erimental economics class: 3an" all !uyer values from high to low and

find the median =middle> value. hen ran" all seller costs from low to high and

find the median cost. he #rice should !e the average of the median value and the

median cost. 7y giving !uyers a lot more units than sellers& it is #ossi!le to

create a design where the median cost and the median value are !oth e'ual to the

lowest of all values and costs. Use this idea to set u# a su##ly and demand array

where the com#etitive #rice #rediction is considera!ly higher than the #rediction

!ased on medians. =f your design has a range of #ossi!le com#etitive #rices& you

may assume for #ur#oses of discussion that the average #rice #rediction is at the

mid+#oint of this range.>

/. 1arge earnings asymmetries !etween !uyers and sellers may slow convergence to

theoretical #redictions. his tendency will !e es#ecially #ronounced if the traders

on one side of the mar"et are #redicted to have )ero earnings. f either of thee#eriment designs that you suggested in your answer to 'uestion involve <ero

earnings for traders on one side of the mar"et& how could you alter the design to

ensure that all trades made !y those on the disadvantaged side of the mar"et in

each treatment generate earnings of a!out 50 cents #er unitN

%



Cha#ter . 6ome 6im#le ames: Com#etition&

Coordination& and uessing

A game with two #layers and two decisions can !e re#resented !y a 22 #ayoff

ta!le or matri. 6uch games often highlight the conflict !etween incentives to

com#ete or coo#erate. his cha#ter introduces two such games& the #risoner s

dilemma and the coordination game. (iscussion can !e #receded with an

e#eriment using #laying cards& with the instructions #rovided in the A##endi.

he main #ur#ose of this cha#ter is to develo# the idea of a ;ash e'uili!rium&

which is a somewhat a!stract and mathematical notion. A sim#le uessing

ame& discussed last& serves to highlight the etent to which ordinary #eo#lemight come to such an e'uili!rium. 7oth the 4atri ame and the uessing

ame can !e run !y hand =with instructions #rovided in the A##endi> or from

the Veconla! site& where they are listed under the ames 4enu.

I. Game heor! and the Prisoner9s Di%emma

he reat (e#ression& which was the defining economic event of the 20 th

century& caused a ma9or rethin"ing of eisting economic theories that re#resented

the economy as a system of self+correcting mar"ets needing little in the way of

active economic #olicy interventions. In the macroeconomic side& Fohn 4aynard

Deynes 8he Genera# 8heor of 4mp#oment, Interest, and Mone focused on #sychological elements = animal s#irits > that could cause a whole economy to

!ecome mired in a low+em#loyment e'uili!rium& with no tendency for

selfcorrection. An e'uili!rium is a state where there are no net forces causing

further change& and Deynes message im#lied that such a state may not necessarily

!e good. In the microeconomics side& *dward Cham!erlin argued that mar"ets

may not yield efficient& com#etitive outcomes& as noted in the #revious cha#ter.

At a!out the same time& von ;eumann and 4orgenstern =,//> #u!lished 8heor

of Games and 4conomic Behavior & which was motivated !y the o!servation that a

ma9or #art of economic activity involves !ilateral and small+grou# interactions&

where the classical assum#tion of non+strategic& #rice+ta"ing !ehavior =used in the #revious cha#ter> is not realistic. n light of the #rotracted (e#ression& these new

theories met with a 'uic" acce#tance. Cham!erlin s models of mono#olistic

com#etition were 'uic"ly incor#orated into tet!oo"s as alternative models& and

-




von ;eumann and 4orgenstern s !oo" on game theory received front+#age

coverage in the Fe; Eork 8imes.

A game is a mathematical model of a strategic situation in which #layers

#ayoffs de#end on their own and others decisions. A game is characteri<ed !y the

#layers& their sets of feasi!le decisions& the information availa!le at each decision #oint&

and the #ayoffs =as functions of all decisions and random events>. A "ey notion is that

of a strateg& which is essentially a com#lete #lan of action that covers all

contingencies. ?or eam#le& a strategy in an auction could !e an amount to !id for each

#ossi!le estimate of the value of the #ri<e. 6ince a strategy covers all contingencies&

even those that are unli"ely to !e faced& it could !e given to a hired em#loyee to !e

#layed out on !ehalf of the #layer in the game. An e'uili!rium is a set of strategies that

is sta!le in some sense& i.e. with no inherent tendency for change. *conomists are

interested in notions of e'uili!rium that will #rovide good #redictions after !ehavior

has had a chance to settle down.As noted in Cha#ter ,& a 8rinceton graduate student& Fohn ;ash& corrected a

ma9or shortcoming in the von ;eumann and 4orgenstern analysis !y develo#ing

a notion of e'uili!rium for non+<ero+sum games. ;ash showed that this

e'uili!rium eisted under general conditions& and this #roof caught the attention

of researchers at the 3A;( Cor#oration head'uarters in 6anta 4onica. n fact&

two 3A;( mathematicians immediately conducted a la!oratory e#eriment

designed to test ;ash s new theory. ;ash s thesis advisor was in the same

!uilding when he noticed the #ayoffs for the e#eriment written on a !lac"!oard.

He found the game interesting and made u# a story of two #risoners facing a

dilemma of whether or not to ma"e a confession. his story was used in a

#resentation to the #sychology de#artment at 6tanford& and the #risoner sdilemma !ecame the most commonly discussed #aradigm in the new field of

game theory.

n a #risoner s dilemma& the two sus#ects are se#arated and offered a set of

threats and rewards that ma"e it !est for each to confess and essentially rat on the

other #erson& whether or not the other #erson confesses. ?or eam#le& the

#rosecutor may say: if you do not confess and the other #erson rats on you& then

can get a conviction and will throw the !oo" at you& so you are !etter off ratting

on them. $hen the #risoner as"s what ha##ens if the other does not confess& the

#rosecutor re#lies: *ven without a confession& can frame you !oth on a lesser

charge& which will do if no!ody confesses. f you do confess and the other #erson does not& then will reward you with immunity and !oo" the holdout on

the greater charge. hus each #risoner is !etter off im#licating the other #erson

even if the other remains silent. 7oth #risoners& aware of these incentives& decide

to rat on the other& even though they would !oth !e !etter off if they could



somehow form a !inding code of silence. his analysis suggests that two

#risoners might !e !ullied into confessing to a crime that they did not commit&

which is a scenario from Murder at the Margin& written !y two mysterious

economists under the #seudonym 4arshall Fevons =,-->.

A game with a #risoner s dilemma structure is shown in the ta!le !elow.

he row #layer s 7ottom decision corres#onds to confession& as does the column

#layer s 3ight decision. he 7ottom3ight outcome& which yields #ayoffs of for

each& is worse than the o#1eft outcome& where !oth receive #ayoffs of . =A

high #ayoff here corres#onds to a light #enalty.> he dilemma is that the !ad

confession outcome is an e'uili!rium in the following sense: if either #erson

e#ects the other to tal"& then their own !est res#onse to this !elief is to confess as

well. ?or eam#le& consider the 3ow #layer& whose #ayoffs are listed on the left

side of each outcome cell in the ta!le. f Column is going to choose 3ight& then

3ow either gets 0 for #laying o# or for #laying 7ottom& so 7ottom is the !estres#onse to 3ight. 6imilarly& column s 3ight decision is the !est res#onse to a

!elief that 3ow will #lay 7ottom. here is no other cell in the ta!le with this

sta!ility #ro#erty. ?or eam#le& if 3ow thin"s column will #lay 1eft& then 3ow

would want to #lay 7ottom& so o#1eft is not sta!le. t is straightforward to

show that the diagonal elements& 7ottom1eft and o#3ight are also unsta!le.

a!le ., A 8risoner s (ilemma =3ow s #ayoff& Column s #ayoff>

Column 8layer:

3ow 8layer: 1eft

3ight

o#

7ottom

II. - Prisoner9s Di%emma "(periment

he #ayoff num!ers in the #revious #risoner s dilemma e#eriment can !e

derived in the contet of a sim#le eam#le in which !oth #layers must choose

!etween a low effort =0> and a higher effort =,>. he cost of eerting the higher

effort is ,0& and the !enefits to each #erson de#end on the total effort for the twoindividuals com!ined. 6ince each #erson can choose an effort of 0 or ,& the total

effort must !e 0& ,& or 2& and the !enefit #er #erson is given:

& 0& ,0

,0& 0 &



"otal e#ort 0 1 2ene%t per person $3 $10 $18


o see the connection !etween this #roduction function and the #risoner s

dilemma #ayoff matri& let 7ottom and 3ight corres#ond to <ero effort. hus the

7ottom3ight outcome results in 0 total effort and #ayoffs of for each #erson& as

shown in the #ayoff matri in a!le .,. he o#1eft cell in the matri is

relevant when !oth choose efforts of , =at a cost of ,0 each>. he total effort is 2&

so each earns , ,0 J & as shown in the o#1eft cell of the matri. he

o#3ight and 7ottom1eft #arts of the #ayoff ta!le #ertain to the case where one

#erson receives the !enefit of ,0 at no cost& the other receives the !enefit of ,0 at

a cost of ,0& for a #ayoff of 0. ;otice that each #erson has an incentive to free

ride on the other s effort& since one s own effort is more costly =,0> than the

marginal !enefit of effort& which is - =J ,0 > for the first unit of effort and =J, ,0> for the second unit.

Coo#er& (eFong& ?orsythe& and 3oss =,%> conducted an e#eriment using the

#risoner s dilemma #ayoffs in the a!ove matri& with the only change !eing that

the #ayoffs were =.5& .5> at the ;ash e'uili!rium. heir matching #rotocol

#revented individuals from !eing matched with the same #erson twice& or from

!eing matched with anyone who had !een matched with them or one of their #rior

#artners. his no+contageon #rotocol is analogous to going down a receiving line

at a wedding and telling everyone the same !it of gossi# a!out the !ride and

groom. *ven if this story is so curious that everyone you meet in line re#eats it to

everyone they meet& you will never encounter anyone who has heard the story&

since all of the #eo#le who have heard the story will !e !ehind you in the line. nan e#eriment& the elimination of any ty#e of re#eated matching& either direct or

indirect& means that no!ody is a!le to send a message or to #unish or reward

others for coo#erating. *ven so& coo#erative decisions =o# or 1eft> were fairly

common in early rounds =/K>& and the incidence of coo#eration declined to

a!out 20K in rounds ,5+20. A recent classroom e#eriment using the Veconla!

#rogram with the a!le ., #ayoffs and random matching #roduced coo#eration

rates of a!out K& with no downward trend. A second e#eriment with a

different class #roduced coo#eration rates starting at a!out K and declining to

<ero !y the fourth #eriod. his latter grou# !ehaved 'uite differently when they

were matched with the same #artner for 5 #eriodsB coo#eration rates stayed level

in the +50K range until the final #eriod. he end+game effect is not sur#rising

since coo#eration in earlier #eriods may consist of an effort to signal good

intentions and stimulate reci#rocity. hese forward+loo"ing strategies are not

availa!le in the final round.

/0



he results of these #risoner s dilemma e#eriments are re#resentative.

here is ty#ically a miture of coo#eration and defection& with the mi !eing

somewhat sensitive to #ayoffs and #rocedural factors& with some variation across

grou#s. n general& coo#eration rates are higher when individuals are matched

with the same #erson in a series of re#eated rounds. n fact& the first #risoner s

dilemma e#eriment run at the 3A;( Cor#oration over fifty years ago lasted for

,00 #eriods& and coo#erative #hases were inter#reted as evidence against the ;ash

e'uili!rium.

A strict game+theoretic analysis would have #eo#le reali<ing that there is

no reason to coo#erate in the final round& and "nowing this& no!ody would try to

coo#erate in the net+to+last round with the ho#e of stimulating final+round

coo#eration. hus& there should !e no coo#eration in the net+to+last round& and

hence there is no reason to try to stimulate such coo#eration. 3easoning

!ac"wards from the end in this manner& one might e#ect no coo#eration even inthe very first round& at least when the total num!er of rounds is finite and "nown.

;ash res#onded to the 3A;( mathematicians with a letter maintaining that it is

unreasona!le to e#ect #eo#le to engage in this many levels of iterative reasoning

in an e#eriment with many rounds. here is an etensive literature on related

to#ics& e.g.& the effects of #unishments& rewards& ada#tive !ehavior& and various

tit+for+tat strategies in re#eated #risoner s dilemma games.

?inally& it should !e noted that there are many ways to #resent the #ayoffs for an

e#eriment& even one as sim#le as the #risoner s dilemma. 7oth the Coo#er et al.

and the Veconla! e#eriment used a #ayoff matri #resentation. Alternatively& the

instructions could !e #resented in terms of the cost of effort and the ta!le showing

the !enefit #er #erson for each #ossi!le level of total effort. his #resentation is #erha#s less neutral& !ut the economic contet ma"es it less a!stract and artificial

than a matri #resentation. he A##endi for this cha#ter ta"es a different

a##roach !y setting u# the #risoner s dilemma game as one where each #erson

chooses which of two cards to #lay =Ca#ra and Holt& 2000>. ?or eam#le&

su##ose that each #erson has #laying cards num!ered and %. 8laying the % #ulls

% =from the e#erimenter s reserve> into one s own earnings& and #laying the

#ushes =from the e#erimenter s reserve> to the other #erson s earnings. f

they !oth #ull the %& earnings are % each. 7oth would !e !etter off if they

#layed & yielding for each. 8ulling %& however& is !etter from a selfish

#ers#ective& regardless of what the other #erson does. A consideration of the

resulting #ayoff matri =Luestion 2> indicates that this is clearly a #risoner s

dilemma. he card #resentation is not neutral enough for a research e#eriment&

!ut it is 'uic" and easy to im#lement in class& where students hold the cards

#layed against their chests& and the instructor #ic"s #eo#le in #airs to reveal their

/,





cards. A ;ash e'uili!rium survives an announcement test in that neither would

wish they had #layed a different card given the card #layed !y the other. n this

case& the ;ash e'uili!rium is for each to #lay %.

III. - Coordination Game

4any #roduction #rocesses have the #ro#erty that one #erson s effort increases

the #roductivity of another s effort. ?or eam#le& a mail order com#any must ta"e

orders !y #hone& #roduce the goods& and shi# them. *ach sale re'uires all three

services& so if one #rocess is slow& the effort of those in other activities is to some

etent wasted. n terms of our two+#erson #roduction function& recall that , unit

of effort #roduced a !enefit of ,0 #er #erson and 2 units #roduced a !enefit of

,. 6u##ose that the second unit of effort ma"es the first one more #roductive in

the sense that the !enefit is more than dou!led when a second unit of effort isadded. n the ta!le that follows& #er+#erson !enefit is 0 for 2 units of effort:

f the cost of effort remains at ,0 #er unit& then each #erson earns 0 + ,0 J

20 in the case where !oth su##ly a unit of effort& as shown in the revised #ayoff

matri !elow:

a!le .2 A Coordination ame

=3ow s #ayoff& Column s #ayoff>

Column 8layer:

3ow 8layer: 1eft

3ight

o#

7ottom

As !efore& the <ero effort outcome =7ottom3ight> is a ;ash e'uili!rium& since

neither #erson would want to change from their decision unilaterally. ?or

eam#le& if 3ow "nows that Column will choose 3ight& then 3ow can get 0 fromo# and from 7ottom& so 3ow would want to choose 7ottom& as indicated !y

the downward #ointing arrow in the lower+right !o. 6imilarly& 3ight is a !est

/2

& 20& 20 ' 0& ,0

,0& 0 ( & )



res#onse to 3ow s decision to use 7ottom& as indicated !y the right arrow in the

lower+right !o.

here is a second ;ash e'uili!rium in the o#1eft !o of this modified game.

o verify this& thin" of a situation in which we start in that !o& and consider

whether either #erson would want to deviate& so there are two things to chec".

6te# ,: if 3ow is thought to choose o#& then Column would want to choose 1eft&

as indicated !y the left arrow in the o#1eft !o. 6te# 2: if 3ow e#ects Column

to choose 1eft& then 3ow would want to choose o#& as indicated !y the u# arrow.

hus o#1eft survives an announcement test and would !e sta!le. his second

e'uili!rium yields #ayoffs of 20 for each& far !etter than the #ayoffs of each in

the 7ottom3ight e'uili!rium outcome. his is called a coordination game& since

the #resence of multi#le e'uili!ria raises the issue of how #layers will guess

which one will !e relevant. =n fact& there is a third ;ash e'uili!rium in which

each #layer chooses randomly& !ut we will not discuss such strategies untilCha#ter 5.>

t used to !e common for economists to assume that rational #layers

would somehow coordinate on the !est e'uili!rium& if all could agree which is the

!est e'uili!rium. $hile it is a##arent that the o#1eft outcome is li"ely to !e the

most fre'uent outcome& one eam#le does not 9ustify a general assum#tion that

#layers will always coordinate on an e'uili!rium that is !etter for all. his

assum#tion can !e tested with la!oratory e#eriments& and it has !een shown to

!e false. n fact& #layers sometimes get driven to an e'uili!rium that is ;orst for

all concerned =Van Huyc"& 7attalio& and 7eilB ,0>. *am#les of data from such

coordination games will !e #rovided in a su!se'uent cha#ter on coordination

#ro!lems. ?or now& consider the intuitive effect of increasing the cost of effortfrom ,0 to ,. $ith this increase in the cost of effort& the #ayoffs in the

higheffort o#1eft outcome are reduced to 0 + , J ,,. 4oreover& the

#erson who eerts effort alone only receives a ,0 !enefit and incurs a , cost&

so the #ayoffs for this #erson are +& as shown:

a!le . A Coordination ame with High *ffort Cost


Column 8layer:

3ow 8layer: 1eft3ight

o#

7ottom

/

& ,,& ,, ' +& ,0

,0& + ( & )




;otice that o#1eft is still a ;ash e'uili!rium: if Column is e#ected to #lay

1eft& then 3ow s !est res#onse is o# and vice versa& as indicated !y the

directional arrows in the u##er+left corner. 7ut o# and 1eft are very ris"y

decisions& with #ossi!le #ayoffs of ,, and +& as com#ared with the ,0 and

#ayoffs associated with the 7ottom and 3ight strategies that yield the other ;ash

e'uili!rium. $hile the a!solute #ayoff is higher for each #erson in the o#1eft

outcome& each #erson has to !e very sure that the other one will not deviate. n

fact& this is a game where !ehavior is li"ely to converge to the ;ash e'uili!rium

that is worst for all. hese #ayoffs were used in a Veconla! e#eriment that

lasted 5 #eriods with random matching of ,2 #artici#ants. 8artici#ants were

individuals or #airs of students at the same com#uter. 7y the fifth #eriod& only a

'uarter of the decisions were o# or 1eft. he same #eo#le then #layed the less

ris"y coordination game shown in a!le .2& again with random matching. heresults were 'uite differentB all !ut one of the #airs ended u# in the good =o#&

1eft> e'uili!rium outcome in all #eriods.

o summari<e& it is not a##ro#riate to assume that !ehavior will somehow

converge to the !est outcome for all& even when this is a ;ash e'uili!rium as in

the coordination games considered here. $ith multi#le e'uili!ria& the one with

the most attraction may de#end on #ayoff features =e.g.& the effort cost> that

determine which e'uili!rium is ris"ier. hese issues will !e revisited in the

cha#ter on coordination.

I. - G$essing Game

A "ey as#ect of most games is the need for #layers to guess what others will do in

order to determine their own !est decision. his as#ect is somewhat o!scured in a

22 game since a wide range of !eliefs may lead to the same decision& and

therefore& it is not #ossi!le to say much a!out the !eliefs that stand !ehind any

#articular o!served decision. n this section& we turn to a game with a continuum

of decisions& from 0 to ,00& and any num!er of #layers& F . *ach #layer selects a

num!er in this interval& with the advance "nowledge that the #erson whose

decision is closest to ,2 of the average of all F decisions will win a money #ri<e.

=he #ri<e is divided e'ually in the event of a tie.> hus each #erson s tas" is to

ma"e a guess a!out the average& and then su!mit a decision that is half of that

amount.

f the game is only #layed once& then there is no way to learn from #ast

decisions& and each #erson must learn !y thin"ing introspective# a!out what

//



others might do. ?or eam#le& someone might reason that since the decisions

must !e !etween 0 and ,00& without further "nowledge it might !e o" to guess

that the average would !e at the mid#oint& 50. his #erson would su!mit a

decision of 25 in order to !e at half of the antici#ated average. 7ut then the

#erson might reali<e that others could !e thin"ing in the same way& and also

su!mitting decisions of 25& so that a decision of ,2.5 would !e closest to the

target level of ,2 of the average. A third level of iteration along these lines

suggests a decision of %.25& and even higher levels of iterated thin"ing will lead

the #erson to choose a decision that is closer and closer to 0. 6o the !est decision

to ma"e de#ends on one s !eliefs a!out the etent to which others are thin"ing

iteratively in this manner& i.e. on !eliefs a!out the others rationality. 6ince the

!est res#onse to any common decision a!ove 0 is to go down to one half of that

level& it follows that no common decision a!ove 0 can !e a ;ash e'uili!rium.

Clearly a common decision of 0 is a ;ash e'uili!rium& since each #erson gets a #ositive share =, F > of the #ri<e& and to deviate and choose a higher decision

would result in a <ero #ayoff. t turns out that this is the only ;ash e'uili!rium

for this game =#ro!lem 5>.

?igure ., shows data for a classroom guessing game run with the

Veconla! software. here were 5 #artici#ants& who made first round decisions of

55& 2& 22& 2,& and %.25& which average to a!out 2-. ;otice that the lowest #erson

is one who engaged in three levels of iterated reasoning a!out what the others

might do& and this #erson was closest to the target level of a!out ,-2 J ,.5.

his #erson su!mitted the same decision in round 2 and won againB the other

decisions were 20& ,5& ,,.5& and ,0. he other #eo#le do not seem to !e engaging

in any #recise iterated reasoning& !ut it is clear the #erson with the lowestdecision is !eing rewarded& which will tend to #ull decisions down in su!se'uent

rounds. hese decisions converge to near+;ash levels !y the fifth round& although

no!ody selected 0 eactly.

/5




0ig$re /.' Data from a C%assroom G$essing Game

he second treatment changes the calculation of the target from ,2 of the

average to 20 O ,2 of the average. he instructor =a.".a. author> who ran this in

class had selected the #arameters 'uic"ly& with the con9ecture that adding 20 to

the target would raise the ;ash e'uili!rium from 0 to 20. $hen the average came

in at 2& the instructor thought that it would then decline toward 20& 9ust as it had

declined toward 0 in the first treatment. 7ut the average rose to in the net

round& and continued to rise to a level that is near /0. ;otice that if all choose /0&

then the average is /0& half of the average is 20& and 20 O half of the average is 20

O 20 J /0& which is the ;ash e'uili!rium for this treatment ='uestion %>.

his game illustrates a cou#le of things. 6ome #eo#le thin" iteratively a!out

what others will do& !ut this is not uniform& and the effect of such intros#ection is

not enough to move decisions to near+;ash levels in a single round of #lay.

$hen #eo#le are a!le to learn from e#erience& !ehavior does converge to the

;ash e'uili!rium in this game& even in some cases when the e'uili!rium is not

a##arent to the #erson who designed the e#eriment.

. "(tensions

he #risoner s dilemma game discussed in this cha#ter can !e given an

alternative inter#retation& i.e. that of a #u!lic good. his is !ecause each #erson s

!enefit de#ends on the total effort& including that of the other #erson. n effect&

/%



neither #erson can a##ro#riate more than half of the !enefit of their own effort&

and in this sense the !enefit is #u!licly availa!le to !oth& 9ust as national defense

or #olice #rotection are freely availa!le to all. $e will consider the #u!lic goods

#rovision #ro!lem in more detail in a later cha#ter in the 8u!lic Choice section of

this !oo". he setu# allows a large num!er of effort levels& and the effects of

changes in costs and other incentives will !e considered.

he #risoner s dilemma has a ;ash e'uili!rium that is worse than the

outcome that results when individuals coo#erate and ignore their #rivate

incentives to defect. he dilemma is that the e'uili!rium is the !ad outcome& and

the good outcome is not an e'uili!rium. n contrast& the coordination game

considered a!ove has a good outcome that is also an e'uili!rium. *ven so& there

is no guarantee that the !etter e'uili!rium will !e reali<ed. A more formal analysis

of the ris" associated with alternative e'uili!ria is #resented in a su!se'uent

cha#ter on coordination. he main #oint is that there may !e multi#le ;ashe'uili!ria& and which e'uili!rium is o!served may de#end on intuitive factors li"e

the cost of effort. n #articular& a high effort cost may cause #layers to get stuc" in

a ;ash e'uili!rium that is worse for all concerned& as com#ared with an

alternative ;ash e'uili!rium. he coordination game #rovides a #aradigm in

which an e'uili!rium may not !e desira!le. 6uch a #ossi!ility has concerned

macroeconomists li"e Deynes& who worried that a mar"et economy may !ecome

mired in a low+em#loyment& low+effort state.

he guessing game is eamined in ;agel =,5& ,->& who shows that

there is a wides#read failure of !ehavior to converge to the ;ash e'uili!rium in a

one+round e#eriment& regardless of su!9ect #ool. ;agel discusses the degree to

which some of the #eo#le go through iterations of thin"ing that may lead them tochoose lower and lower decisions =e.g.& 50& 25& ,2.5& etc.>. he game was

originally motivated !y one of Deynes remar"s& that investors are ty#ically in a

#osition of trying to guess what stoc" others will find attractive =first iteration>& or

what stoc" that others thin" other investors will find attractive =second iteration>&

etc.

n all of the e'uili!ria considered thusfar& each #erson chooses a decision without

any randomness. t is easy to thin" of games where it is not good to !e #erfectly

#redicta!le. ?or eam#le& a soccer #layer in a #enalty "ic" situation should

sometimes "ic" to the left and sometimes to the right& and the goalie should also

avoid a statistical #reference for diving to one side or the other. 6uch !ehavior in

a game is called a randomi<ed strategy. 7efore considering randomi<ed strategies

=in Cha#ter 5>& it is useful to introduce the notions of #ro!a!ility& e#ected value&

and other as#ects of decision ma"ing in uncertain situations. his is the to#ic of

/-




"otal e#ort 0 1 2ene%t per person 3 5 18


the net cha#ter& where we consider the choice !etween sim#le lotteries over cash

#ri<es.

?or a !rief& non+technical discussion of Fohn ;ash s original ,/ Proceedings of

the Fationa# !cadem of Sciences #a#er& its content and su!se'uent im#act and

#olicy a##lications& see Holt and 3oth =200/>.

<$estions:

,. 6u##ose that the cost of a unit of effort is raised from ,0 to 25 for the

eam#le !ased on the effort+value ta!le shown !elow. s the resulting

game a coordination game or a #risoner s dilemmaN *#lain& and find the

;ash e'uili!rium =or e'uili!ria> for the new game.

2. $hat is the #ayoff ta!le for the #risoner s dilemma game descri!ed in

section of the tet !ased on #ulling % or #ushing N

. *ach #layer is given a % of Hearts and an of 6#ades. he two #layers

select one of their cards to #lay. f the suit matches& then they each are

#aid % for the case of matching Hearts and for the case of matching

6#ades. *arnings are <ero in the event of a mismatch. s this a

coordination game or a #risoner s dilemmaN 6how the #ayoff ta!le and

find all ;ash e'uili!ria =that do not involve random #lay>.

/. 6u##ose that the #ayoffs for the original #risoner s dilemma are altered as

follows. he cost of effort remains at ,0& !ut the #er+#erson !enefits are

given in the ta!le !elow. 3ecalculate the #ayoff matri& find all ;ash

e'uili!ria =that do not involve random #lay>& and e#lain whether the

game is a #risoner s dilemma or a coordination game.

5. 6u##ose that 2 #eo#le are #laying a guessing game with a #ri<e going tothe #erson closest to ,2 of the average. uesses are re'uired to !e

!etween 0 and ,00. 6how that none of the following are ;ash e'uili!ria:

a> 7oth choose ,. =Hint& consider a deviation to 0 !y one #erson& so that

the average is ,2& and half of the average is ,/.> !> Ine #erson chooses

/



, and the other chooses 0. c> Ine chooses 5 and the other chooses &

where 5 P P 0. =Hint: f they choose these decisions& what is the

average& what is the target& and is the target less than the mid#oint of the

range of guesses !etween 5 and N hen use these calculations to figure

out which #erson would win& and whether the other #erson would have an

incentive to deviate.>

%. Consider a guessing game with ; #eo#le& and the #erson closest to 20 #lus ,2

of the average is awarded the #ri<e& which is s#lit in the event of a tie.

he range of guesses is from 0 to ,00. 6how that /0 is a ;ash

e'uili!rium. =Hint: 6u##ose that F +, #eo#le choose /0 and one #erson

deviates to 5 Q /0. Calculate the target as a function of the deviant s

decision& 5. he deviant will lose if the target is greater than the mid#oint

!etween 5 and /0& i.e. if the target is greater than =/0 O 5>2. Chec" to see

if this is the case.>-. Consider a guessing game with ; #eo#le& and the #erson closest to ,0 #lus 2

of the average is awarded the #ri<e. ?ind the ;ash e'uili!riumN

. A more general formulation of the guessing game might !e for the target level

to !e: ! O BR=average decision>& where ! P 0 and 0 Q B Q ,& and decisions

must !etween 0 and an u##er limit - P 0. ?ind the ;ash e'uili!rium. $hen is

it e'ual to -N

Cha#ter /. 3is" and (ecision 4a"ingn this cha#ter& we consider decisions in ris"y situations. *ach decision has a set

of money conse'uences or #ri<es and the associated #ro!a!ilities. n such cases&

it is straightforward to calculate the e#ected money value of each decision. A

#erson who is neutral to ris" will select the decision with the highest e#ected

#ayoff. A ris"+averse #erson is willing to acce#t a lower e#ected #ayoff in order

to reduce ris". A sim#le lottery choice e#eriment is used to illustrate the

conce#ts of e#ected value maimi<ation and ris" aversion. Variations on the

e#eriment can !e conducted #rior to class discussions& either with the

instructions included in the a##endi or with the Veconla! software =select the

1ottery Choice e#eriment listed under the (ecisions 4enu>.

I. Who Wants to Be a Millionaire?

/




4any decision situations involve conse'uences that cannot !e #redicted #erfectly

in advance. ?or eam#le& su##ose that you are a contestant on the television

game show =ho =ants to Be a Mi##ionaire Mou are at the 500&000 #oint and

the 'uestion is on a to#ic that you "now nothing a!out. ?ortunately& you have

saved the fifty+fifty o#tion that rules out two answers& leaving two that turn out to

!e unfamiliar. At this #oint& you figure you only have a one+half chance of

guessing correctly& which would ma"e you into a millionaire. f you guess

incorrectly& you receive the safety level of 2&000. Ir you can fold and ta"e the

sure 500&000. n thin"ing a!out whether to ta"e the 500&000 and fold& you

decide to calculate the e#ected money value of the guess o#tion. $ith

#ro!a!ility 0.5 you earn 2&000& and with #ro!a!ility 0.5 you earn ,&000&000&

so the e#ected value of a guess is:

0.5=2&000> O 0.5=,&000&000> J ,%&000 O 500&000 J 5,%&000.

his e#ected #ayoff is greater than the sure 500&000 one gets from sto##ing& !ut

the trou!le with guessing is that you either get 2D or one million dollars& not the

average. 6o the issue is whether the ,%&000 increase in the average #ayoff is

worth the ris"& which is considera!le if you guess. f you love ris"& then there is

no #ro!lemB ta"e the guess. f you are neutral to ris"& the etra ,%D in the

average #ayoff should cause you to ta"e the ris". Alternatively& you might reason

that the 2D would !e gone in % months& and only a #ri<e of 500D or greater

would !e large enough to !ring a!out a change in your lifestyle =new 6UV&

tro#ical vacation& etc.>& which may lead you to fold. o an economist& the #erson

who folds would !e classified as !eing risk averse in this case& !ecause the ris" for this #erson is sufficiently !ad that it is not worth the etra ,%D in average

#ayoff associated with the guess.

;ow consider an even more etreme case. Mou have the chance to secure a sure

, million dollars. he alternative is to ta"e a coin fli#& which #rovides a #ri<e of

million dollars for Heads& and nothing for ails. f you !elieve that the coin is

fair& then you are choosing !etween two lotteries:

Safe -otter$ , million for sureB

'isk -otter$ million with #ro!a!ility 0.5&0 with #ro!a!ility 0.5.

As !efore& the e#ected money value for the ris"y lottery can !e calculated !y

multi#lying the #ro!a!ilities with the associated #ayoffs& yielding an e#ected

50



0

1

2

0 1 2 3

utility

#ayoff of ,.5 million dollars. $hen as"ed a!out this =hy#othetical> choice& most

students will select the safe million. ;otice that for these #eo#le& the etra

500&000 in e#ected #ayoff is not worth the ris" of ending u# with nothing. An

economist would call this ris" aversion& !ut to a layman the reason is intuitive: the

first million #rovides a ma9or change in lifestyle. A second million is an e'ually

large money amount& !ut it is hard for most of us to imagine how much additiona#

!enefit this would #rovide. 3oughly s#ea"ing& the additional =marginal> utility of

the second million dollars is much lower than the utility of the first million& and

the marginal utility of the third million is li"ely to !e even lower. A utility

function with a diminishing marginal utility is one with a curved u#hill sha#e& as

shown in ?igure /.,.

his utility function is the familiar s'uare root function& where utility =in

millions> is the s'uare root of the #ayoff =in millions>. hus the utility of 0 is 0&

the utility of , million dollars is ,& the utility of / million dollars is 2& the utility of ,% million is /& etc. n fact& each time you multi#ly the #ayoff !y /& the utility

only increases !y a factor of 2& which is indicative of the diminishing marginal

utility. his feature is also a##arent from the fact that the slo#e of the utility

function near the origin is high& and it diminishes as we move to the right. he

more curved the utility function& the faster the utility of an additional million

diminishes.

millions of dollars

5,




0ig$re 3.'. - =S#$are Root= >ti%it! 0$nction

he diminishing+marginal+utility hy#othesis was first suggested !y (aniel

7ernoulli =,->. here are many functions with this #ro#erty. Ine is the class

of #ower functions: > = 5> J 5,+r & where 5 is money income and r is a measure of

ris" aversion that is less than ,. ;otice that when r J 0& the e#onent in the utility

function is ,& so we have the linear function: > = 5> J 5, J 5. his function has no

curvature& and hence no diminishing marginal utility for income. ?or a #erson

whose choices are re#resented !y this function& the second million is 9ust as good

as the first& etc.& so the #erson is neutral to ris". Alternatively& if the ris" aversion

measure r is increased from 0 to 0.5& we have > = 5> J 50.5& which is the s'uare root

function in the figure. ?urther increases in r result in more curvature& and in this

sense r is a measure of the etent to which marginal utility of additional moneyincome decreases. his measure is often called the coefficient of re#ative risk

aversion.

n all of the eam#les considered thusfar in this cha#ter& you have !een as"ed to

thin" a!out what you would do if you had to choose !etween gam!les involving

millions of dollars. he #ayoffs were =unfortunately> hy#othetical. his raises

the issue of whether what you say is what you would actually do if you faced the

real situation& e.g. if you were really in the final stage of the series of 'uestions on

=ho =ants to +e a Mi##ionaireN t would !e fortunate if we did not really have to

#ay large sums of money to find out how #eo#le would !ehave in high+#ayoff

situations. he #ossi!ility of a hpothetica# +ias& i.e. the #ro#osition that !ehavior

might !e dramatically different when high hy#othetical sta"es !ecome real& isechoed in the film !n Indecent Proposa# :

Fohn =a.".a. 3o!ert>: 6u##ose were to offer you one million dollars for one night with

your wife.

(avid: =a.".a. $oody> d assume you were "idding.

Fohn: 1et s #retend m not. $hat would you sayN

(iana =a.".a. (emi>: He d tell you to go to hell.

Fohn: didn t hear him.

(avid: d tell you to go to hell.

Fohn: hat s 9ust a refle answer !ecause you view it as hy#othetical. 7ut let s saythere were real money !ehind it. m not "idding. A million dollars. ;ow&

the night would come and go& !ut the money could last a lifetime. hin" of

it a million dollars. A lifetime of security for one night. And don t answer

right away. 7ut consider it seriously.

52



n the film& Fohn s #ro#osal was ultimately acce#ted& which is the Hollywood

answer to the incentives 'uestion. In a more scientific note& incentive effects are

an issue that can !e investigated with e#erimental techni'ues. 7efore returning

to this issue& it is useful to consider the results of a lottery choice e#eriment

designed to evaluate ris" attitudes& which is the to#ic of the net section.

II. - Simp%e Lotter!?Choice "(periment

n all of the cases discussed a!ove& the safe lottery is a sure amount of money. n

this section we consider a case where !oth lotteries have random outcomes& !ut

one is ris"ier than the other. n #articular& su##ose that I#tion A #ays either /0

or 2& each with #ro!a!ility one half& and that I#tion 7 #ays either -- or 2&

each with #ro!a!ility one half. ?irst& we calculate the e#ected values:

/ption !$ 0.5=/0> O 0.5=2> J 20 O ,% J %.00

/ption B$ 0.5=--> O 0.5=2> J .50 O , J .50.

n this case& o#tion 7 has a higher e#ected value& !y .50& !ut a lot more ris"

since the #ayoff s#read from -- to 2 is almost ten times as large as the s#read

from 2 to /0.

his choice was on a menu of choices used in an e#eriment =Holt and 1aury&

200,>. here were a!out 200 #artici#ants from several universities& including

undergraduates& a!out 0 47A students and a!out 20 !usiness school faculty.

*ven though I#tion 7 had a higher e#ected #ayoff when each #ri<e is e'uallyli"ely& eighty+four #ercent of the #artici#ants selected the safe o#tion& which

indicates some ris" aversion.

a!le /.,. A 4enu of 1ottery Choices Used to *valuate 3is" 8references

I#tion A I#tion 7

Mour

Choice

A or 7

(ecision ,

.

/0.00 if throw of die is ,

2.00 if throw of die is 2+,0

--.00 if throw of die is ,

2.00 if throw of die is 2+,0 SSSSS

(ecision /

/0.00 if throw of die is ,+/

2.00 if throw of die is 5+,0--.00 if throw of die is ,+/

2.00 if throw of die is 5+,0 SSSSS

5




(ecision 5 /0.00 if throw of die is ,+5

2.00 if throw of die is %+,0--.00 if throw of die is ,+5

2.00 if throw of die is %+,0 SSSSS

(ecision %

.

/0.00 if throw of die is ,+%

2.00 if throw of die is -+,0

--.00 if throw of die is ,+%2.00 if throw of die is -+,0 SSSSS

(ecision ,0 /0.00 if throw of die is ,+,0 --.00 if throw of die is ,+,0 SSSSS

he #ayoff #ro!a!ilities in the 1aury and Holt e#eriment were im#lemented !y

the throw of a ten+sided die. his allowed the researchers to alter the #ro!a!ility

of the high #ayoff in one+tenth increments. A #art of the menu of choices is

shown in a!le /.,& where the #ro!a!ility of the high #ayoff =/0 or --> is onetenth in (ecision ,& four tenths in (ecision /& etc. ;otice that (ecision ,0 is a

"ind of rationality chec"& where the #ro!a!ility of the high #ayoff is ,& so it is a

choice !etween /0 for sure and -- for sure. he su!9ects indicated a #reference

for all ten decisions& and then one decision was selected at random& e5 post & to

determine earnings. n #articular& after all decisions were made& we threw a ,0+

sided die to determine the relevant decision& and then we threw the ten+sided die

again to determine the su!9ect s earnings for the selected decision. his

#rocedure has the advantage of #roviding data on all ten decisions without any

wealth effects. 6uch wealth effects could come into #lay& for eam#le& if a #erson

wins -- on one decision and this ma"es them more willing to ta"e a ris" on thesu!se'uent decision. he disadvantage is that incentives are diluted& which was

com#ensated for !y raising #ayoffs.

he e#ected #ayoffs associated with each #ossi!le choice are shown under 3is"

;eutrality in the second and third columns of the a!le /.2 =the other columns

will !e discussed later>. ?irst& loo" in the fifth row where the #ro!a!ilities are

0.5. his is the choice #reviously discussed in this cha#ter& with e#ected values

of % and .50 as calculated a!ove. he other e#ected values are calculated

in the same manner& !y multi#lying #ro!a!ilities !y the associated #ayoffs& and

adding u# these #roducts.

a!le /.2. I#timal (ecisions for 3is" ;eutrality and 3is" Aversion

8ro!a!ility 3is" ;eutrality 3is" Aversion =r J 0.5>

5/



of High *#ected 8ayoffs for U= 5 > J 5 *#ected Utilities for U= 5 > J 5 ,2

8ayoff 6afe 3is"y 6afe 3is"y

/0 or 2 -- or 2 /0 or 2 -- or 2

0., /.2+ .50 *.) 2.,50.2 //.1+ ,-.00 *.)4 2.0. /3.3+ 2/.50 *.21 .%20./ /*.+ 2.00 *.4 /.%0.5 %.00 /4.*+ *.44 5.0

0.% %.0 3).++ 1.+1 5.0.- -.%0 *3.*+ %.,2 1.*)

0. ./0 1.++ %., )./+

0. .20 14.*+ %.2% 2.+3

,.0 /0.00 )).++ %.2 2.))

he !est decision when the #ro!a!ility of the high #ayoff is only 0., =in the to#

row of a!le /.2> is o!vious& since the safe decision also has a higher e#ectedvalue& or 2.0 =versus .50 for the ris"y lottery>. n fact& #ercent of the

su!9ects selected the safe lottery =I#tion A> in this choice. 6imilarly& the last

choice is !etween sure amounts of money& and all su!9ects chose I#tion 7 in this

case. he e#ected #ayoffs are higher for the to# four choices& as shown !y the

!olded num!ers in columns 2 and . hus a ris"+neutral #erson& who !y

definition only cares a!out e#ected values regardless of ris"& would ma"e / safe

choices in this menu. n fact& the average num!er of safe choices was %& not /&

which indicates some ris" aversion. As can !e seen from the 0.% row of the ta!le&

the ty#ical safe choice for this o#tion involves giving u# a!out ,0 in e#ected

value in order to reduce the ris". A!out two+thirds of the #eo#le made the safe

choice for this decision& and /0 #ercent chose safe in (ecision -.

he #ercentages of safe choices are shown !y the thic" solid line in ?igure /.2&

where the decision num!er is listed on the hori<ontal ais. he !ehavior

#redicted for a ris"+neutral #erson is re#resented !y the dashed line& which stays

at the to# =,00 #ercent safe choices> for the first four decisions& and then shifts to

the !ottom =no safe choices> for the last si decisions. he thic" line re#resenting

actual !ehavior is generally a!ove this dashed line& which indicated the tendency

to ma"e more safe choices. here is some randomness in actual choices&

however& so that choice #ercentages do not 'uite get to ,00 #ercent on the left

side of the figure.

55



0102030405060708090

100

101 2 3 4 5 6 7 8 9

Per centa*e of Saf e +,oices

+as, Payo#s

-ypot,etical Payo#s

.is/ Neutrality


(ecision

0ig$re 3.. Percentages of Safe Choices 6ith Rea% Incentives and 7!pothetica% Incentives

@So$rce: 7o%t and La$r! @++'A.

his real+choice e#eriment was #receded !y a hpothetica# choice tas"&

in which su!9ects made the same ten choices with the understanding that they

would not !e #aid their earnings for that #art. he data averages for thehy#othetical choices are #lotted as #oints on the thin line in the figure. 6everal

differences !etween the real and hy#othetical choice data are clear. he thin line

lies !elow the thic" line& indicating less ris" aversion when choices have no real

im#act. he average num!er of safe choices was a!out % with real #ayments& and

5%



a!out 5 with hy#othetical #ayments. *ven without #ayments& su!9ects were a

little ris" averse as com#ared with the ris"+neutral #rediction of / safe choices&

!ut they could not imagine how they really would !ehave when they had to face

real conse'uences. 6econd& with hy#othetical incentives& there may !e a tendency

for #eo#le to thin" less carefully& which may #roduce noise in the data. n

#articular& for decision ,0& the thin line is a little higher& corres#onding to the fact

that 2 #ercent of the #eo#le chose the sure /0 over the sure =!ut hy#othetical>

--.

he issue of whether or not to #ay su!9ects is one of the "ey issues that divides

research in e#erimental economics from some wor" on similar issues !y

#sychologists =see Hertwig and Irtmann& 200,& for a #rovocative survey on

#ractices in #sychology& with a!out 0 comments and an authors re#ly>. Ine

9ustification for using high hy#othetical #ayoffs is realism. wo #rominent

#sychologists& Dahneman and vers"y& 9ustify the use of hy#othetical incentives:

*#erimental studies ty#ically involve contrived gam!les for small

sta"es& and a large num!er of re#etitions of very similar #ro!lems.

hese features of la!oratory gam!ling com#licate the inter#retation of

the results and restrict their generality. 7y default& the method of

hy#othetical choices emerges as the sim#lest #rocedure !y which a

large num!er of theoretical 'uestions can !e investigated. he use of

the method relies on the assum#tion that #eo#le often "now how they

would !ehave in actual situations of choice& and on the further

assum#tion that the su!9ects have no s#ecial reason to disguise their

true #references. =Dahneman and vers"y& ,-& #. 2%5>

If course& there are many documented cases where hy#othetical and realincentive

choices coincide 'uite closely =one such eam#le will !e #resented in the net

section>. 7ut in the a!sence of a widely acce#ted theory a!out when they do and

do not coincide& it is dangerous to assume that real incentives are not needed.

III. Risk -version: Incentive "ffects, Demographics, and 5rder "ffects

he intuitive effect of ris" aversion is to diminish the utility associated with

higher earnings levels& as can !e seen from the curvature for the s'uare root utility

function in ?igure /.,. $ith nonlinear utility& the calculation of a #erson s

e#ected utility is analogous to the calculation of e#ected money value. ?or

eam#le& the safe o#tion A for (ecision 5 is a ,2 of /0 and a ,2 chance of 2.

5-




he e#ected #ayoff is found !y adding u# the #roducts of money #ri<e amounts

and the associated #ro!a!ilities:

45pected paoff 0safe option2 J 0.5=/0> O 0.5=2> J 20 O ,% J %.

he e#ected uti#it of this o#tion is o!tained !y re#lacing the money amounts&

/0 and 2& with the utilities of these amounts& which we will denote !y > =/0>

and > =2>. f the utility function is the s'uare+root function& then > =/0> J =/0>,2

J %.2 and > =2> J =2>,2 J 5.%%. 6ince each #ri<e is e'ually li"ely& we ta"e the

average of the two utilities:

45pected uti#it 0safe option2 J 0.5 > =/0> O 0.5 > =2>

J 0.5=%.2> O 0.5=5.%%> J 5..

his is the e#ected utility for the safe o#tion when the #ro!a!ilities are 0.5& as

shown in the 0.5 row of a!le /.2 in the column under 3is" Aversion for the 6afe

I#tion. 6imilarly& it can !e shown that the e#ected utility for the ris"y o#tion&

with #ayoffs of -- and 2& is 5.0. he other e#ected utilities for all ten

decisions are shown in the two right+hand columns of a!le /.2. he safe o#tion

has the higher e#ected utility for the to# si decisions& so the theoretical

#rediction for someone with this utility function is that they choose si safe

o#tions !efore crossing over to the ris"y o#tion. 3ecall that the analogous

#rediction for ris" neutrality =see the left side of the ta!le> is / safe choices& and

that the data for this treatment ehi!it % safe choices on average. n this sense& the

s'uare+root utility function #rovides a !etter fit to the data than the linear utility

function that corres#onds to ris" neutrality.

Ine interesting 'uestion is whether the order in which a choice is made matters.

n #articular& the Holt and 1aury =2002> e#eriment involved su!9ects who first

went through a lottery choice trainer involving a choice !etween a safe and a

menu of gam!les with #ayoffs of , or %& to familiari<e them with the dice+

throwing #rocedure for #ic"ing one decision at random and then for determining

the #ayoff for that decision. hen all #artici#ants made ,0 decisions for a low

real+#ayoff choice menu& where all money amounts were ,20 of the level shown

in a!le /.,. hus the #ayoffs for the ris"y o#tion were .5 and 0.,0& and the

#ossi!le #ayoffs for the safer o#tion were 2.00 and ,.%0. hese low #ayoffswill !e referred to as the , treatment& and the #ayoffs in a!le /., will !e called

the 20 #ayoffs. Ither treatments are designated similarly as multi#les of the low

#ayoff level. he initial low+#ayoff choice was followed !y a choice menu with

high hpothetica# #ayoffs =20& /0& or 0>& followed !y the same menu with

5



high rea# #ayoffs =20& /0& or 0>& and ending with a second , real choice.

he average num!ers of safe choices& shown in the to# two rows of a!le /.&

indicate that the num!er of safe choices increases steadily as real #ayoffs are

scaled u#& !ut this incentive effect is not o!served as hy#othetical #ayoffs are

scaled u#.

a&%e 3./. Average ;um!ers of 6afe Choices: Irder and ncentive *ffects

Dey: 6u#erscri#ts indicate Irder =a J ,st& + J 2nd& c J rd& d J /th>

*#eriment ncentives , ,0 20 50 0

Holt and 1aury =2002>

3eal 5.2a

5.d

%.0c %.c -.2c

U.C.?.& a. 6t.& U. 4iami20 su!9ects

Hy#othetical /. !

5., !

5. !

Harrison et al. =2005>

6outh Carolina

,- su!9ects

3eal 5.a %.0a

%./ !

3eal 5.-a %.-a

Holt and 1aury =2005>

U. of Virginia

,% su!9ectsHy#othetical 5.%a 5.-a

Harrison et al. =2005> #oint out this design mingles order and #ayoff scale effects& #roducing a #ossi!le fatal flaw =loss of control> as discussed in Cha#ter ,. he

letter su#erscri#ts in the ta!le indicate the order in which the decision was made

=a first& + second& etc.>& so a com#arison of the , decisions with those of higher

scales is called into 'uestion !y the change in order& as are the com#arisons of

high hy#othetical and high real #ayoffs =done in orders 2 and res#ectively>. he

#resence of order effects is su##orted !y evidence summari<ed in the third row of

the ta!le& where ris" aversion seems to !e higher when the ,0 scale treatment

follows the , treatment than when the ,0 treatment is done first. he order

effect #roduces a difference of 0./ safe choices. 6uch order effects call into

'uestion the real versus hy#othetical com#arisons& although it is still #ossi!le toma"e inferences a!out the scale effects from 20 to /0 to 0 in the Holt and

1aury e#eriment& since these were made in the same order.

n res#onse to these issues of inter#retation& Holt and 1aury =2005> ran a

followu# e#eriment with a 22 design: real or hy#othetical #ayoffs and , or

5




20 #ayoffs. *ach #artici#ant !egan with a lottery choice trainer =again with

choices !etween and gam!les involving , or %. hen each #erson made a

single menu of choices from one of the / treatment cells& so all decisions are

made in the same order. Again& the scaling u# of real #ayoffs causes a shar#

increase in the average num!er of safe choices =from 5.- to %.->& whereas this

incentive effect is not o!served with a scaling u# of hy#othetical choices. ?inally&

the data from the lottery+choice trainers can !e used to chec" whether o!served

differences in the / treatments are somehow due to unantici#ated demogra#hic

differences or other uno!served factors that may cause #eo#le in one cell to !e

intrinsically more ris" averse than those in other cells. A chec" of data from the

trainers shows virtually no differences across treatment cells& and if anything& the

trainer data indicate slightly lower ris" aversion for the grou# of #eo#le who were

su!se'uently given the high+real #ayoff menu.

he main conclusions from the second study are that #artici#ants are ris" averse&the ris" aversion increases with higher real #ayoffs& and that loo"ing at high

hy#othetical #ayoffs may !e very misleading. hese conclusions are a##arent

from the gra#h of the distri!utions of the num!er of safe choices for each of the

,0 decisions& as shown in ?igure /.. n this contet& it does not seem to matter

much whether or not one used real money when the scale of #ayoffs is low& !ut

o!serving no difference and then inferring that money #ayoffs do not matter in

other contets would !e incorrect. ?inally& note that #ayoff+scale effects may not

!e a##arent in classroom e#eriments where #ayoffs are ty#ically small or

hy#othetical.

%0



01020

30405060708090

100

12345678910

Per centa*e of Saf e +,oices

20

.eal

20

-ypot,etical

.eal11 -ypot,etical

.is/ Neutrality

%,




(ecision

0ig$re 3./. Res$%ts for 8e6 "(periment 6ith Sing%e Decision 6ith 8o 5rder "ffects:Percentages of Safe Choices for 7igh @+(A and Lo6 @'(A Pa!offs,

>nder Rea% and 7!pothetica% Incentives @/1 S$&ects per reatmentA

So$rce: 7o%t and La$r! @++*A

Another interesting 'uestion is whether there are systematic demogra#hic effects.

he su!9ects in the original Holt and 1aury =2002> e#eriment included a!out %0

47A students& a!out thirty !usiness school faculty =and a (ean>& and over a

hundred undergraduates from three universities =University of 4iami& University

of Central ?lorida& and eorgia 6tate University>. here is a slight tendency for

#eo#le with higher incomes to !e less ris" averse. he women were more ris"

averse than men in the low+#ayoff =,> condition& as o!served in some #reviousstudies& !ut this effect disa##eared for higher #ayoff scales. All of that !ravado

went away with high sta"es =e.g. 20 scale>. here was no whitenonwhite

difference& !ut His#anics in the sam#le were a little less ris" averse. t is not

a##ro#riate to ma"e !road inferences a!out demogra#hic effects from a single

study. ?or eam#le& the His#anic effect could !e an artifact of the fact that most

His#anic su!9ects were 4.7.A. students in 4iami& many of whom are from

families that emigrated from Cu!a.

I. "(tensions

here are& of course& other #ossi!le values for the ris" aversion coefficient of the #ower function& which corres#ond to more or less curvature. And there are

functions with the curvature of ?igure /., that are not #ower functions& e.g. the

natural logarithm. ?inding the function that #rovides the !est fit to the data

involves a statistical analysis !ased on s#ecifying an error term that incor#orates

un+modeled random variations due to emotions& #erce#tions& inter#ersonal

differences& etc. =hese differences might !e controlled !y including demogra#hic

varia!les.> Various functional forms for the utility function will !e discussed in

later cha#ters.

Another issue that is !eing deferred is whether utility should !e a function of

final wealth or of gains from the current wealth #osition. Here we treat utility as

a function of gains only. here is some e#erimental and theoretical evidence for

this #ers#ective that will also !e discussed in a su!se'uent cha#ter.

%2



<$estions

,. ?or the s'uare root utility function& find the e#ected utility of the ris"y

lottery for (ecision %& and chec" your answer with the a##ro#riate entry

in a!le /.2.

2. Consider the 'uadratic utility function > = 5> J 52. 6"etch the sha#e of this

function in a figure analogous to ?igure /.,. (oes this function ehi!it

increasing or decreasing marginal utilityN s this sha#e indicative of ris"

aversionN Calculate the e#ected utilities for the safe and ris"y o#tions in

(ecision / for a!le /.,& i.e. when the #ro!a!ilities are e'ual to 0./ and

0.%.

. 6u##ose that a dec" with all face cards =Ace& Ding& Lueen& and Fac">

removed is used to determine a money #ayoff& e.g. a draw of a 2 of Clu!s

would #ay 2& a draw of a ,0 of (iamonds would #ay ,0& etc. $rite

down the nine #ossi!le money #ayoffs and the #ro!a!ility associated witheach. $hat is the e#ected value of a single draw from the dec"&

assuming that it has !een well shuffledN How much money would you

#ay to #lay this gameN

Cha#ter 5. 3andomi<ed 6trategies

n many situations there is a strategic advantage associated with !eing

un#redicta!le& much as a tennis #layer does not always lo! in res#onse to an

o##onent s charge to the net. his cha#ter discusses randomi<ed strategies in the

contet of sim#le matri games =4atching 8ennies and 7attle of 6ees>. heassociated class e#eriments can !e run using the instructions in the A##endi or

with the Veconla! software =the 4atri ame #rogram on the ames 4enu>.

I. S!mmetric Matching Pennies Games

n a matching #ennies game& each #erson uncovers a #enny showing either Heads

or ails. 7y #rior agreement& one #erson can ta"e !oth coins if the #ennies match

=two Heads or two ails>& and the other can ta"e the coins if the #ennies do not

match. n this case& a #erson cannot afford to have a re#utation of always

choosing Heads& or of always choosing ails& !ecause !eing #redicta!le will result

in a loss every time. ntuition suggests that each #erson #lay Heads half of thetime.

A similar situation may arise in a soccer #enalty "ic"& where the goalie has to

dive to one side or the other& and the "ic"er has to "ic" to one side or the other.

%




he goalie wants a match and the "ic"er wants a mismatch. Again& any tendency

to go in one direction more often than another can !e e#loited !y the other

#layer. f there are no asymmetries in "ic"ing and diving a!ility for one side

versus the other& then each direction should !e selected a!out half of the time. t

is easy to imagine economic situations where #eo#le would not li"e to !e

#redicta!le. ?or eam#le& a la<y manager only wants to #re#are for an audit if

such an audit is li"ely. In the other side& the auditor is rewarded for discovering

#ro!lems& so the auditor would only want to audit if it is li"ely that the manager is

un#re#ared. he 'ualitative structure of the #ayoffs for these situations may !e

re#resented !y a!le 5.,.

a!le 5.,. A 4atching 8ennies ame


3ow 8layer =manager>: Column 8layer =auditor>:

1eft =audit>

3ight =not audit> o# =#re#are>

7ottom =not

#re#are>

?irst su##ose that the manager e#ects an audit& so the #ayoffs on the left side of the

ta!le are more li"ely. hen the manager #refers to #re#are for the antici#ated audit to

o!tain the good outcome with a #ayoff of ,& which is greater than the #ayoff of , for

getting caught un#re#ared. his #referred deviation is re#resented !y the u# arrow inthe lower+left !o. Alternatively& if the manager e#ects no audit =the right side of

the ta!le>& then the manager #refers not to #re#are& which again yields a #ayoff of ,

=notice the down arrow in the to#+right !o>. Conversely& the auditor #refers to audit

when the manager is not #re#ared& and to s"i# the audit otherwise.

iven the intuition a!out !eing un#redicta!le& it is not sur#rising that we

ty#ically o!serve a near+e'ual s#lit on aggregate decisions for the symmetric

matching #ennies game in a!le 5.,& although there can !e considera!le variation

in the choice #ro#ortions from round to round. =n la!oratory e#eriments& the

#ayoffs are scaled u#& and a fied #ayment is added to eliminate the #ossi!ility of

losses& !ut the essential structure of the game is unchanged& as will !e seen in

Cha#ter ,,.> (es#ite the intuitive nature of fifty+fifty s#lits for the game in a!le5.,& it is useful to see what !ehavior is sta!le in the sense of !eing a ;ash

e'uili!rium. U# to now& we have only considered strategies without random

elements& !ut such strategies will not constitute an e'uili!rium in this game. ?irst

consider the o#1eft !o in the ta!le& which corres#onds to audit#re#are. his

%/

,& , ) ( ,& ,

,& , & ' ,& ,



would !e a ;ash e'uili!rium if neither #layer has an incentive to change

unilaterally& which is not the case& since the auditor would not want to audit if the

manager is going to #re#are. n all cells& the #layer with the lower #ayoff would

#refer to switch unilaterally.

As mentioned in Cha#ter ,& ;ash =,/> #roved that an e'uili!rium

always eists =for games in which each #erson has a finite num!er of strategies>.

6ince there is no e'uili!rium in non+random strategies in the matching #ennies

game& there must !e an e'uili!rium that involves random #lay. he earlier

discussion of matching #ennies already indicated that this e'uili!rium involves

using each strategy half of the time. hin" a!out the announcement test. f one

#erson is #laying Heads half of the time& then #laying ails will win half of the

timeB #laying Heads will win half of the time& and #laying a fifty+fifty mi of

Heads and ails will win half of the time. n other words& when one #layer is

#laying randomly with #ro!a!ilities of one half& the other #erson cannot do any !etter than using the same #ro!a!ilities. f each #erson were to announce that

they would use a coin fli# to decide which side to #lay& the other could not do any

!etter than using a coin fli#. his is the ;ash e'uili!rium for this game. t is

called a mied e'uili!rium since #layers use a #ro!a!ilistic mi of each of their

decisions. n contrast& an e'uili!rium in which no strategies are random is called

a #ure strategy e'uili!rium& since none of the strategies are #ro!a!ility mies.

Another #ers#ective on the mied e'uili!rium is !ased on the o!servation

that a #erson is only willing to choose randomly if no decision is any !etter than

another. 6o the row #layer s choice #ro!a!ilities must "ee# the column #layer

indifferent& and vice versa. he only way one #erson will !e indifferent is if the

other is using e'ual #ro!a!ilities of Heads and ails& which is the e'uili!riumoutcome.

*ven though the answer is o!vious& it is useful to introduce a gra#hical device

that will hel# clarify matters in more com#licated situations. his gra#h will

show each #erson s !est res#onse to any given !eliefs a!out the other s decisions.

n ?igure 5.,& the thick so%id %ine shows the !est res#onse for the row #layer

=manager>. he hori<ontal ais re#resents what 3ow e#ects Column to do.

hese !eliefs can !e thought of as a #ro!a!ility of 3ight& going from 0 on the left

to , on the right. f Column is e#ected to choose 1eft& then 3ow s !est res#onse

is to choose o#& so the !est res#onse line starts at the to#+left #art if the figure& as

shown !y the thic" line. f Column is e#ected to choose 3ight& then 3ow should

#lay !ottom& so the !est res#onse line ends u# on the !ottom+right side of the

gra#h. he crossover #oint is where the Column s #ro!a!ility is eactly 0.5& since

3ow does !etter !y #laying o# whenever Column is more li"ely to choose 1eft.

%5



00

01

02

0304

05

06

07

08

09

10

Probability of "op

.osest .esponse

+olumns est .esponse


0 0., 0.2 0. 0./ 0.5 0.% 0.- 0. 0. ,

8ro!a!ility of 3ight

0ig$re *.'. Ro69s Best Response to Be%iefs a&o$t Co%$mn9s DecisionCo%$mn9s Best Response to Be%iefs a&o$t Ro69s Decision

A mathematical derivation of the crossover #oint =where 3ow is

indifferent and willing to cross over> re'uires that we find the #ro!a!ility of 3ight

for which 3ow s e#ected #ayoff is eactly e'ual for each decision. 1et p denote

3ow s !eliefs a!out the #ro!a!ility of 3ight& so , p is the #ro!a!ility of 1eft.

3ecall from the #revious cha#ter that e#ected #ayoffs are found !y adding u# the

#roducts of #ayoffs and associated #ro!a!ilities. ?rom the to# row of a!le 5.,&

we see that if 3ow chooses o#& then 3ow earns , with #ro!a!ility , p and 3ow

earns , with #ro!a!ility p& so the e#ected #ayoff is:

'o;9s 45pected Paoff for 8op J ,=, p> ,= p> J , 2 p.

%%



6imilarly& !y #laying 7ottom& 3ow earns , with #ro!a!ility , p and , with #ro!a!ility

p& so the e#ected #ayoff is:

'o;9s 45pected Paoff for Bottom J =, p> O p J , O 2 p.

hese e#ected #ayoffs are e'ual when:

, 2 p J , O 2 p.

6olving& we see that p J 2/ J 0.5& which confirms the earlier conclusion that 3ow is

indifferent when Column is choosing each decision with e'ual #ro!a!ility.

A similar analysis shows that Column is indifferent when 3ow is using

#ro!a!ilities of one half. herefore the !est res#onse line will cross over when

3ow s #ro!a!ility of o# is 0.5. o see this gra#hically& change the inter#retationof the aes in ?igure 5., to let the vertical ais re#resent Column s +e#iefs a!out

what 3ow will do. And instead of inter#reting the hori<ontal ais as a #ro!a!ility

re#resenting 3ow s !eliefs a!out Column s action& now inter#ret it in terms of

Column s actual !est res#onse. $ith this change& the dashed line that crosses at a

height of one half is Column s !est res#onse to !eliefs on the vertical ais. f

Column thin"s 3ow will #lay 7ottom& then Column wants to #lay 1eft& so this

line starts in the !ottomleft #art of the figure. his is !ecause the high #ayoff of

, for Column is in the !ottomleft #art of the #ayoff ta!le in a!le 5.,. here is

another #ayoff of , for Column in the to#right #art of the ta!le& i.e. when Column

thin"s 3ow will #lay o#& then 3ight is the !est res#onse. ?or this reason& thedashed !est res#onse line in ?igure 5., ends u# in the to#right corner.

n a ;ash e'uili!rium& neither #layer can do !etter !y deviating& so a ;ash

e'uili!rium must !e on the !est res#onse lines for !oth #layers. he only

intersection of the solid and dashed lines in ?igure 5., is at #ro!a!ilities of 0.5 for

each #layer. f !oth #layers thin" the other s move is li"e a coin fli#& they would

!e indifferent themselves& and hence willing to decide !y a fli# of a coin. his

e'uili!rium #rediction wor"s well in the symmetric matching #ennies game& !ut

we will consider 'ualifications in a later cha#ter. he focus here is on calculating

and inter#reting the notion of an e'uili!rium in randomi<ed strategies& not on

summari<ing all !ehavioral tendencies in these games.

II. Batt%e of the Se(es

hus far& we have considered two ty#es of games with uni'ue ;ash

e'uili!ria& the #risoner s dilemma =Cha#ter > and the matching #ennies game. n

%-




contrast& the coordination game discussed in Cha#ter had t;o e'uili!ria in

nonrandom strategies& one of which was #referred !y !oth #layers. ;et we

consider another game with two e'uili!ria in non+random strategies& the #ayoffs

for which are shown in a!le 5.2. hin" of this as a game where two friends who

live on o##osite sides of Central 8ar" wish to meet at one of the entrances& i.e. on

the *ast side or on the $est side. t is o!vious from the #ayoffs that Column

wishes to meet on the $est side& and 3ow #refers the *ast side. 7ut notice the

<ero #ayoffs in the mis+matched =$est and *ast> outcomes& i.e. each #erson would

rather !e with the other than to !e on the #referred side of the #ar" alone. ames

with the structure shown in a!le 5.2 are generally "nown as !attle+of+the+sees

games.

a!le 5.2. A 7attle+of+6ees ame


3ow: Column:

$est *ast

$est *ast

hin" of what you would do in a re#eated situation. Clearly most #eo#le

would ta"e turns. his is eactly what tends to ha##en when two #eo#le #lay the

game re#eatedly in controlled e#eriments with the same #artner. n fact&

coordinated switching often arises even when e#licit communication is not #ermitted. a!le 5. shows a decision se'uence for a #air who were matched with

each other for % #eriods& using #ayoffs from a!le 5.2. here is a match on *ast

in the first #eriod& and the Column #layer then switches to $est in #eriod 2& which

results in a mismatch and earnings of 0 for !oth. 3ow switches to $est in #eriod

& and they alternate in a coordinated manner in each su!se'uent #eriod& which

maimi<es their 9oint earnings. ?our of the si #airs alternated in this manner&

and the other two #airs settled into a #attern where one earned / in all rounds.

a!le 5.. Alternating Choices for a 8air of 6u!9ects in

a 7attle+of+6ees ame with ?ied 8artners

3ound , 3ound 2 3ound 3ound / 3ound 5 3ound %

3ow

8layer

%

,& / 0& 0

0& 0 /& ,



*ast =/> *ast =0> $est =,> *ast =/> $est =,> *ast =/>

Column

8layer *ast =,> $est =0> $est =/> *ast =,> $est =/> *ast =,>

he #ro!lem is harder if the !attle+of+sees game is #layed only once&

without communication& or if there is re#etition with random matchings.

Consider the !attle+of+sees game shown in a!le 5./.

a!le 5./. A 7attle+of+6ees ame


3ow: Column:1eft 3ight

o#

7ottom

he #ayoffs in this ta!le were used in a recent classroom e#eriment

conducted at the College of $illiam and 4ary. here were 0 students located in

several different com#uter la!s& with random matchings !etween those designated

as row #layers and those designated as column #layers. a!le 5.5 shows the

#ercentage of times that #layers chose the #referred location =o# for 3ow and

7ottom for Column>. ntuitively& one would e#ect that the #ercentage of

#referred+location choices to !e a!ove one half& and in fact this #ercentage

converges to %-K. his mi of choices does not corres#ond to either e'uili!rium

in #ure strategies.

he remar"a!le feature of a!le 5.5 is that !oth ty#es seem to !e choosing

their #referred location a!out two+thirds of the time. t is not sur#rising that this

fraction is a!ove one+half& !ut why two thirdsN f you loo" at the #ayoff ta!le&

you will notice that 3ow gets either 2 or 0 for #laying o#& and either 0 or , for

#laying 7ottom& so in a loose sense o# is more attractive unless Column is

e#ected to #lay 3ight with high #ro!a!ility. 6imilarly& from Column s #oint of

view& 3ight =with #ayoffs of 0 or 2> is more attractive than 1eft =with #ayoffs of ,and 0>& unless 3ow is e#ected to #lay o# with high #ro!a!ility. his intuition

only #rovides a 'ualitative #rediction& that each #erson will choose their #referred

decision more often than not. he remar"a!le convergence of the fre'uency of

#referred decisions to 2 cries out for some mathematical e#lanation& es#ecially

%

2& , 0& 0

0& 0 ,& 2




considering that each #erson s 2 and , #ayoffs add u# to & and two thirds of sum

can only !e o!tained with the #referred decision.

a!le 5.5. 8ercentage of 8referred+1ocation (ecisions for

the 7attle+of+6ees ame in a!le 5./

3ound

3ow

8layers

Column

8layers

All

8layers

, 0 - .5

2 - 0

- %0 -.5

/ %- %- %-

5 %- %- %-

;ash %- %- %-

nstead of loo"ing for mathematical coincidences& let us calculate some e#ected

#ayoff e#ressions as was done in the #revious section. ?irst consider 3ow s

#ers#ective when Column is e#ected to #lay 3ight with #ro!a!ility p. Consider

the to# row of the #ayoff matri in a!le 5./& where 3ow thin"s the right column

is relevant with #ro!a!ility p. f 3ow chooses o#& 3ow gets 2 when Column

#lays 1eft =e#ected with #ro!a!ility , p> and 3ow gets 0 when Column #lays

3ight =e#ected with #ro!a!ility p>. $hen 3ow chooses 7ottom& these #ayoffs

are re#laced !y 0 and ,. hus 3ow s e#ected #ayoffs are:

3ow)s e#ected #ayoff for o# 2=, p>0= > p 22 p=5.,>

3ow)s e#ected #ayoff for 7ottom 0=, p>,= > p 0

he e#ected #ayoff for o# is higher when 2 + 2 p P p& or e'uivalently& when p Q 2.

I!viously& the e#ected #ayoffs are e'ual when p J 2& and o# #rovides a lower

e#ected #ayoff when p P 2.

6ince 3ow s e#ected #ayoffs are e'ual when Column s #ro!a!ility of choosing

3ight is 2& 3ow would not have a #reference !etween the two decisions and

would !e willing to choose randomly. At this time& you might 9ust guess that

since the game loo"s symmetric& the e'uili!rium involves each #layer choosing

their #referred decision with #ro!a!ility 2. his guess would !e correct& as we

shall show with ?igure 5.2. As !efore& the solid line re#resents 3ow s !est

res#onse that we analy<ed a!ove. f the hori<ontal ais re#resents 3ow s !eliefs

a!out how li"ely it is that Column will #lay 3ight& then 3ow would want to go to

-0



the to# of the gra#h =#lay o#> as long as Column s #ro!a!ility p is less than 2.

3ow is indifferent when p J 2& and row would want to go to the !ottom when p

P 2.

6o far& we have !een loo"ing at things from 3ow s #oint of view& !ut an

e'uili!rium involves !oth #layers& so let s thin" a!out Column s decision where

the vertical ais now re#resents Column s !elief a!out what 3ow will do. At the

to#& 3ow is e#ected to #lay o#& and Column s !est res#onse is to #lay 1eft in

order to !e in the same location as 3ow. hus Column s dashed !est res#onse

line starts in the u##er+left #art of the figure. his line also ends u# in the

lowerright #art& since when 3ow is e#ected to #lay 7ottom =coming down the

vertical ais>& Column would want to switch to the 3ight location that is

#referred. t can !e verified !y sim#le alge!ra that the switchover #oint is at 2&

as shown !y the hori<ontal segment of the dashed line.

-,



01

02

03

04

05

06

07

08

09

10

10 01 02 03 04 05 06 07 08 09

.o23s Pr obability of " o

p

.os est

.esponse

+olumns est.esponse


-2



Column)s 8ro!a!ility of 3ight

0ig$re *.. Best Responses for the Game in a&%e *.3

6ince a ;ash e'uili!rium is a #air of strategies such that each #layer cannot do

!etter !y deviating& each #layer has to !e ma"ing a !est res#onse to the other s

strategy. n the figure& a ;ash e'uili!rium will !e on !oth 3ow s =solid> !est+

res#onse line and on Column s =dashed> !est+res#onse line. hus the final ste# is

to loo" for e'uili!rium #oints at the intersections of the !est res#onse lines.

here are three intersections. here is one in the u##er+left corner of the figure

where !oth coordinate on 3ow s #referred outcome =3ow earns 2& Column earns

,>. 6imilarly& the lower+right intersection is an e'uili!rium at Column s #referred

outcome =3ow earns ,& Column earns 2>. $e already found these e'uili!ria !yloo"ing at the #ayoff matri directly& !ut the third intersection #oint in the interior

of the figure is new. At this #oint& #layers choose their #referred decisions =o#

for 3ow and 3ight for Column> with #ro!a!ility 2& which is what we see in the

data.

he gra#h shows the e'uili!ria clearly& !ut it is useful to see how the random

strategy e'uili!rium would !e found with only sim#le alge!ra& since the gra#hical

a##roach will not !e #ossi!le with more #layers or more decisions.

Step 1. ?irst we need to summari<e notation:

p J #ro!a!ility that Column chooses 3ight

H J #ro!a!ility that 3ow chooses o#

Step . Calculate e#ected #ayoffs for each decision.

3ow s *#ected 8ayoff for o# J 2=, p> O 0= p> J 2 2 p

3ow s *#ected 8ayoff for 7ottom J 0=, p> O ,= p> J 0 O p

Column s *#ected 8ayoff for 3ight J 0=H> O 2=, H> J 2 2H

Column s *#ected 8ayoff for 1eft J ,=H> O 0=,+H> J H O 0

Step 3. Calculate the e'uili!rium #ro!a!ilities.

*'uate 3ow s e#ected #ayoffs to determine p.

*'uate Column s e#ected #ayoffs to determine H.

-




$e already showed that the first #art yields p J 2& and it is easy to verify that H J

2.

he idea !ehind these calculations is that& in order to randomi<e willingly& a

#erson must !e indifferent !etween the decisions& and indifference is found !y

e'uating e#ected #ayoffs. he tric"y #art is that e'uating 'o;9s e#ected

#ayoffs #ins down Co#umn9s #ro!a!ility& and vice versa.

"(tensions

he data in a!le 5.5 are aty#ical in the sense that such shar# convergence to an

e'uili!rium in randomi<ed strategies is not always o!served. Iften there is a

little more !ouncing around the #redictions& due to noise factors =see Luestion %>.

3emem!er that each #erson is seeing a series of other #eo#le& so #eo#le have

different e#eriences& and hence different !eliefs. his raises the issue of how #eo#le learn after o!serving others decisions& which will !e discussed in a later

cha#ter on 7ayesian learning. 6econd& the games discussed in this cha#ter are

symmetric in some senseB #ayoff asymmetries may cause !iases& as will !e

discussed in a later cha#ter. ?inally& the !attle+of+sees game discussed here was

conducted under very low #ayoff conditions& with only one #erson of 0 !eing

selected e5 post to !e #aid their earnings in cash. High #ayoffs might cause other

factors li"e ris" aversion to !ecome im#ortant& es#ecially when there is a lot more

varia!ility in the #ayoffs associated with one decision than with another. f ris"

aversion is a factor& then the e#ected #ayoffs would have to !e re#laced with

e#ected utility calculations.

<$estions

,. Use the e#ected #ayoff calculations in 6te# 2 a!ove to solve for an

e'uili!rium level of H for the !attle+of+sees game.

2. he a##endi to this cha#ter contains instructions for a game with #laying

cards. a> $rite out the #ayoff matri for round , of this game. !> ?ind all

e'uili!ria in non+random strategies and say what "ind of game this is. c>

?ind e#ressions for 3ow s e#ected #ayoffs& one for each of 3ow s

decisions. d> ?ind e#ressions for Column s e#ected #ayoffs. e> ?ind the

e'uili!rium with randomi<ed strategies& using alge!ra. f> llustrate your

answer with a gra#h.

-/



. 6u##ose that the #ayoffs in a!le 5./ are changed !y raising the 2 #ayoff

to a & for !oth #layers. Answer #arts c>& d> and e> of 'uestion 2 for this

game.

/. ?ind the ;ash e'uili!rium in mied strategies for the coordination game in

a!le .2& and illustrate your answer with a gra#h.

5. ra#h the !est+res#onse lines for the #risoner s dilemma game in a!le

.,& and indicate why there is a uni'ue ;ash e'uili!rium.

%. he data in the ta!le !elow were for the !attle+of+sees game shown in

a!le 5.2. he ,2 #layers were randomly matched& and the final 5 #eriods

out of a ,0+#eriod se'uence are shown. he game was run with an

e#erimental economics class at the University of Virginia. *ach #layer

consisted of one or two #eo#le at the same 8C& and one #layer was

selected at random e5 post to !e #aid a third of earnings. Calculate the

#ercentage of times that 3ow #layers chose *ast in all 5 #eriods& the #ercentage of times that Column #layers chose $est& and the average of

these two num!ers. hen calculate the mied+strategy ;ash e'uili!rium.

3ound % - ,0

3ow % *ast *ast& $est 5 *ast& , $est 5 *ast& , $est % *ast

Column 2 *ast& / $est 2 *ast& / $est , *ast& 5 $est *ast& $est 2 *ast& / $est

-5




-%



8art . ndividual (ecision *#eriments

he net #art of the !oo" contains a series of cha#ters on games involving

decision ma"ing when the #ayoff outcomes cannot !e "nown for sure in advance.

Uncertainty a!out outcomes is re#resented !y #ro!a!ilities& which can !e used to

calculate the e#ected value of some #ayoff or utility function. As indicated in

Cha#ter /& a ris"+neutral #erson will choose the decision with the highest

e#ected money value. ;on+neutral attitudes towards ris" can !e re#resented as

the maimi<ation of a nonlinear utility function.

he notion of e#ected utility !ecame widely acce#ted in economics after

von ;eumann and 4orgenstern =,//> #rovided a set of #lausi!le assum#tions or

aioms which ensure that decisions will corres#ond to the maimi<ation of the

e#ected value of some utility function. his is& of course& an as if claimB you

may sometimes see #eo#le calculate e#ected values& !ut it is rare to see a #erson

actually multi#lying out utilities and the associated #ro!a!ilities. oday& e#ected

utility is widely used in economics =and related areas li"e finance>& des#ite some

well+"nown situations where !ehavior is sensitive to !iases and !ehavioral

factors.

Cha#ter % #ertains to the sim#lest two+way #rediction tas"& e.g. rain or

shine& when the underlying #ro!a!ilities are fied !ut un"nown in re#eated

rounds. *ach new o!servation #rovides more information a!out the relative

li"elihood of the two events. he issue of #ro!a!ility matching concerns the

relationshi# !etween the #redictions and the underlying fre'uency of each eventBit is shown that sim#le matching #atterns ty#ically re#resent deviations from

o#timal !ehavior. 8ro!a!ility matching e#eriments are also used to introduce

some sim#le learning rules& e.g. reinforcement learning.

Anomalies in lottery+choice situations are discussed in Cha#ter -& e.g. the

Allais #arado& which is a #attern of choices that cannot !e e#lained !y standard

e#ected utility theory.

All decision #ro!lems considered u# this #oint will have !een static& i.e.

without any time+de#endent elements. he eighth cha#ter is organi<ed around a

dynamic situation involving costly search =e.g.& for a high wage or a low #rice>.

(es#ite the com#le nature of se'uential search #ro!lems& o!served !ehavior is

often sur#risingly consistent with search rules derived from an analysis of o#timaldecision+ma"ing.

--




-



Cha#ter %. 8ro!a!ility 4atching

8erha#s the sim#lest #rediction #ro!lem involves guessing which of two random

events will occur. he #ro!a!ilities of the two events are fied !ut not "nown& so

a #erson can learn a!out these #ro!a!ilities !y o!serving the relative fre'uencies

of the two events. Ine of the earliest !iases recorded in the #sychology literature

was the tendency for individuals to #redict each event with a fre'uency that

a##roimately matches the fraction of times that the event is o!served. his !ias&

"nown as #ro!a!ility matching& has !een widely acce#ted as !eing evidence of

irrationality des#ite 6iegel s e#eriments in the ,%0 s. hese e#eriments

#rovide an im#ortant methodological lesson for how e#eriments should !e

conducted. he results are also used to !egin a discussion of learning models that

may e#lain #aths of ad9ustment to some steady state where systematic changes in

!ehavior have ceased. 7inary #rediction tas"s can !e run !y hand =with

instructions in the A##endi> or using the Veconla! 8ro!a!ility 4atching

#rogram =listed under the (ecisions 4enu>.

I. Being reated Like a Rat

7efore the days of com#uters& the #rocedures for !inary #rediction tas"s in

#sychology e#eriments sometimes seemed li"e a setu# for rats or #igeons that

had !een scaled u# for human su!9ects. will descri!e the setu# used !y 6iegel

and oldstein =,5>. he su!9ect was seated at a des" with a screen thatse#arated the wor"ing area from the e#erimenter on the other side. he screen

contained two light !ul!s& one on the left and one on the right& and a third& smaller

light in the center that was used to signal that the net decision must !e made.

$hen the signal light went on& the su!9ect recorded a #rediction !y #ressing one

of two levers =left and right>. hen one of the lights was illuminated& and the

su!9ect would receive reinforcement =if any> !ased on whether or not the

#rediction was correct. $hen the net trial was ready& the signal light would

come on& and the #rocess would !e re#eated& #erha#s hundreds of times. ;o

information was #rovided a!out the relative li"elihood of the two events =1eft and

3ight>& although sometimes #eo#le were told how the events were generated& e.g.

!y using a #rinted list. n fact& one of the events would !e set to occur more often&e.g. -5 #ercent of the time. he #rocess generating these events was not always

random& e.g. sometimes the events were rigged so that in each !loc" of 20 trials&

eactly ,5 would result in the more li"ely event.

-




7y the time of the 6iegel and oldstein e#eriments in the late ,50 s&

#sychologists had already !een studying #ro!a!ility matching !ehavior for over

twenty years. he results seemed to indicate a curious #attern: the #ro#ortion of

times that su!9ects #redicted each event roughly matched the fre'uency with

which the events occurred. ?or eam#le& if 1eft occurred three+fourths of the

time& then su!9ects would come to learn this !y e#erience and then would tend to

#redict 1eft three+fourths of the time.

II. -re 7$ngr! Rats Rea%%! More Rationa% then 7$mans

he astute reader may have already figured out what is the !est thing to do

in such a situation& !ut a formal analysis will hel# ensure that such a conclusion

does not rely on hidden assum#tions li"e ris" neutrality. n order to evaluate the

rationality of this matching !ehavior& let us assume that the events really were

inde#endent random reali<ations& and that the more li"ely event occurred with #ro!a!ility that is greater than ,2. 1et > C denote the utility of the reward for a

correct #rediction& and let > denote the utility of the reward for an incorrect

#rediction. he rewards could !e eternal =money #ayments& food>& internal

=#sychological self+reinforcement>& or some com!ination. he only assum#tion is

that there is some #reference for ma"ing a correct #rediction: > C P > . hese

utilities may even !e changing over time& de#ending on the rewards received thus

farB the only assum#tion is that an additional correct #rediction is #referred.

Ince a num!er of trials have #assed& the #erson will have figured out

which event is more li"ely& so let p denote the su!9ective #ro!a!ility that

re#resents those !eliefs& with p P ,2. here are two decisions: #redict the more

li"ely event and #redict the less li"ely event. *ach decision yields a lottery:

Predict more #ike# event$ > C with #ro!a!ility p

U I with #ro!a!ility ,+ p

Predict #ess #ike# event$ > I with #ro!a!ility p

> C with #ro!a!ility ,+ p

hus the e#ected utility for #redicting the more li"ely event is higher if

p> C =, p > > I p> I =, p > > C &

or e'uivalently&

0



=2 p ,>> C =2 p ,>> I &

which is always the case since # P ,2 and > C P > . Although animals may !ecome satiated with food #ellets and other #hysical rewards& economists have no

trou!le with a non+satiation assum#tion for money rewards. ;ote that this

argument does not de#end on any assum#tion a!out ris" attitudes& since the two

#ossi!le #ayoffs are the same in each case& i.e. there is no more s#read in one case

than in the other. =$ith only two #ossi!le outcomes& the only role of #ro!a!ility

is to determine whether the !etter outcome has a higher #ro!a!ility& so ris"

aversion does not matter.> 6ince the higher reward is for a correct #rediction& the

more li"ely event should always !e #redicted& i.e. with #ro!a!ility ,. n this

sense& #ro!a!ility matching is irrational as long as there is no satiation& so > C P

> .

(es#ite the clear #rediction that one should #redict the more li"ely event,00 #er cent of the time after it !ecomes clear which event that is& this !ehavior is

often not o!served in la!oratory e#eriments. ?or eam#le& in a recent summary

of the #ro!a!ility matching literature& the #sychologist ?antino =,& ##.

%0%,> concludes: human su!9ects do not !ehave o#timally. nstead they match

the #ro#ortion of their choices to the #ro!a!ility of reinforcement . his !ehavior

is #er#leing given that non+humans are 'uite ade#t at o#timal !ehavior in this

situation. As evidence for the higher degree of rational !ehavior in animals& he

cites a ,% study that re#orted choice fre'uencies for the more li"ely event to !e

well a!ove the #ro!a!ility matching #redictions in most treatments conducted

with chic"s and rats. 7efore concluding that the animal su!9ects are more rational

than humans& it will !e instructive to review a #articularly well+designed

#ro!a!ility matching e#eriment.

III. Siege% and Go%dstein9s "(periments

6idney 6iegel is #erha#s the #sychologist who has had the largest direct and

indirect im#act on e#eriments in economics. His early wor" #rovides a high

standard of careful re#orting and #rocedures& a##ro#riate statistical techni'ues&

and the use of financial incentives where a##ro#riate. His e#eriments on

#ro!a!ility matching are a good eam#le of this wor". n one e#eriment& %male 8enn 6tate students were allowed to ma"e #redictions for ,00 trials& and

then ,2 of these were !rought !ac" on a later day to ma"e #redictions in 200 more

trials. he #ro#ortions of #redictions for the more li"ely event are gra#hed in

?igure %.,& with each #oint !eing the average over 20 trials.

,



prm

0

01

02

03

04

05

06

07

0809

1

500 100 150 200 250 300

Pre!iction .ate f or 8or e 9i/ely :;ent

no pay

pay<lossno pay =-olt>pay<loss =-olt>


he ,2 su!9ects in the no+#ay treatment were sim#ly told to do your !est to

#redict which light !ul! would !e illuminated. hese averages are #lotted as the

heavy dashed line& which !egins at a!out 0.5 as would !e e#ected in early trials

with no information a!out which event is more li"ely. ;otice that the #ro#ortion

of #redictions for the more li"ely event converges to the level of 0.-5 =shown !y a

hori<ontal line on the right> #redicted !y #ro!a!ility matching& with a leveling off

at a!out trial ,00.

8eriods Com#leted

0ig$re 1.'. Prediction Proportions for the "vent 6ith 0re#$enc! +.)*

So$rce: Siege%, Siege%, and -ndre6s @'413A for Dark Lines and 7o%t @'44A for Light Lines

n the #ay+loss treatment& ,2 #artici#ants received 5 cents for each correct

#rediction& and they lost 5 cents for each incorrect decision. he 20+trial averagesare #lotted as the dar" solid line in the figure. ;otice that the line converges to a

level of a!out 0.. A third #ay treatment offered a 5+cent reward !ut no loss for

an incorrect #rediction& and the results =not shown> are in !etween the other two

2



treatments& and clearly a!ove 0.-5. Clearly& incentives matter& and #ro!a!ility

matching is not o!served with incentives in this contet.

t would !e misleading to conclude that incentives always matter& or that

#ro!a!ility matching will !e o!served in no+#ay treatments using different

#rocedures. he two thin lines in ?igure %., show 20+trial averages for an

e#eriment =Holt& ,2> in which % University of Virginia students in no+#ay and

#ayloss treatments made decisions using com#uters& where the events were

determined !y a random num!er generator. he instructions matched those

re#orted in 6iegel& 6iegel& and Andrews =,%/>& ece#t that colored !oes on the

screen were used instead of light !ul!s. ;otice that #ro!a!ility matching is not

o!served in either the #ayloss treatment =with a reward of ,0 cents and a #enalty

of ,0 cents& shown !y the thin solid line> or the no+#ay treatment =thin dashed

line>. he results for a third treatment with a 20+cent reward and no #enalty =not

shown> were similar. he reason that matching was not o!served in the Holtno#ay treatment is unclear. Ine con9ecture is that the com#uter interface ma"es

the situation more anonymous and less li"e a matching #ennies game. n the

6iegel setu# with the e#erimenter on one side of the screen& the su!9ect might

incorrectly #erceive the situation as having some as#ects of a game against the

e#erimenter. 3ecall that the e'uili!rium in a matching #ennies game involves

e'ual #ro!a!ilities for each decision. his might e#lain the lower choice

#ercentages for the more li"ely event re#orted in the non+com#uteri<ed setu#& !ut

this is only a guess.

?inally& note that 6iegel s findings suggest a resolution to the #aradoical

finding that rats are smarter than humans in !inary #rediction tas"s. 6ince you

cannot 9ust tell a rat to do your !est& animal e#eriments are always run with foodor drin" incentives. As a result& the o!served choice #ro#ortions are closer to

those of financially motivated human su!9ects. n a recent survey of over fifty

years of #ro!a!ility matching e#eriments& Vulcan =,> concluded that

#ro!a!ility matching is generally not o!served with real #ayoffs& although

humans can !e sur#risingly slow learners in this sim#le setting.

I. - Simp%e Mode% of Be%ief Learning

Although the #ro!a!ility matching !ias is not ta"en seriously these days& at least

!y economists& the e#eriments #rovide a useful data set for the study of learning

!ehavior. iven the symmetry of the #ro!lem& a #erson s initial !eliefs ought to

!e that each event is e'ually li"ely& !ut the first o!servation should raise the

#ro!a!ility associated with the event that was 9ust o!served. Ine way to model




this learning #rocess is to let initial !eliefs for the #ro!a!ility of events 1 and 3

!e calculated as:

? ?=%.,> 8r=1> and 8r=3> =#riors>& ?? ??

where ? is a #ositive #arameter to !e e#lained !elow. If course& ? has no role yet&

since !oth of the a!ove #ro!a!ilities are e'ual to ,2.

f event 1 is o!served& then 8r=1> should increase& so let us add , to the

numerator for 8r=1>. o ma"e the two #ro!a!ilities sum to ,& we must add , to

the denominators for each #ro!a!ility e#ression:

?, ?=%.2> 8r=1> and 8r=3>

=after o!serving 1>. ? ? , ? ? ,

;ote that ? determines how 'uic"ly the #ro!a!ilities res#ond to the new

informationB a large value of ? will "ee# these #ro!a!ilities close to ,2.

Continuing to add , to the numerator of the #ro!a!ility for the event 9ust

o!served& and to add , to the denominators& we have a formula for the

#ro!a!ilities after F - o!servations of event 1 and F ' o!servations of event 3. 1et

F !e the total num!er of o!servations to date. hen the resulting #ro!a!ilities

are:

=%.> 8r=1> ? F - and 8r=3> ?

F ' =after F o!servations>.

2? F 2? F

where F J F - O F '.

n the early #eriods& the totals& F - and F '& might switch in terms of which one is

higher& !ut the more li"ely event will soon dominate& and therefore 8r=1> will !e

greater than ,2. he #rediction of this learning model =and the earlier analysis of

#erfect rationality> is that all #eo#le will eventually start to #redict the more li"ely

event every time. Any une#ected #rediction switches would have to !ee#lained !y adding some randomness to the decision ma"ing =oeree and Holt&

200>& or !y adding recency effects which ma"e #ro!a!ility assessments more

sensitive to the most recently o!served outcomes. hese "inds of modeling

changes will !e deferred until a later cha#ter. he #oint here is that a sim#le and

/



intuitive learning model can !e constructed& and that this model can e#lain a

high #ro#ortion of #redictions of the more li"ely event.

. Reinforcement Learning

n a #sychology e#eriment& the rewards and #unishments are referred to as

reinforcements. Ine #rominent theory of learning associates changes in !ehavior

to the reinforcements actually received. ?or eam#le& su##ose that the #erson

earns a reinforcement of 5 for each correct #rediction& nothing otherwise. f one

#redicts event 1 and is correct& then the #ro!a!ility of choosing 1 should increase&

and the etent of the !ehavioral change may de#end on the si<e of the

reinforcement& 5. Ine way to model this is to let the choice #ro!a!ility !e:

? 5 ?=%./> 8r=choose 1> and 8r=choose 3> .

? 5 ? ? ? 5

(es#ite the similarity with e'uation =%.2>& there are two im#ortant differences.

he left side of =%./> is a choice #ro!a!ility& not a #ro!a!ility that re#resents

!eliefs. $ith reinforcement learning& !eliefs are not e#licitly modeled& as is the

case for the !elief learning models of the ty#e discussed in the #revious section.

he second difference is that the 5 in =%./> re#resents a reinforcement& not an integer

count as in =%.2>.

If course& reinforcement is a !road term& which can include !oth #hysical

things li"e food #ellets given to rats& as well as #sychological feelings associated

with success or failure. Ine way that this model is im#lemented for e#eriments

with money #ayments is to ma"e the sim#lifying assum#tion that reinforcement is

measured !y earnings. 6u##ose that event 1 has !een #redicted F - times and that

the #redictions have sometimes !een correct and sometimes not. hen the total

earnings for #redicting 1& denoted e -& would !e less than 5F -. 6imilarly& let e ' !e

the total earnings from the correct 3 #redictions. he choice #ro!a!ilities would

then !e:

=%.5> 8r=choose 1> ?e - and 8r=choose 3> ?e '

.

2?e - e ' 2?e - e '

5




;otice that the ? #arameters again have the role of determining how 'uic"ly learning

res#onds to the stimulus& which is the money reinforcement in this case.

his "ind of model might also e#lain some as#ects of !ehavior in

#ro!a!ility matching e#eriments with financial incentives. he choice

#ro!a!ilities would !e e'ual initially& !ut a #rediction of the more li"ely event

will !e correct -5K of the time& and the resulting asymmetries in reinforcement

would tend to raise #rediction #ro!a!ilities for that event& and the total earnings

for this event would tend to !e larger than for the other event. hen e - would !e

growing faster& so that e ' e - would tend to get smaller as e - gets larger. hus the

#ro!a!ility of choosing 1 in =%.5> would tend to converge to ,. his convergence

may !e 'uite slow& as evidenced !y some com#uter simulations re#orted !y

oeree and Holt =200>. =6ee 'uestion % for some details on how such

simulations might !e structured.> he simulations normali<ed the #ayoff 5 to !e ,&

and used a value of 5 for ?. ?igure %.2 shows the 20+#eriod averages for simulations run for ,000 #eo#le& each ma"ing a series of 00 #redictions. he

simulated averages start somewhat a!ove the o!served averages for 6iegel s

su!9ects =more li"e the Holt data in ?igure %.,>& !ut similar 'ualitative #atterns are

o!served.

%



0

01

02

0304

05

06

07

08

09

1

0 50 100 150 200 250

Pre!iction .ate for t,e 8or e 9i/ely :;ent

Siegel's data

reinforcement learning simula

8eriod

0ig$re 1.. Sim$%ated and 5&served Prediction Proportions for the More Like%! "vent So$rce:Goeree and 7o%t @++/A for the sim$%ated data.

I. "(tensions

7oth of the learning models discussed here are somewhat sim#le& which is

#art of their a##eal. he reinforcement model incor#orates some randomness in

!ehavior and has the a##ealing feature that incentives matter. 7ut it has less of a

cognitive elementB there is no reinforcement for decisions not made. ?or

eam#le& su##ose that a #erson chooses 1 three times in a row =!y chance> and is

wrong each time. 6ince no reinforcement is received& the choice #ro!a!ilities stayat 0.5& which seems li"e an unreasona!le #rediction. I!viously& #eo#le learn

something in the a!sence of #reviously received reinforcement& since they reali<e

that ma"ing a good decision may result in higher earnings in the net round.

-




Camerer and Ho =,> have develo#ed a generali<ation of reinforcement

learning that contains some elements of !elief learning. 3oughly s#ea"ing&

o!served outcomes receive #artial reinforcement even if nothing is earned.

he !elief+learning model in section V can !e derived from statistical

#rinci#les =7ayes s rule to !e discussed a later cha#ter>. hen !eliefs determine the

e#ected #ayoffs =or utilities> for each decision& which in turn determine the decisions

made. n theory& the decision with the highest e#ected #ayoff is selected with

certainty. n an e#eriment& however& some randomness in decision ma"ing might !e

e#ected if the e#ected #ayoffs for the two decisions are not too different. his

randomness may !e due to random emotions& calculation errors& selective forgetting of

#ast e#erience& etc. 7uilding some randomness into models that use !elief learning

=as in Ca#ra& et al.& ,> is a useful way to go a!out e#laining data from la!oratory

e#eriments& as we shall see in later cha#ters.

hese learning models can !e enriched in other ways to o!tain !etter #redictions of !ehavior. ?or eam#le& the sums of event o!servations in the

!elief+learning model weigh each o!servation e'ually. t may !e reasona!le to

allow for forgetting in some contets& so that the o!servation of an event li"e 1 in

the most recent trial may carry more weight than something o!served a long time

ago. his is done !y re#lacing sums with weighted sums. ?or eam#le& if event 1

were o!served three times& F - in =%.2> would !e & which can !e thought of as

,O,O,. f the most recent o!servation =listed on the right in this sum> is twice as

#rominent as the one !efore it& then the #rior event would get a weight of one half&

and the one !efore that would get a weight of one+fourth& etc. hese and other

enrichments will !e discussed in later cha#ters.

<$estions

,. he initial !eliefs im#lied !y =%.,> are that each event is e'ually li"ely.

How might this e'uation !e altered in a situation where a #erson has some

reason to !elieve that one event is more li"ely than another& even !efore

any draws are o!servedN

2. A recent class e#eriment used the #ro!a!ility matching =84> Veconla!

software with #ayoffs of 0.20 for each correct #rediction& and in+class

earnings averaged several dollars. here were % teams of ,+2 students&

who made #redictions for 20 trials only. =8artici#ants were University of

Virginia undergraduates in an e#erimental economics class who had not

seen a draft of this cha#ter& !ut who had read drafts of the earlier cha#ters.>

he more li"ely event was #rogrammed to occur with #ro!a!ility 0.-5.

Calculate the e#ected earnings #er trial for a team that follows #erfect



#ro!a!ility matching. How much more would a team earn #er trial !y

!eing #erfectly rational after learning which event is more li"elyN

. Answer the two #arts of 'uestion 2 for the case where a correct answer

results in a gain of 0.,0 and an incorrect answer results in a loss of 0.,0.

/. ?or the gain treatment descri!ed in 'uestion 2& the more li"ely event

actually occurred with #ro!a!ility 0.-- in the first 20 trials& averaged over

all % teams. his event was #redicted with a fre'uency of 0.. $here would a

data #oint re#resenting this average !e #lotted in ?igure %.,N

5. ?or the gainloss treatment descri!ed in 'uestion & the more li"ely event

was o!served with a fre'uency of a!out 0.-5 in the first ten trials and 0.-

in the second ,0 trials. 8redictions were made !y % teams of ,+2 students&

and their earnings averaged a!out 25 cents #er team. =o cover losses&

each team !egan with a cash !alance of ,& as did the % other teams in the

#arallel gains treatment.> n the gainloss treatment& the more li"ely eventwas #redicted with a fre'uency of 0.5 in the first ,0 trials and 0.-0 in

trials ,,+20. o what etent do these results #rovide evidence in su##ort

of #ro!a!ility matchingN

%. =A ,0+sided die is re'uired.> he discussion of long+run tendencies for the

learning models was a little loose& since there are random elements in

these models. Ine way to #roceed is to simulate the learning #rocesses

im#lied !y these models. Consider the reinforcement learning model& with

initial choice #ro!a!ilities of one half each. Mou can simulate the initial

choice !y throwing the ,0+sided die twice& where the first throw

determines the tens digit and the second determines the ones digit. ?or

eam#le& throws of a % and a 2 would determine a %2. f the die is mar"edwith num!ers 0& ,& & then any integer from 0 to is e'ually li"ely. he

simulation could #roceed !y letting the #erson #redict 1 if the throw is less

than 50 =i.e.& one of the 50 outcomes: ,& 2& & ... />& which would occur

with #ro!a!ility 0.5. he determination of the random event& 1 or 3&

could !e done similarly& with the outcome !eing 1 if the net two throws

determine a num!er that is less than -5. iven the #rediction& the event&

and the reward& say ,0 cents& you can use the formula in =%./> to determine

the choice #ro!a!ilities for the net round. his whole #rocess can !e

re#eated for a num!er of rounds& and then one could start over with a new

simulation& which can !e thought of as a simulation of the decision #attern

of a second #erson. 6imulations of this ty#e are easily done with

com#uters& or even with the random num!er features of a s#readsheet

#rogram. However you choose to generate the random num!ers& your tas"

is to simulate the #rocess for four rounds& showing the choice




#ro!a!ilities& the actual choice& the event o!served& and the reinforcement

for each round.

Cha#ter -. 1ottery Choice Anomalies

Choices !etween lotteries with money #ayoffs may #roduce aniety and other

emotional reactions& es#ecially if these choices may result in significant monetary

gains and losses. *#ected utility theory im#lies that such choices& even the

difficult and stressful ones& can !e modeled as the maimi<ation of a

mathematical function =a sum of #roducts of #ro!a!ilities and utilities>. As would

!e e#ected& actual decisions sometimes deviate from these mathematical

#redictions& and this cha#ter considers one of the more common anomalies: the

Allais #arado. Ither !iases& such as the mis#erce#tion of large and small

#ro!a!ilities are discussed. he 8airwise 1ottery Choice #rogram on the

Veconla! site can !e used to evaluate many of these anomaliesB the default setu#

#rovides #aired lottery choices that are !ased on Allais #arado and reflection

effects.

I. Introd$ction

he #redominant a##roach to the study of individual decision ma"ing in ris"y

situations is the e#ected utility model introduced in Cha#ter /. *#ected utility

calculations are sums of #ro!a!ilities times =#ossi!ly nonlinear> utility functionsof the #ri<e amounts. n contrast& the #ro!a!ilities enter linearly& which #recludes

over+weighting of low #ro!a!ilities& for eam#le. he nonlinearities in utility

#ermit an e#lanation of ris" aversion. his model& which dates to 7ernoulli

=,->& received an formal foundation in von ;eumann and 4orgensetern s

=,//> !oo" on game theory. his !oo" s#ecified a set of assum#tions = aioms >

that im#ly !ehavior consistent with the maimi<ation of the e#ected value of a

utility function.

Almost from the very !eginning& economists were concerned that !ehavior

in some situations seemed to contradict the #redictions of this model. he most

famous contradiction& the Allais #arado& is #resented in the net section. 6uch

anomalies have stimulated a lot of wor"& theoretical and e#erimental& ondevelo#ing alternative models of choice under ris". he most commonly

mentioned alternative& 8ros#ect heory& is #resented and discussed in the sections

that follow.

0



A reader loo"ing for a resolution of the "ey modeling issues will !e

disa##ointedB some #rogress has !een made& !ut much of the evidence is mied.

he #ur#ose of this cha#ter is to introduce the issues& and to hel# the reader

inter#ret seemingly contradictory results o!tained with different #rocedures. ?or

eam#le& it is not uncommon for the estimates of the environmental !enefits of

some #olicy to differ !y a factor of 2& which may !e attri!uted to the way the

'uestions were as"ed and to a willingness+to+#aywillingness+to+acce#t !ias. A

familiarity with this and other !iases is crucial for anyone interested in

inter#reting the results of e#erimental and survey studies of situations where the

outcomes are un"nown in advance.

II. he -%%ais Parado(

Consider a choice !etween a sure &000 and a 0. chance of winning /&000. hischoice can !e thought of as a choice !etween two lotteries that yield random

earnings:

a!le -.,. A 1ottery Choice *#eriment =Dahneman and vers"y& ,->

Lotter! S =selected !y 0 K> Lotter! R =selected !y 20K>

&000 with #ro!a!ility ,.0 /&000 with #ro!a!ility 0.

0 with #ro!a!ility 0.2

A lottery is an economic item that can !e owned& given& !ought& or sold. 8eo#le

may #refer some lotteries to others& and economists assume that these #referencescan !e re#resented !y a utility function& i.e. that there is some mathematical

function with an e#ected value that is higher for the lottery selected than for the

lottery not selected. his is not an assum#tion that #eo#le actually thin" a!out

utility or do such calculations& !ut rather& that choices can !e re#resented !y =and

are consistent with> ran"ings #rovided !y the utility function.

he sim#lest utility function is the e#ected money value& which would re#resent

the #references of someone who is ris" neutral. An e#ected #ayoff com#arison

would favor 1ottery 3& since 0.=/&000> J &200& which is higher than the &000

for the safe o#tion. n this situation& Dahneman and vers"y re#orted that 0K of

the su!9ects chose the safe o#tion& which indicates some ris" aversion. =8ayoffs&

in sraeli #ounds& were hy#othetical.> A #erson who is not neutral to ris" would

have #references re#resented !y a utility function with some curvature. he

decision of an e#ected utility maimi<er who #refers the safe o#tion could !e

re#resented:

,




=-.,> > =&000> 0.> =/&000>0.2> =0>&

where the utility function re#resents #references over money income.

;et consider some sim#le mathematical o#erations that can !e used to

o!tain a #rediction for how such a #erson would choose in a different situation.

?or eam#le& su##ose that there is a three+fourths chance that all gains from either

lottery will !e confiscated& i.e. that is a 0.-5 chance of earning 0. o analy<e this

#ossi!ility& multi#ly !oth sides of =-.,> !y 0.25& to o!tain:

=-.2> 0.25> =&000> 0.2> =/&000>0.05> =0>.

n order to ma"e the #ro!a!ilities on each side sum to ,& we need to add the 0.-5 > =0>

that corres#onds to confiscation. his amount is added to !oth sides of-.2> to o!tain:

=-.> 0.25> =&000>0.-5> =0> 0.2> =/&000>0.> =0>.

he ine'uality im#lies that the same #erson =who initially #referred 1ottery 6 to

1ottery 3 in a!le -.,> would #refer a one+fourth chance of &000 to a one+fifth

chance of /&000. Any reversal of this #reference #attern& e.g. #referring the sure

&000 in the first choice and the 0.2 chance of /&000 in the second choice would

violate e#ected utility theory. A ris"+neutral #erson& for eam#le& would #refer

the lottery with the #ossi!ility of winning /&000 in !oth cases =we alreadychec"ed the first case& and the second is assigned in 'uestion ,>.

he intuition underlying these #redictions can !e seen !y reeamining e'uation

=-.>. he left side is the e#ected utility of a one+fourth chance of &000 and a

three+fourths chance of 0. *'uivalently& we can thin" of the left side as a one+

fourth chance of 1ottery 6 =which gives &000> and a three+fourths chance of 0.

Although it is not so trans#arent& the right side of =-.> can !e e#ressed

analogously as a one fourth chance of 1ottery 3 and a three+fourths chance of 0.

o see this& note that 1ottery 3 #rovides /&000 with #ro!a!ility 0.& and one

fourth of 0. is 0.2& as indicated on the right side of =-.>. hus the im#lication of

the ine'uality in =-.> is that a one+fourth chance of 1ottery 6 is #referred to a

one+fourth chance of 1ottery 3. he mathematics of e#ected utility im#lies thatif you #refer 1ottery 6 to 1ottery 3 as in =-.,>& then you #refer a one+fourth

chance of 1ottery 6 to a one+fourth chance of 1ottery 3 as in =-.>. he intuition

for this #rediction is that an etra three+fourths chance of 0 was added to +oth

sides of the e'uation in going from =-.2> to =-.>. his etra #ro!a!ility of 0

2



dilutes the chances of winning in !oth 1ottery 6 and 1ottery 3& !ut this added

#ro!a!ility of 0 is a common& and hence irrelevant addition. ndeed& one of the

!asic aioms used to motivate the use of e#ected utility is the assum#tion of

inde#endence with res#ect to irrelevant alternatives.

As intuitive as the argument in the #revious #aragra#h may sound& a significant

fraction of the Dahneman and vers"y su!9ects violated this #rediction. *ighty

#ercent chose 1ottery 6 over 1ottery 3& !ut sity+five #ercent chose the diluted

version of 1ottery 3 over the diluted version of 1ottery 6. his !ehavior is

inconsistent with e#ected utility theory& and is "nown as an Allais 8arado&

named after the ?rench economist& Allais =,5>& who first #ro#osed these ty#es

of #aired lottery choice situations. n #articular& this is the commonratio version

of the Allais #arado& since #ro!a!ilities of #ositive #ayoffs for !oth lotteries are

diluted !y a common ratio. Anomalous !ehavior in Allais #arado situations has

also !een re#orted for e#eriments in which the money #ri<es were #aid in cash=e.g.& 6tarmer and 6ugden& ,& ,,>. 7attalio& Dagel& and 4acdonald =,5>

even o!served similar choice #atterns with rats that could choose !etween levers

that #rovided food #ellets on a random !asis.

III. Prospect heor!: Pro&a&i%it! Misperception

he Allais #arado results stimulated a large num!er of studies that were

intended to develo# alternatives to e#ected utility theory. he alternatives are

ty#ically more com#licated to use& and none of them show a clear advantage over

e#ected utility in terms of #redictive a!ility. he #ossi!le ece#tion to this

conclusion is #ros#ect theory #ro#osed !y Dahneman and vers"y =,->.8ros#ect theory is really a collection of elements that s#ecify how a #erson

evaluates a ris"y #ros#ect in relation to some status Huo #osition for the

individual. 3oughly s#ea"ing& there is a reference #oint& e.g. current wealth& from

which gains and losses are evaluated& and gains are treated differently from

losses. 8eo#le are assumed to !e averse to losses& !ut when ma"ing choices

where #ayoffs are all losses& they are thought to !e ris" see"ing. In the other

hand& when #ayoffs are all gains& #eo#le are assumed to !e ris" averse in general.

A final element is a notion that #ro!a!ilities are not always correctly #erceived&

i.e. that low #ro!a!ilities are over+weighted and high #ro!a!ilities are

underweighted. hese elements can !e un!undled and evaluated one at a time&

and modifications of e#ected utility theory that include one or more of these

elements can !e considered& which is the #lan for the remainder of this cha#ter.

he #ossi!ility of #ro!a!ility mis#erce#tion will !e suggested !y an e#eriment&




re#orted in Cha#ter 0& in which #eo#le are #rovided some information and are

as"ed to re#ort the #ro!a!ility that some event has occurred.

he incentives in this e#eriment are set so that su!9ects should re#ort the truth&

e.g.. when the true #ro!a!ility is 0.-5 they should re#ort 0.-5. n a gra#h with the

true #ro!a!ility on the hori<ontal ais and the re#orted #ro!a!ility on the vertical

ais& a /5+degree line would re#resent the correct re#ort from which deviations

could !e o!served. As will !e seen& data show a reverse 6+sha#ed #attern to the

deviations =over+weighting of low #ro!a!ilities and under+weighting of high

#ro!a!ilities>& although the !iases there are somewhat small =see a!le 0.,

!elow>. 8ros#ect theory #redicts a #ro!a!ility weighting function with this

general sha#e& starting at the origin& rising a!ove the /5+degree line for low

#ro!a!ilities& falling !elow for high #ro!a!ilities& and ending u# on the /5+degree

line in the u##er+right corner. he notion that the #ro!a!ility weighting function

should cross the /5+degree line at the u##er+right corner is !ased on the intuitionthat it is difficult to mis#erceive a #ro!a!ility of one.

o see how #ros#ect theory may e#lain the Allais #arado& note that 1ottery 6

on the left side of a!le -., is a sure thing& so no mis#erce#tion of #ro!a!ility is

#ossi!le. he 0. chance of a /&000 #ayoff on the right& however& may !e

affected. ;ow reconsider 1ottery 3 on the right side of a!le -.,& when the 0.

#ro!a!ility of /&000 is treated as if it were lower& say 0.-. his mis#erce#tion

would enhance the attractiveness of 1ottery 6& so that even a ris" neutral #erson

would #refer 6 if the #ro!a!ility was mis#erceived in this manner ='uestion 2>.

;et consider the choice that results when !oth lotteries are diluted !y a three+

fourths chance of a 0 #ayoff. t is a##arent from =-.> that the 0.25 #ro!a!ility of

the &000 gain on the left is a!out the same as the 0.2 #ro!a!ility of the /&000 onthe right. n other words& 0.25 and 0.2 are located close to each other& so that a

smooth #ro!a!ility weighting function will overweight them more or less !y the

same amount. Here a #ro!a!ility weighting function will have little effect& and a

ris"+neutral #erson will #refer the diluted version of 1ottery 3& even though the

same #erson might #refer the undiluted version of 1ottery 6 when the high

#ro!a!ility of the /&000 #ayoff for lottery 3 is under+weighted. 8ut differently&

the certain #ayoff in the original 1ottery 6 is not mis#erceived& !ut the 0.

#ro!a!ility of /&000 in 1ottery 3 is under+weighted. 6o #ro!a!ility weighting

tends to !ias choices towards 6. 7ut the diluted 6 and 3 lotteries have

#ro!a!ilities in the same range =0.25 and 0.2> so no such !ias would !e introduced

!y a smooth #ro!a!ility weighting function.

his e#lanation of the Allais #arado is #lausi!le& !ut it is not universally

acce#ted. ?irst& the evidence on #ro!a!ility weighting functions is mied. ;ote

that the reverse+6 #attern of deviations =raising low #ro!a!ilities and lowering

/



high #ro!a!ilities> could also !e due to the tendency for random errors in the

elicitation #rocess to s#read out on the side where there is more room for error. A

num!er of other recent studies have failed to find the reverse+6 #attern of

#ro!a!ility mis#erce#tions =oeree& Holt& and 8alfrey& 2002& 200>. Har!augh&

Drause& and Vesterlund =2002> found this reverse 6 #attern when #ro!a!ilities

were elicited !y as"ing for su!9ects to assign #rices to the lotteries& !ut the

o##osite #attern =6+sha#ed> was o!served when the #ro!a!ilities were elicited !y

giving #eo#le choices !etween lotteries. As intuitive as the #ro!a!ility weighting

e#lanation for the Allais #arado seems& one would have to say that the evidence

is inconclusive at this time.

I. Prospect heor!: Gains, Losses, and Ref%ection "ffectsE

A second #art of #ros#ect theory #ertains to the notion of a reference #oint fromwhich gains and losses are evaluated. n e#eriments with money #ayments& the

most o!vious candidate for the reference #oint is the current level of wealth&

which includes earnings u# to the #resent. he reference #oint is the !asis for the

notion of loss aversion& which im#lies that losses are given more weight in

choices with outcomes that involve !oth gains and losses. A second #ro#erty of

the reference #oint is "nown as the ref#ection effect & which #ertains to cases where

#ositive #ayoffs are multi#lied !y minus one in a manner that reflects them

around 0. he reflection effect #ostulates that the ris" aversion ehi!ited !y

choices when all outcomes are gains will !e transformed into a #reference for ris"

when all outcomes are losses.

A com#arison of !ehavior in the gain and loss domains is difficult in a la!oratorye#eriment for a num!er of reasons. ?irst& the notion of a reference #oint is not

#recisely defined. ?or eam#le& would #a#er earnings recorded u# to the #resent

#oint =!ut not actually #aid> in cash !e factored into the current wealth #ositionN

A second #ro!lem is that human su!9ects committees do not allow researchers to

collect losses from su!9ects in an e#eriment& who cannot wal" out with less

money that they started with. Ine solution is to give #eo#le an initial sta"e of

cash !efore they face losses& !ut it is not clear that this #rocess will really change

a #erson s reference #oint. he notion that an initial sta"e is treated differently

than hard+earned cash is called the house+money effect. here is some evidence

that gifts =e.g. candy> tend to ma"e #eo#le more willing to ta"e ris"s in some

contets and less willing in others =Ar"es et al. ,& ,/>. t is at least #ossi!le

that the warm glow of a house+money effect may cause #eo#le to a##ear ris"

see"ing for losses when this may not ordinarily !e the case with earned cash. Ine

solution is to give #eo#le identical sta"es !efore !oth gain and loss treatments&

5




which holds the house+money effect constant. And ma"ing #eo#le earn the initial

sta"e through a series of e#erimental tas"s is #ro!a!ly more li"ely to change the

reference #oint.

Dahneman and vers"y =,-> #resented strong e#erimental evidence for a

reflection effect. he design involved ta"ing all gains in a choice #air li"e those

in a!le -., and reflecting them around <ero to get losses& as in a!le -.2. ;ow

the safe lottery& 6R& involves a sure loss& whereas the ris"y lottery& 3R& may yield a

worse loss or no loss at all. he choice #attern in a!le -.,& with 0K safe

choices& is reversed in a!le -.2& with only K safe choices.

a!le -.2. A 3eflection *ffect *#eriment =Dahneman and vers"y& ,->

Lotter! SF =selected !y K> Lotter! RF =selected !y 2K>

minus &000 with #ro!a!ility ,.0 minus /&000 with #ro!a!ility 0.

0 with #ro!a!ility 0.2

he Dahneman and vers"y e#eriments used hy#othetical #ayoffs& which

raises the issue of whether this reflection effect will #ersist with economic

incentives. =3ecall from Cha#ter / that ris" aversion was strongly affected !y the

use of high economic incentives& as com#ared with hy#othetical #ayoffs>.

Holt and 1aury =2002> evaluate the etent of the hy#othetical !ias in a

reflection e#eriment. hey too" a menu of #aired lottery choices similar to that

in a!le /., and reflected all #ayoffs around 0. 3ecall that this menu has safe

lotteries on one side and ris"y lotteries on the other& and that the #ro!a!ility of the

higher #ayoff num!er increases as one moves down the menu. 3is" aversion is

inferred !y loo"ing at the num!er of safe choices relative to the num!er of safe

choices that would !e made !y a ris"+neutral #erson =in this case 5>. All

#artici#ants made decisions in !oth the gains menu and in the losses menu& with

the order of menu #resentation alternated in half of the sessions. n their

hy#othetical #ayoff treatment& su!9ects were #aid a fied amount /5 in echange

for #artici#ating in a series of tas"s =search& #u!lic goods> in a different

e#eriment& and afterwards they were as"ed to indicate their decisions for the

lottery choice menus with the understanding that all gains and losses would !e

hy#othetical. $hen all #ayoffs were hy#othetical gains& a!out half of the su!9ectswere ris" averse& and slightly more than 50K of those who showed ris" aversion

for gains were ris" see"ing for losses. he modal #attern in this treatment was

reflection& although other #atterns =e.g. ris" aversion for gains and losses> were

also o!served with some fre'uency. he choice fre'uencies for the hy#othetical

%



000

005

010

015

020

025

030

.is/ @;erse

.is/ See/in*

Real

Gains

000

005

010

015

020

025

030

.is/ @;erse

.is/ See/in*

Hypothetcal

Gains

choices are shown in the left #anel of ?igure -.,. he modal #attern of reflection

is re#resented !y the tall s#i"e in the !ac"+right corner of the left #anel.

he real+incentive treatments for gains and losses were run in a #arallel

manner with the same choice menus. 8artici#ants were allowed to !uild u#

earnings of a!out /5 in a different e#eriment using the same tas"s used under

the hy#othetical treatment. n contrast with earlier results& the modal #attern of

!ehavior with real incentives did not involve reflection. he most common

#attern was for #eo#le to ehi!it ris" aversion for !oth gains and losses. he

real#ayoff choice fre'uencies are shown in the right #anel of ?igure -.,. he

modal #attern of ris" aversion in !oth cases is re#resented !y the s#i"e in the

!ac"+left side of this #anel. here is a little more ris" aversion with real #ayoffs

than with hy#othetical #ayoffsB sity #ercent of the su!9ects ehi!it ris" aversion

in the gain condition& and of these only a!out a fifth are ris" see"ing for losses.

he rate of reflection with real #ayoffs is less than half of the reflection rateo!served with hy#othetical #ayoffs.

(es#ite the a!sence of a clear reflection effect& there is some evidence that

gains and losses are treated differently. In average& #eo#le tended to !e

essentially ris" neutral in the loss domain& !ut they were generally ris" averse in

the gain domain. his result #rovides some su##ort for the notion of a reference

#oint& around which gains and losses are evaluated& which suggests that la!oratory

data should !e analy<ed using utility as a function of earnings =gains and losses>&

not final wealth. n other words& if the net worth of a #erson s assets is ;& and if a

decision may #roduce earnings or losses of 5& then the analysis of e#ected utility

=with or without the #ro!a!ility weighting of #ros#ect theory> should !e

e#ressed in terms of U= 5>& not U=;O 5>.

-

3is" 3is"

Averse 6ee"ing

7!pothetica%

Losses

3is" 3is"

Averse 6ee"ing

Rea% Losses




0ig$re ).'. Inferred Risk -version for 7!pothetica% Pa!offs @LeftA and Rea% Pa!offs @RightA

So$rce: 7o%t and La$r! @++A

he a!sence of a clear reflection effect in the Holt and 1aury data is a little sur#rising

given the results of several other studies that found reflection with real money incentives

=Camerer& ,B 7attalio et al. ,0>. Ine difference is that instead of holding initial

wealth constant in !oth treatments =at a level high enough to cover losses>& these studies

#rovided a high initial sta"e in the loss treatment& so the final wealth #osition is constant

across treatments. ?or eam#le& a lottery over gains of /&000 and 0 could !e re#laced

with an initial #ayoff of /&000 and a choice involving #osses of /&000 and 0. *ach

#resentation or frame #rovides the same #ossi!le final wealth #ositions =0 or /&000>& !ut

the framing is in terms of gains in one treatment and in terms of losses in the other. A

setu# li"e this is eactly what is needed to document a framing effect. 6uch an effect is #resent since !oth studies re#ort a tendency for su!9ects to !e ris" averse in the gain

frame and ris" see"ing in the loss frame. $hether these results indicate a reflection

effect is less clear& since the higher sta"e #rovided in the loss treatment may itself have

induced more ris" see"ing !ehavior& 9ust as gifts of candy and money tend to increase

ris" see"ing in e#eriments re#orted !y #sychologists.

. "(tensions and 0$rther Reading

he dominant method of modeling choice under ris" in economics and finance involves

e#ected utility& either a##lied to gains and losses or to final wealth. he final wealth

a##roach involves a stronger ty#e of rationality in the sense that #eo#le can see #astgains and losses and focus on the varia!le that determines consum#tion o##ortunities

=final wealth>. he Camerer =,> and 7attalio et al. =,> e#eriments #rovide

strong evidence that decisions are framed in terms of gains and losses& and that #eo#le

do not integrate gains and losses into a final asset #osition. ndeed& there is little if any

e#erimental evidence for such asset integration. 3a!in =2000> and 3a!in and haler

=200,> also #rovide a theoretical argument against the use of e#ected utility as a

function of final wealthB the argument !eing that the ris" aversion needed to e#lain

choices involving small amounts of money im#lies a!surd levels of ris" aversion for

choices involving large amounts of money. 4ost analyses of ris" aversion in la!oratory

e#eriments are& in fact& already done in terms of gains and losses =7inswanger& ,0B

Dachelmeyer and 6hehata& ,2B oeree& Holt& and 8alfrey& 2002& 200>.*ven if e#ected utility is modeled in terms of gains and losses& there is the issue of

whether to incor#orate other elements li"e nonlinear #ro!a!ility weighting that

re#resents systematic mis#erce#tions of #ro!a!ilities. As noted a!ove& the evidence on



this method is mied& as is the evidence for the reflection effect. 6ome& li"e Camerer

=,5>& have urged economists to give u# on e#ected utility theory in favor of #ros#ect

theory and other alternatives. 4ore recently& 3a!in and haler =200,> have e#ressed

the ho#e that they have written the final #a#er that discusses the e#ected utility

hy#othesis& referring to it as the ehy#othesis with the same tone that is sometimes used

in tal"ing a!out an es#ouse. Ither economists li"e Hey =,5> maintain that the

e#ected+utility model out#erforms the alternatives& es#ecially when decision errors are

e#licitly modeled in the #rocess of estimation. n s#ite of this controversy& e#ected

utility continues to !e widely used& either im#licitly !y assuming ris" neutrality or

e#licitly !y modeling ris" aversion in terms of either gains and losses or in terms of

final wealth.

6ome may find the mied evidence on some of these issues to !e worrisome& !ut

to an e#erimentalist it #rovides an eciting area for further research. ?or eam#le&

there remains a lot of wor" to !e done in terms of finding out how #eo#le !ehave inhigh+sta"es situations. Ine way to run such e#eriments is to go to countries where

using high incentives would not !e so e#ensive. ?or eam#le& 7inswanger =,0>

studied the choices of farmers in 7angladesh when the #ri<e amounts sometimes

involved more than a month s salary. =He o!served considera!le ris" aversion& which

was more #ronounced with very high sta"es.> 6imilarly& Dachelmeier and 6hehata

=,2> #erformed lottery choice e#eriments in rural China. hey found that the method

of as"ing the 'uestion has a large im#act on the way #eo#le value lotteries. ?or

eam#le& if you as" for a selling #rice =the least amount of money you would acce#t to

sell the lottery>& #eo#le tend to give a high answer& which would seem to indicate a high

value for the ris"y lottery& and hence a #reference for ris". 7ut if you as" them for the

most they would !e willing to #ay for a ris"y lottery& they tend to give a much lower num!er& which would seem to indicate ris" aversion. he incentive structure was such

that the o#timal decision was to #rovide a true money value in !oth treatments =much as

the instructions for the #revious cha#ter #rovided an incentive for #eo#le to tell the truth

a!out their #ro!a!ilities in a 7ayes rule e#eriment>. t seems that #eo#le go into a

!argaining mode when #resented with a #ricing tas"& demanding high selling #rices and

offering low !uying #rices. he nature of this willingness+to+#aywillingness+to+acce#t

!ias is not well "nown& at least !eyond the sim#le !argaining mode intuition #rovided

here. ;evertheless& it is im#ortant for #olicy ma"ers to !e aware of the $8$A !ias&

since studies of non+mar"et goods =li"e air and water 'uality> may have estimates of

environmental !enefits that vary !y ,00K de#ending on how the 'uestion is as"ed.

iven the strong nature of this $8$A !ias& it is usually advisa!le to avoid using

#ricing tas"s to elicit valuations. ?or more discussion of this to#ic& see 6hogren& et al.

=,/>.




A num!er of additional !iases have !een documented in the #sychology

literature on 9udgment and decision ma"ing. ?or eam#le& there may !e a tendency for

#eo#le to !e overconfident a!out their 9udgments in some contets. 6ome of the

systematic ty#es of 9udgmental errors will !e discussed at length in later cha#ters& such

as the winner s curse in auctions for #ri<es of un"nown value. ?or further discussion of

these and other anomalies& see Camerer =,5>.

<$estions

,. 6how that a ris"+neutral #erson would #refer a 0. chance of winning /&000 to a

sure #ayment of &000& and that the same #erson would #refer a 0.2 chance of

winning /&000 to a 0.25 chance of winning &000.

2. f the #ro!a!ility of the /&000 #ayoff for lottery 3 in a!le -., is re#laced !y 0.-&

show that a ris" neutral #erson would #refer 1ottery 6.

Cha#ter . 6I =n 6earch of >

$hen you go out to ma"e a #urchase& you #ro!a!ly do not chec" #rices at all #ossi!le

locations. n fact& you might sto# searching as soon as you find an acce#ta!le offer&

even if you "now that a !etter offer would #ro!a!ly turn u# eventually. 6uch !ehavior

may !e o#timal if the search #rocess is costly& which ma"es it worthwhile to com#are

the costs and !enefits of additional searching. he game discussed in this cha#ter is a

search #ro!lem& where each additional offer costs a fied amount of money. I!served !ehavior in such situations is often sur#risingly rational& and the classroom game can !e

used to introduce a discussion of o#timal search. As usual& the game can !e done using

ten+sided dice to generate the random offers& following the instructions in the A##endi&

or it can !e run on the Veconla! software =select 6e'uential 6earch from the (ecisions

4enu>.

I. Introd$ction

4any economists !elieved that the rise of e+commerce would lead to

dramatically lower search costs and a conse'uent reduction in #rice levels and

dis#ersion. *vidence to date& however& suggests that there is still a fair amount of #rice

varia!ility on the internet& and most of us can attest to the time costs of searching for low

#rices. If course& a reduction in the cost of o!taining a #rice 'uote would cause one to

get more 'uotes !efore deciding on a ma9or #urchase& !ut the total search cost may go

,00



either way& since this total is the #roduct of the cost #er search and the num!er of

searches.

ssues of search and #rice dis#ersion are central to the study of how mar"ets

#romote efficiency !y connecting !uyers and sellers. 6earch is also im#ortant on a

macroeconomic level& since much of what is called frictional unem#loyment is due to

wor"ers searching for an acce#ta!le wage offer. his cha#ter #resents a #articularly

sim#le search #ro!lem& in which a #erson #ays a constant cost for o!taining each new

o!servation =e.g. a wage offer or #rice 'uote>. I!servations are inde#endent draws from

a #ro!a!ility distri!ution that is "nown. ?rom a methodological #oint of view& the

search #ro!lem is interesting !ecause it is dynamic& i.e. decisions are made in se'uence.

II. Search from a >niform Distri&$tion

he setu# used here is !ased on a #articular #ro!a!ility framewor"& the uniformdistri!ution. 4any situations of interest involve e'ual #ro!a!ilities for the relevant

events. ?or eam#le& consider an air#ort with continuously circulating !uses num!ered

from , to 20& which all #ass !y a certain central #ic"u# #oint. f there is no #articular

#attern& the net !us to #ass might !e modeled as a uniform random varia!le. he

uniform distri!ution is easy to e#lain since all #ro!a!ilities are e'ual& and the

theoretical #ro#erties of models with uniform distri!utions are often 'uite sim#le.

herefore& the uniform distri!ution will !e used re#eatedly in other #arts of this !oo"&

for eam#le& in the cha#ters on auctions.

A uniform distri!ution is easily generated with a random device& li"e drawing

#ing #ong !alls with different num!ers written on them& so that each num!er is e'ually

li"ely to !e drawn. *ven if a com#uter random num!er generator is used& it is im#ortantto #rovide a #hysical eam#le to e#lain what the distri!ution im#lies. Ine way to

e#lain a uniform distri!ution on some interval& say from 0 to & is to imagine a

roulette wheel with ,00 sto#s& la!eled 0& ,& 2& & so that a hard s#in is 9ust as li"ely to

sto# at one #oint as at any other. A convenient and ine#ensive way to generate the

reali<ation of a uniform distri!ution is with throws of a ,0+sided die& which is generally

la!eled 0& ,& 2&

. he first throw can determine the tens digit& and the second throw can determine

the ones digit. n this manner& each of the ,00 integers on the interval from 0 to is

e'ually li"ely.

7esides !eing easy to e#lain and im#lement& the uniform distri!ution has the

useful #ro#erty that the e#ected value is the mid#oint of the interval. his mid#oint

#ro#erty is due to the a!sence of any asymmetry around the mid#oint that would #ull

e#ected value in one direction or the other. o see this& consider the sim#lest case& i.e.

with two e'ually li"ely outcomes& 0 and ,. 6ince each occurs with #ro!a!ility ,2& the

,0,




e#ected value is =,2>=0> O =,2>=,> J ,2& which is the mid#oint of the interval. ?or a

distri!ution from 0 to determined !y the die throws& each integer in the interval has

the same #ro!a!ility& 0.0,& and the e#ected value is 0.0,=0> O 0.0,=,> O 0.0,=2> . O

0.0,R=>& which is the mid#oint& /.5.

he search instructions for the e#eriment discussed in this cha#ter #resent

su!9ects with an o##ortunity to #urchase draws from a distri!ution that is uniform on the

interval from 0 to 0 =#ennies>. his distri!ution is generated !y a com#uter random

num!er generator for the we! version =63>& and it can !e determined !y the throws of

dice !y ignoring all outcomes a!ove 0. *ach draw costs 5 cents& and there is no limit

on the num!er of draws. he su!9ect may decline to search& there!y earning <ero. f the

first draw is (, #ennies& then sto##ing at that #oint would result in earnings of ( , 5

#ennies. 6u##ose that the second draw is (2. hen the o#tions are: ,> #ay another

nic"el and search again& 2> sto# and earn ( 2 ,0& or if going !ac" is allowed& > acce#t

the first draw and earn (, ,0. $e will say that there is recall if going !ac" to ta"e any #reviously re9ected draw is #ermittedB otherwise we say that there is no recall. 3ecall

may not !e #ossi!le in some situations& e.g. when going through the classified =or

#ersonal 6I> ads.

6ome ty#ical search se'uences for one #erson are shown in a!le .,. he round

num!er is in the left column& and the draws are listed in se'uence in the middle column&

with the acce#ted draw shown in !oldface ty#e. he search cost was 5 cents& and the

resulting earnings are shown in the right column. his #erson& who had no #rior

#ractice& seems to have sto##ed as soon as a draw of a!out /5 cents or a!ove was

o!tained.

a!le .,. ndividual 6earch with a 5 Cent Cost

3ound (raw 6e'uence =cents> *arnings

, )+ %52 ,& /& 1/ / 2,& & /& /& *' 2%

/ */ /5 2) 2% 14 %/- & & 5& 3/ 2

2* 0

31 /,,0 22& ,2& 1* 50

,02



he analysis of the o#timal way to search is sim#lified if we assume that the #erson is

ris" neutral& which lets us determine a !enchmar" from which the effects of ris"

aversion can !e evaluated. n addition& assume that the su!9ect faces no cash constraint

=wouldn t it !e nice> on the num!er of nic"els =well that s not so unreasona!le> availa!le

to #ay the search costs. n this case& the future always loo"s the same !ecause the

hori<on is infinite and the #ayoff #arameters =search cost& #ri<e distri!ution> are fied

over time. 6ince the future o##ortunities are the same regardless of how many draws

have !een re9ected thus far& any draw re9ected at one #oint in time should never !e ta"en

at a later #oint. hus each #ossi!le draw in the interval from 0 to 0 should either !e

always acce#ted or always re9ected& and the !oundary that se#arates the acce#ta!le and

unacce#ta!le draws is a goal or reservation va#ue. he #erson who made the choices

shown in a!le ., seems to have a reservation value level of a!out /5 cents. t is useful

to see a !roader sam#le of how #eo#le !ehave in this situation !efore further analysis of

the o#timal manner of search.

III. "(perimenta% Data

?igure ., shows the results of a classroom e#eriment with two different search cost

treatments. *ach search se'uence was a series of draws made until the su!9ect sto##ed

and acce#ted a draw& and there were ten such search se'uences in each treatment. =he

data are from the last 5 #eriods for the second and third se'uences& so some learning has

already occurred. Using the final 5 #eriods ta"es out the inertia effects caused !y slow

convergence that may show u# when the search cost is changed !etween treatments. As

a result& the data in the figure are a little cleaner than is normally the case.>

,0




0ig$re 2.'. - C%assroom Se#$entia% Search "(periment

Consider the 5+cent search cost treatment& shown on the left side of the figure&

where the draw num!er is indicated on the hori<ontal ais. he mar"s directly over the

,st la!el on the ais re#resent the first+round draws& one #er search se'uence& which are

fairly uniformly distri!uted from 0 to 0 as would !e e#ected. hose acce#ted in the

first round are la!eled as O signs& and the re9ected draws are la!eled as dots. ?or each of

the dots in the ,st column& there is a mar" in the 2nd column& either a O if the second

draw is acce#ted or a dot if not. he dots in the second column are associated with

mar"s =#lus or dot> in the third column. he ma9ority of search se'uences ended !y the

fourth draw& so all data that ended in the fourth #eriod and afterwards are shown in the

/+,2th column =no!ody made more than ,2 draws>. As !efore& this last column contains

#lus mar"s for acce#ted draws and dots for re9ected draws. Hence a draw that is re9ectedin #eriod / and acce#ted in #eriod 5 will show u# as two mar"s in the /,2 th column. n

addition& the /+,2th column contains several asteris" mar"s& which re#resent acce#ted

draws that were initially re9ected. 6ince the draw was !oth re9ected and acce#ted& thin"

of the asteris" as a com!ination of a #lus sign and a minus sign. 6uch recalls are

,0/



occasionally o!served& even though they are irrational under the assum#tions made

a!ove.

he divide !etween acce#ted and re9ected draws for the 5+cent cost treatment is in the

range from 55 to %0 cents for most #eo#le in this grou#. n contrast& the dramatic

increase in search cost to 20 cents results a much larger acce#tance region& as shown on

the right side of the figure. $ith high search costs& there are fewer searches& and the

divide is somewhere !etween 25 and 0 cents. he infre'uency of recalls =asteris"s>

and the relatively clean divide !etween acce#tance and re9ection regions is roughly

consistent with the notion of a reservation value or goal level. he net ste# is to

consider whether the o!served divides can !e e#lained with sim#le e#ected value

calculations of !enefits and costs.

I. 5ptima% Search?irst& consider the low+search+cost treatment. I!viously any amount a!ove 5

cents should !e acce#ted& since the #otential gain from searching again is at most 5 cents

=if a very luc"y draw of 0 is o!tained> and the cost of the new draw is 5 cents. hus

there is only a very low chance =,,> that the new draw will cover the cost. At the

o##osite etreme& a low draw& say 0& would !e re9ected since the cost of another draw is

only 5 cents and the e#ected value of the net draw is /5 cents =the mid#oint of the

distri!ution from 0 to 0>. ;otice the use of an e#ected value here& which is 9ustified

!y the assum#tion of ris" neutrality. o summari<e& the !enefits of further search eceed

the cost when the !est current draw is very low& and the cost eceeds the e#ected

!enefit when the !est current draw is near the to# of the range. he o#timal reservation

#ri<e level is found !y locating the #oint at which the e#ected !enefits of another search are e'ual to the search cost.

6u##ose that the !est current draw is %0. here is essentially a 2 chance that the net

draw is at %0 or !elow& in which case the net gain is 0. his situation is re#resented in

?igure .2& where the flat hori<ontal line re#resents the fact that each of the draws will

!e o!served with the same #ro!a!ility. he area !elow this line re#resents #ro!a!ility.

?or eam#le& two+thirds of the area !elow the dashed line is to the left of %0& and one

third is to the right of %0. n other words& starting from %0& there is a , chance of

o!taining !etter draw.

,05



0

0005001

0015

0 10 20 30 40 50 60 70 80 90

Probability


(raw in 8ennies

0ig$re 2.. - >niform Distri&$tion on the Interva% +, 4+H

@ith 5ne hird of the Pro&a&i%it! to the Right of 1+A

;et consider the e#ected gains from search when the !est current draw is %0

=with recall and a 5+cent search cost>. A draw !elow %0 will not #roduce any gain& so the

e#ected value of the gain will have a term that is the #roduct: =2>0 J 0. he region of

gain is to the right of the vertical line at %0 in ?igure .2& and the area !elow the dashed

line is one third of the total area. hus& there is essentially a , chance of an

im#rovement& which on average will !e half of the distance from %0 to 0& i.e. half of

0. he e#ected value of an im#rovement& therefore& is ,5 cents. An im#rovement

occurs with a #ro!a!ility a!out ,& so the e#ected im#rovement is a##roimately =,>

=,5> J 5 cents. Any lower current draw will #roduce an e#ected im#rovement from

further search that is a!ove 5 cents& and any higher current draw will #roduce a lower

e#ected im#rovement. t follows that %0 is the reservation draw level for which the

cost of another draw e'uals the e#ected im#rovement. he same reasoning a##lied tothe 20+cent search cost treatment im#lies that the o#timal reservation draw is 0 cents

=see 'uestion 2>.

he #redicted reservation values of %0 and 0 are gra#hed as hori<ontal lines in

?igure .,. he #ower of these #redictions is 'uite clear. here is a !rea" !etween the

#lus signs indicating acce#tance and the dots indicating re9ection& with little overla#.

his !rea" is slightly !elow the 0 and %0 levels #redicted for a ris"+neutral #erson.

Actually& adding ris" aversion lowers the #redicted cutoff. he intuition !ehind ris"

aversion effects is clear. $hile a ris"+neutral #erson is a##roimately indifferent

!etween searching again at a cost of 5 cents and sto##ing with a draw of %0& a ris"+

averse #erson would #refer to sto# with %0& since it is a sure thing. n contrast& drawing

again entails the significant ris" =two thirds> of having to #ay the cost without gettingany im#rovement.

Ine im#lication of o#timal !ehavior is that the reservation value is essentially

the value of having the o##ortunity to go through this search #rocess. his is !ecause

,0%



one re9ects all draws !elow the reservation value& there!y indicating that the value of

#laying the search game is greater than those draws.

Conversely& draws a!ove the reservation draw level are acce#ted& indicating a #reference

for those sure amounts of money over a continuation of the search #rocess.

o summari<e& the value of continuing to search is greater than any num!er

!elow the reservation value& and it is less than any num!er a!ove the reservation value&

so the value of continuing to search must !e e'ual to the reservation value. his

im#lication was tested !y 6chotter and 7raunstein =,,>& who elicited a #rice at which

individuals would !e willing to sell the o#tion to search. n one of their treatments& the

draws were from a uniform distri!ution on the interval from 0 to 200 with a search cost

of 5 cents. he o#timal reservation value for a ris" neutral #erson is ,55 =see 'uestion

>. his should !e the value of !eing a!le to #lay the search game& assuming that there

is no en9oyment derived from doing an additional search se'uence after several have

already !een com#leted. he mean re#orted selling #rice was ,5-& and the averageacce#ted draw was ,-0& which is a!out halfway !etween the theoretical reservation

value of ,55 and the u##er !ound of 200. hese and other results lead to the conclusion

that the o!served !ehavior is roughly consistent with the #redictions of o#timal search

theory in this e#eriment.

0$rther Reading and "(tensions

he search game discussed here is sim#le& !ut it illustrates the main intuition !ehind the

determination of the reservation draw level in a se'uential search #ro!lem. here are

many interesting and realistic variations of this #ro!lem. he #lanning hori<on may not

!e infinite& e.g. when you have a deadline for finding a new a#artment !efore you haveto move out of the current one. n this case& the reservation value will tend to fall as the

deadline nears and des#eration ta"es over. n the Co and Iaaca =,> e#eriments

with a finite hori<on& su!9ects sto##ed at the #redicted #oint a!out three+fourths of the

time& and the deviations were in the direction of sto##ing too soon& which would !e

consistent with ris" aversion.

n all situations considered thusfar& the #erson searching has !een assumed to "now the

#ro!a!ility distri!ution from which draws were !eing made. his is a strong

assum#tion& and therefore& it is interesting to consider cases in which #eo#le learn a!out

the distri!ution of draws as they search. 6u##ose that you thin" the draws will !e in the

range from 0 to ,00& and you see a draw of ,0. his would have !een well a!ove your

reservation value if the distri!ution from which draws were made had an u##er limit of

,00& !ut now you reali<e that you do not "now what the u##er limit really is. n this

case& a high draw may !e re9ected in order to find out whether even higher draws are

#ossi!le =Co and Iaaca& 2000>.

,0-




Although the aggregate data discussed here are roughly consistent with theoretical

#redictions& this leaves o#en the issue of how #eo#le learn to !ehave in this manner.

Certainly they do not generally do any mathematical analysis. Hey =,,> has analy<ed

individual !ehavior in search of heuristics and ada#tive #atterns that may e#lain

!ehavior at the individual level. Also& see Hey =,-& when he is still searching >. He

s#ecified a num!er of rules of thum!& such as:

/neBounce 'u#e$

7uy at least two draws& and sto# if a draw received is less than the #revious draw.

Modified /neBounce 'u#e$

7uy at least two draws& and sto# if a draw received is less than the sum of the

#revious draw and the search cost.

$ith a search cost of 5 and initial draws of 0 and 50& for eam#le& the modified one+

!ounce rule would #redict that the #erson would sto# if the third draw turned out to !e

less than 55. he e#eriment involved treatments with and without recall& and with and

without information a!out the distri!ution =a truncated normal distri!ution> from which

draws were drawn. 7ehavior of some su!9ects in some rounds corres#onded to one or

more of these rules& !ut !y far the most common #attern of !ehavior was to use a

reservation draw level. n the fifth and final round& a!out three+fourths of the

#artici#ants ehi!ited !ehavior that conformed to the o#timal reservation+value rule

alone.

<$estions

,. he com#uteri<ed e#eriments discussed in this cha#ter were set u# so that

draws were e'ually li"ely to !e any amount on the interval from 0 to 0. $hen

throwing ,0+sided dice with num!ers from 0 to & the throws will !e 0& ,& 2& ...

. n order to truncate this distri!ution at 0& is it is convenient to ignore the

first throw if it is a & so that all of the 0 integers from 0 through are e'ually

li"ely& i.e. each has a #ro!a!ility of ,0. =his is the a##roach ta"en for the

hand+run e#eriment instructions in the a##endi to this cha#ter.> f the current

draw is %0& then the chances that the net draw will !e as good or !etter are: a

,0 chance of a draw of %0& for a gain of 0& a ,0 chance of a draw of %, for again of , centB a ,0 chance of a draw of %2 for a gain of 2B etc. Use a

s#readsheet to find the e#ected value of the gain over %0 =not the e#ected

value of the draw>& and com#are this with the search cost of 5 cents. $ould a

,0



draw of %0 !e re9ectedN $ould a draw of 5 !e re9ectedN Hint: ma"e a column

of integers from %0 u# to & and then ma"e a second column of gains over

the current draw =of either 5 or %0>. ?inally& weight each gain !y the #ro!a!ility

=,0> of that draw& and add u# all of the weighted gains to get the e#ected gain&

which can then !e com#ared with the search cost of 5 cents.

2. $ith a search cost of 20 cents #er draw from a distri!ution that is uniform from 0

to 0& show that the #erson is a##roimately indifferent !etween searching again

and sto##ing when the current !est draw is 0.

. $ith a search cost of 5 cents for draws from a uniform distri!ution on the

interval from 0 to 200& show that the reservation value is a!out ,55.

/. *valuate the data in a!le ., in terms of the Ine+7ounce 3ule and the 4odified

Ine+7ounce 3ule. (o these rules wor" well& and if not& what seems to !e going

wrongN

5. =I#en+ended> 3ules of thum! have more a##eal in limited+information situationswhere rational !ehavior is less li"ely to !e o!served. Can you thin" of other

rules of thum! that might !e good when the distri!ution of draws is not "nown

and the #erson searching has not had time to learn much a!out itN

,0




,,0



8art . ame heory *#eriments: reasures

and ntuitive Contradictions

As one might e#ect& human !ehavior will not always conform tightly to sim#le

mathematical models& es#ecially in interactive situations. hese models& however& can

!e etremely useful if deviations from the ;ash e'uili!rium are systematic and

#redicta!le. his #art #resents several games in which !ehavior is influenced !y

intuitive economic forces in ways that are not ca#tured !y !asic game theory. 4any of

the a##lications are ta"en from oeree and Holt =200,>& en 1ittle reasures of ame

heory and en ntuitive Contradictions. he treasures are treatments where data

conform to theory& and the contradictions are #roduced !y #ayoff changes with strong

!ehavioral effects& even though these changes do not affect the ;ash #redictions. 6incegame theory is so widely used in the study of strategic situations li"e auctions& mergers&

and legal dis#utes& it is fortunate that there is #rogress in understanding these anomalies.

$e will consider a modification of the ;ash e'uili!rium that in9ects some randomness

or noise into #layers !est res#onse functions. Mou might thin" of the randomness as

!eing due to un+modeled effects of emotions& attention la#ses& #artial calculations& and

other factors that may vary from #erson to #erson and from time to time for the same

#erson.

4any games involve decisions !y different #layers made in se'uence. n these

situations& #layers who ma"e initial decisions must #redict what those with su!se'uent

decisions will do. hese se'uential games& which are re#resented !y an etensive form

or decision tree structure& will !e considered in Cha#ter .

he generali<ed matching #ennies game in Cha#ter ,0 shows that !ehavioral

deviations can !e strongly influenced !y #ayoff asymmetries. *isting data indicate that

the ;ash e'uili!rium only #rovides good& un!iased #redictions !y coincidence& i.e. when

the #ayoffs for each decision ehi!it a "ind of !alance or symmetry. his effect is even

more dramatic in the raveler s (ilemma game discussed in Cha#ter ,,& where the data

#atterns and the ;ash e'uili!rium may !e on opposite ends of the range of #ossi!le

decisions. ?inally& the Coordination ame in cha#ter ,2 has a whole series of ;ash

e'uili!ria& and the issue is which one will have more drawing #ower. his game is

im#ortant in develo#ing an understanding of how #eo#le may !ecome stuc" in an

e'uili!rium that is !ad for all concerned.here has !een a considera!le amount of research devoted to e#laining why !ehavior

in e#eriments might !e res#onsive to #ayoff conditions that do not affect the ;ash

e'uili!rium =or e'uili!ria>. he resulting models often rela the strong game+theoretic

assum#tions of #erfect rationality and #erfect #redictions of others decisions&

,,,




assum#tions which im#ly that one always chooses the decision with the highest

e#ected #ayoff& even if #ayoff differences are small. 6uch rationality leaves no room

for com#utation errors& emotional or im#ulsive actions& and other random factors that

are not em!odied in the formal model of the game. n a sim#le decision& such noise

elements tend to s#read decisions around the o#timal choice& with the most li"ely choice

!eing near the o#timal level& !ut with some #ro!a!ility of error in either direction. his

would create a "ind of !ell curve #attern. $ith strategic interaction& however& even

small amounts of noise in one #erson s decisions may cause large !iases in others

decisions& es#ecially if there is a lot of ris". magine a #erson trying to find the highest

#oint on a hill& where the #ea" is on the edge of a stee# cliff. he effect of this

asymmetry in ris" is that any slight chance of wind gusts may cause the #erson to move

off of the #ea". n a game& the #ayoff #ea"s de#end on others decisions& i.e. the #ea"s

move. n such cases& the effects of noise or small un+modeled factors may have a

snow!all effect& moving all decisions well away from the ;ash #redictions. Ine su!#lotin the following cha#ters is an analysis of the effects of introducing some randomness or

noise into the gamesB this a##roach is #resented in a series of o#tional sections at the end

of each cha#ter.

,,2



Cha#ter . 4ulti+6tage ames in *tensive

?orm

he games considered in Cha#ters and 5 each involved simultaneous decisions.

4any interesting games& however& are se'uential in nature& so that the first mover must

try to antici#ate how the su!se'uent decision ma"er=s> will react. ?or eam#le& the first

#erson may ma"e a ta"e+it+or+leave+it #ro#osal for how to s#lit a sum of money& and the

second #erson must either acce#t the #ro#osed s#lit or re9ect. his is an eam#le of an

ultimatum !argaining game& which will !e considered in Cha#ter 2. Another eam#le

is a la!or+mar"et interaction& where the em#loyer first chooses a wage or other set of

contract terms& and the wor"er& seeing the wage& selects an effort level. hese two+stage

#rinci#al+agent games are considered in Cha#ter 2/. here are several common #rinci#les that are used in the analysis of such games& and these will !e covered in the

current cha#ter. 6omewhat #aradoically& the first #erson s decision is sometimes more

difficult& since the o#timal decision may de#end on a forecast of the second #erson s

res#onse& whereas the #erson who ma"es the final decision does not need to forecast the

other s action if it has already !een o!served. n this case& we !egin !y considering the

final decision+ma"er s choice& and then we ty#ically wor" !ac"ward to consider the first

#erson s decision& in a #rocess that is "nown as !ac"ward induction. If course& the

etent to which this method of !ac"ward induction yields good #redictions is a

!ehavioral 'uestion& which will !e evaluated in the contet of la!oratory e#eriments.

6everal of the two+stage games discussed !elow are the default settings for the

Veconla! wo+6tage ame #rogram. here is a se#arate Centi#ede ame #rogram&

with setu# o#tions that allow any num!er of stages =even ,00>. A handrun version of the

Centi#ede game can !e run with a roll of 'uarters and a tray ='uestion % at the end of the

cha#ter>.

I. "(tensive 0orms and Strategies

his section will #resent a sim#le two+stage !argaining game& in which the first move is

a #ro#osal of how do divide /& which must either !e for the #ro#oser and , for the

other& or 2 for each. he other #layer =res#onder> sees this #ro#osal and either acce#ts&

which im#lements the s#lit& or re9ects& which results in 0 for each. As defined inCha#ter & a strateg for such a game is a com#lete #lan of action that covers each

#ossi!le contingency. n other words& a strategy for a #layer tells the #layer what

decision to ma"e at each stage& and these instructions are so detailed that a strategy

could !e carried out !y an em#loyee or agent& who would never need to as" 'uestions

,,




a!out what to do. n the two+stage !argaining game& a strategy for the first #erson to

ma"e a decision is which #ro#osal to offer& i.e. =& ,> or =2& 2>& where the #ro#oser s

#ayoff is listed on the left in each #ayoff #air. A res#onder s strategy must s#ecify a

reaction to each of these #ro#osals. hus the res#onder has a num!er of o#tions: a>

acce#t !oth& !> re9ect !oth& c> acce#t =& ,> and re9ect =2& 2>& or d> re9ect =& ,>

and acce#t =2& 2>. hese strategies will !e referred to as: AA& 33& A3& and 3A

res#ectively& where the first letter indicates the res#onse& Acce#t or 3e9ect& to the

une'ual #ro#osal. 6imilarly& the #ro#oser s strategies will !e referred to as *'ual =e'ual

s#lit> or Une'ual =une'ual& favoring the #ro#oser>. A ;ash e'uili!rium is a #air of

strategies& one for each #layer& with the #ro#erty that neither can increase their own

#ayoff !y deviating to a different strategy under the assum#tion that the other #layer

stays with the e'uili!rium strategy.

Ine ;ash e'uili!rium for this game is =*'ual& 3A>& where the #ro#oser offers an e'ual

s#lit& and the res#onder re9ects the une'ual #ro#osal and acce#ts the e'ual #ro#osal.(eviations are not #rofita!le& since the res#onder is already receiving the #ro#osal that

offers the higher of the two #ossi!le #ayoffs& and the #ro#oser would not want to switch

to the une'ual #ro#osal given that the res#onder s strategy re'uires that it !e re9ected.

his ;ash e'uili!rium& however& involves a threat !y the res#onder to re9ect an une'ual

#ro#osal = for the #ro#oser& , for the res#onder>& there!y causing !oth to earn 0.

6uch a re9ection would reduce the res#onder s #ayoff form , to 0& so it violates a

notion of se'uential rationality that re'uires #lay to !e rational in all #arts of the game

= su!games >. he reader might wonder how this "ind of irrationality can !e #art of a

;ash e'uili!rium& and the answer is that the sim#le notion of a ;ash e'uili!rium only

re'uires rationality in terms of considering unilateral deviations from the e'uili!rium !y

one #layer& under the assum#tion that the other #layer s strategy is unchanged. n thee'uili!rium !eing considered& =*'ual& 3A>& the une'ual #ro#osal is not made& so the

res#onder never has to carry out the threat to re9ect this #ro#osal. 6elten =,%5>

#ro#osed that the rationality re'uirement !e e#anded to cover all #arts of the game&

which he defined as su!games. He called the resulting e'uili!rium a su+game perfect

;ash e'uili!rium. I!viously& not all ;ash e'uili!ria are su!game #erfectB as the

e'uili!rium =*'ual& 3A> demonstrates.

n order to find the e'uili!rium for this game& consider starting in the final stage& where

the res#onder is considering a s#ecific #ro#osal. *ach of the #ossi!le #ro#osals involve

strictly #ositive amounts of money for the res#onder& so a rational res#onder who only

cares a!out own #ayoff would acce#t either #ro#osal. his analysis re'uires that the

res#onder acce#t either #ro#osal& so the only res#onder strategy that satisfies rationality

in all su!games is AA. ;et consider the first stage. f the #ro#oser #redicts that the

res#onder is rational and will acce#t either offer& then the #ro#oser will demand the

larger share& so the e'uili!rium will !e =Une'ual& AA>. his is a ;ash e'uili!rium& and

,,/



:Aual BneAualProposer

.espon!er

@ . @ .

.espon!er

$2C $2 $0C $0 $0C $0$3C $1

from the way it was constructed& we "now that it satisfies se'uential rationality& i.e. is

su!game #erfect.

he reasoning #rocess that was used in the #revious #aragra#h is an eam#le of

!ac"ward induction& which sim#ly means analy<ing a se'uence of decisions !y starting

at the end and wor"ing !ac"wards toward the !eginning. he conce#ts& !ac"ward

induction& se'uential rationality& and su!game #erfection& can all !e defined more

#recisely& !ut the goal here is for the reader to o!tain an intuitive understanding of the

#rinci#les involved& !ased on s#ecific eam#les. his understanding is useful in

constructing #redictions for outcomes of sim#le multi+stage games& which can serve as

!enchmar"s for evaluating o!served !ehavior in e#eriments. As we shall see& these

!enchmar"s do not always yield very good #redictions& for a variety of reasons to !e

discussed !elow.

7efore #roceeding with eam#les& it is instructive to show how the twostage !argaining

game would !e re#resented as a decision tree& which is "nown as the etensive form of agame. n ?igure .,& the game !egins at the to#& at the node that is la!eled 8ro#oser.

his #layer either chooses an *'ual #ro#osal =2&2>& or an Une'ual #ro#osal =&,>.

he 3es#onder has a decision node following each of these #ro#osals. In the left& the

3es#onder may either choose A =acce#t> or 3 =re9ect> in res#onse to the *'ual #ro#osal.

he same o#tions are also availa!le on the right. ;otice that each decision node is

la!eled with the name of the #layer who ma"es a decision at that node& and each !ranch

emanating from a node is la!eled with the name of one of the feasi!le decisions. he

!ranches end at the !ottom =terminal nodes>& which show the #ayoffs& with the #ayoff of

the 8ro#oser listed on the left.

0ig$re 4.'. -n "(tensive 0orm Representation for the Bargaining Game

,,5




II. 6o?Stage r$st Games

he first two e#eriments to !e considered are !ased on the two games shown in ?igure

.2. Consider the to# #art& where the first #layer must !egin !y ma"ing a safe choice =6>

or a ris"y choice =3>. (ecision 6 is safe in the sense that the #ayoffs are deterministic:

0 for the first #layer and 50 for the second =the first #layer s #ayoff will !e listed on the

left in each case>. he #ayoffs that may result from choosing 3 de#end on the second

#layer s res#onse& 8 or ;. ?or the game shown in the to# #art of ?igure .2& this is an

easy choice& since the second #layer would earn ,0 from choosing 8 and -0 from

choosing ;. he first #layer should ma"e the ris"y choice 3 as long as the first #layer

trusts the second #layer to !e rational. his game was #layed only once& without

re#etition& !y #airs of su!9ects& with #ayoffs were in #ennies =see oeree and Holt&

200,& for details>. As indicated in the to# #art of the figure& / #ercent of the first

movers were confident enough to choose 3& and all of these received the antici#ated ;

res#onse. ;ote that the second movers would have reduced their earnings !y %0 cents if any of them had selected 8 in res#onse to 3.

0ig$re 4.. - 6o?Stage Game here Mistakes Matter

So$rce: Goeree and 7o%t @++'A

,,%



he game shown in the !ottom #art of ?igure .2 is almost identical& ece#t that the

second mover only loses 2 cents !y ma"ing a 8 res#onse& which generates a #ayoff of %

for the second mover instead of -0. n this case& more than half of the first movers were

sufficiently concerned a!out the #ossi!ility of a 8 res#onse that they chose decision the

safe decision& 6. his fear was well founded& since a fourth of the first movers who chose

3 encountered the 8 res#onse in the second stage. n fact& the first #layers who chose 6

in the !ottom game in ?igure .2 earned more on average than those who chose 3

='uestion ,>. he standard game+theoretic analysis of the two games in ?igure .2

would yield the same #rediction for each game& since this analysis is !ased on the

assum#tion that !oth #layers are rational and are concerned with maimi<ing their own

#ayoffs. his assum#tion im#lies that the second #layer will choose ;& regardless of

whether it increases earnings !y %0 cents& as in the to# game& or !y only 2 cents& as in the

!ottom game. n each of these games& we !egin !y analy<ing the second #layer s !est

choice =choose ;>& and "nowing this& we can calculate the !est choice for the first #layer =choose 3>. he #air of choices& ; and 3& constitutes a ;ash e'uili!rium& since neither

#layer can increase earnings !y deviating.

he ;ash outcome =3 and ;> that was identified for the games in ?igure .2 involves

little tension in the sense that it maimi<es the #ayoffs for each #layer. here is another

;ash outcome& =6 and 8>& which can !e verified !y showing that neither #erson has an

incentive to deviate unilaterally ='uestion 2>. n #articular& the second #layer earns 50 in

this e'uili!rium& and a unilateral deviation form 8 to ; would not change the outcome&

since the outcome is fully determined !y the first #layer s 6 decision. his second

e'uili!rium is ty#ically ruled out !ecause it im#lies a ty#e of !ehavior for the second

#layer that is not rational in a se'uential sense. (es#ite these arguments& the 6 outcome

#redicted !y second e'uili!rium is the most commonly o!served outcome for the !ottomgame in ?igure .2.

here is more tension for the games shown in ?igure .. As !efore& there are two ;ash

e'uili!ria: =6 and 8> and =3 and ;>. ?irst consider the =3 and ;> outcome& which yields

0 for the first #layer and 50 for the second. f the second #layer were to deviate to 8&

then the outcome would !e =3 and 8> with lower #ayoffs for the second #layer& since we

are considering a unilateral deviation& i.e. a deviation !y one #layer given that the other

continues to use the e'uili!rium decision. 6imilarly& given that the second #layer is

using ;& the first #layer cannot deviate and increase the #ayoff a!ove 0& which is the

maimum for the first #layer in any case. his e'uili!rium =3 and ;> is the one

#referred !y the first #layer. here is another e'uili!rium =6 and 8>B see 'uestion .

Here& you can thin" of 8 as indicating #unishment& since a 8 decision in the second stage

reduces the first #layer s #ayoff from 0 to %0. he reason that such a #unishment might

!e enacted is that the second #layer actually #refers the outcome on the left side of the

figure for each of the two games in ?igure ..

,,-




he difference !etween the two games in the figure is that the cost of #unishment is /0

cents for the to# game& and only 2 cents for the !ottom game.

he standard game+theoretic analysis of the games in ?igure . would !e the same& i.e.

that the =6 and 8> e'uili!rium is im#lausi!le since it im#lies that the second mover !e

ready to use a #unishment that is costly& and hence not credi!le. 6o the #redicted

outcome would !e the e'uili!rium =3 and ;> in each case. his #rediction is accurate

for the game shown in the to# #art of the figure& where #ercent of the outcomes are as

#redicted. he outcomes deviate shar#ly from this #rediction for the game shown in the

!ottom #art of the figure& since only a!out a third of the o!served outcomes are at the =3

and ;> node.

0ig$re 4./. - 6o?Stage Game ith a hreat hat Is 8ot Credi&%eE

So$rce: Goeree and 7o%t @++'A

o summari<e& the notion of se'uential rationality rules out the #ossi!ility that the

second mover will ma"e a mista"e in either of the games in ?igure .2& or that the

second mover will carry out a costly =and hence non+credi!le> threat for either of thegames in ?igure .. hus the #redicted outcome would !e the same =3 and ;> in all

four games. his #rediction wor"s well for only one of the games in each figure& !ut not

for the other one. he #ro!lem with the assum#tion of #erfect se'uential rationality is

,,



that it does not allow room for random deviations or mista"es& which are more li"ely if

the cost of the mista"e is small =2 cents for each of the !ottom games in the figures>. A

more systematic analysis of the costs of mista"es and the effects on !ehavior will !e

underta"en in the three cha#ters that follow.

III. he Centipede Game

n the two+stage games considered thusfar& the first mover ty#ically has to consider how

the second mover will react to the initial decision. his thought #rocess may involve the

first mover thin"ing intros#ectively: $hat would do on the second stage& having 9ust

seen this #articular initial decisionN his #rocess of seeing a future situation through the

eyes of someone else may not !e easy or natural& and the difficulty will increase greatly

if there are more than two stages& so an initial decision ma"er might have to thin" a!out

reactions several stages later. n such cases& it is still often straightforward& !ut tedious&to !egin with the last decision+ma"er& consider what is rational for that #erson& and wor"

!ac"wards to the second+to+last decision ma"er& and then to the third+to+last decision

ma"er& etc. in a #rocess of !ac"ward induction. he noise o!served in some of the two+

stage games already considered in this cha#ter ma"es it #lausi!le that #layers in multi+

stage games may not !e very #redicta!le& at least in terms of their a!ility to reason via

!ac"ward induction. 3osenthal =,2> #ro#osed a se'uential game with ,00 stages&

which serves as an etreme stress test of the !ac"ward induction reasoning #rocess.

*ach of the ,00 stages has a terminal node in the etensive form that hangs down& so the

etensive form loo"s li"e an an insect with ,00 legs& which is why this game is "nown

as the centi#ede game.

?igure ./ shows a truncated version of this game& with only / legs. At each node& the #layer =3ed or 7lue> can either move down& which sto#s the game& or continue to the

3ight& which #asses the move to the other #layer& ece#t in the final stage. he #lay

!egins on the left side& where the 3ed #layer must decide whether to sto# the game =the

downward arrow> or continue =the right arrow>. A sto#down move in this first stage

results in #ayoffs of /0 cents for 3ed and ,0 cents for the other #layer& 7lue. ;ote that

the #ayoffs for this first terminal node are listed as =/0& ,0>& with #ayoffs for the 3ed

#layer shown on the left. f 3ed decides to continue to the right& then the net move is

made !y 7lue& who can either continue or sto#& yielding #ayoffs of =20& 0>& where the

left #ayoff is for 3ed& as !efore. 7lue might reason that sto##ing gives 0 for sure&

whereas a decision to continue will either yield #ayoffs of /0 =if 3ed sto#s in the net

stage>& 20 =if 7lue sto#s !y moving down in the final stage>& or ,%0 =if 7lue moves

right in the final stage>. I!viously& it is !etter for 7lue to move down in the final stage&

getting 20 instead of ,%0. Having figured out the !est decision in the final stage& we

can !egin the #rocess of !ac"ward induction. f 7lue is e#ected to go down in the final

,,



=640C 160>2 D

RedBlue Blue


stage& then 3ed will e#ect a #ayoff of 0 if #lay reaches the final stage& instead of %/0

if #lay sto#s in the #revious stage. hus is would !e !etter for 3ed to ta"e the ,%0 and

sto# the game in the third stage& rather than #ass to 7lue and end u# with only 0 in the

final stage.

o summari<e thusfar& we have concluded that 7lue will sto# in the final stage& and that

3ed& antici#ating this& will sto# in the third stage. 6imilar reasoning can !e used to

show that 7lue will want to sto# in the second stage& if #lay reaches this #oint& and

hence that 3ed will want to sto# in the first stage& yielding #ayoffs of /0 for 3ed and

only ,0 for 7lue. hus the #rocess of !ac"ward induction yields a very s#ecific

#rediction& i.e. that the game will sto# in the initial stage& with relatively low #ayoffs for

!oth.

Red

=3+& '+> =+& 2+> ='1+& 3+> =2+& /+>

K /, K K ,0 K

0ig$re 4.3. - Centipede Game

So$rce: McJe%ve! and Pa%fre! @'44A

he e#eriment& conducted !y 4cDelvey and 8alfrey =,2>& involved grou#s of 20

#artici#ants& with ,0 #eo#le in each role =3ed or 7lue>. *ach #artici#ant was matchedwith all ,0 #eo#le in the other role& in a series of ,0 games& each game !eing the

centi#ede game shown in ?igure ./. he aggregate #ercentages for each outcome are

shown in the figure& !elow the #ayoffs at each node. Inly K of the 3ed #layers

sto##ed in the first stage& so the e#eriment #rovides a shar# re9ection of #redictions

!ased on !ac"ward induction. 3ates of continuation were somewhat lower toward the

end of the session& !ut the incidence of sto##ing in the initial stage remained relatively

low throughout. 4ost of the games ended in the second and third stages& !ut 2K of the

games went on until the final stage.

he failure of !ac"ward induction #redictions in this game is #ro!a!ly due to the fact

that even a small amount of un#redicta!ility of others decisions in later stages may ma"eit o#timal to continue in early stages& since the high #ayoffs are concentrated at the

terminal nodes on the right side of the figure. 4cDelvey suggested that a small

#ro#ortion of #eo#le were somewhat concerned with others #ayoffs = altruists >& and that

even if these ty#es are not #redominant& it would !e o#timal for selfish individuals to

,20



continue in early stages. n fact& any ty#e of noise or un#redicta!ility& regardless of its

source& will have similar effects.

;otice that a!out one in si 7lue #layers continue in the final stage& which cuts their

#ayoff in half !ut raises the 3ed #layer s #ayoff !y a factor of & from 0 to %/0.

$hether this is due to altruism& to randomness from another cause& or to miscalculation

is an o#en 'uestion. ?or an analysis of the effects of randomness in centi#ede games&

see Tauner =,>.

I. "(tensions

Ine #ossi!le e#lanation of the data #attern in the centi#ede game e#eriment is that

the failure of !ac"ward induction may !e due to low incentives& and that !ehavior would

!e more rational in high sta"es e#eriments. 3ecent e#eriments with #otential sta"es

of thousands of dollars indicate that this is not the case. Ither recent wor" hasconsidered how su!9ects in e#eriments learn and ad9ust in centi#ede games& see ;agel

and ang =,> and 8onti =2000>. n addition& 7ornstein& Dugler& and Tiegelmeyer

=2002> com#are !ehavior of individuals and grou#s in centi#ede games. ?ey& 4cDelvey

and 8alfrey =,%> re#ort results of a variation of the centi#ede game& where the #ayoffs

sum to a constant.

<$estions

,. 6how that the first #layers who chose 3 earned more on average than those who

chose 6 in the game shown in the !ottom #art of ?igure .2. =Hint: you have to

use the res#onse #ercentages shown !elow each outcome.>2. 6how that =6 for the first #layer& 8 for the second #layer> is a ;ash e'uili!rium

for the games in ?igure .2. o do this& you have to chec" to see whether a

unilateral deviation !y either #layer will not increase that #layers #ayoff& under

the assum#tion that the other #layer stays with the e'uili!rium decision.

. 6how that =6 for the first #layer& 8 for the second #layer> is a ;ash e'uili!rium

for each of the games in ?igure .B i.e. chec" the #rofita!ility of unilateral

deviations for each #layer.

/. Consider a game in which there is / to !e divided& and the first mover is only

#ermitted to ma"e one of three #ro#osals: a> for the first mover and , for the

second mover& !> 2 for each& or c> , for the first mover and for the second

mover. he second mover is shown the #ro#osal and can either acce#t& in whichcase it is im#lemented& or re9ect and cause each to earn 0. 6how this game in

etensive form. =Hint: the decision node for the first mover has three arrows&

one for each decision& and each of these arrows leads to a node with two arrows.>

,2,




7e sure and show the #ayoffs for each #erson& with the first mover listed on the

left& for each of the % terminal nodes.

5. 6how that #ro#osal =c> in 'uestion / is a #art of a ;ash e'uili!rium for the game

in 'uestion /. o finish s#ecifying the decisions for this e'uili!rium& you have to

say what the second mover s res#onse is to each of the three #ro#osals. (oes

!ehavior in this e'uili!rium satisfy se'uential rationalityN

%. Consider a game in which the instructor divides the class into two grou#s and

ta"es a ,0 roll of 'uarters. he instructor then #uts a 'uarter into a collection

tray and allows those on one side of the room to ta"e the 'uarter =and somehow

divide or allocate it among themselves> or to #ass. A #ass results in dou!ling the

money =to 2 'uarters> and giving those on the other side of the room the o#tion to

ta"e or #ass. his #rocess of dou!ling the num!er of 'uarters in the tray

continues until one side ta"es the 'uarters& or until one side s #ass forces the

instructor to #ut all remaining coins into the #late& which are then given to theside of the room whose decision would !e the net one. (raw a figure that

shows the etensive form for this game. 1a!el the #layers A and 7& and show

#ayoffs for each terminal node as an ordered #air: = for A& for 7>. $hat is the

#redicted outcome of the game& on the !asis of !ac"ward induction& #erfect

rationality& and selfish !ehaviorN

Cha#ter ,0. enerali<ed 4atching 8ennies

here are many situations in which a #erson does not want to !e #redicta!le. he

e'uili!rium in such cases involves randomi<ation& !ut to !e willing to randomi<e& each #layer must !e indifferent !etween the decisions over which they are randomi<ing. n

#articular& each #layer s decision #ro!a!ilities have to "ee# the other #layer indifferent.

?or this reason& changes in a #layer s own #ayoffs are not #redicted to affect the #layer s

own decisions. his counterintuitive feature is contradicted !y data from matching

games with #ayoff im!alances& where own+#ayoff effects are systematic. hese games

can !e run with minor modifications to the instructions used for Cha#ter 5& or with the

Veconla! 4atri ame #rogram.

I. he Case of Ba%anced Pa!offs

n a classic game of matching #ennies& each #erson #laces a coin on a ta!le& covering itso that the other #erson cannot see which side is u#. 7y #rior agreement& one #erson

ta"es !oth #ennies if the #ennies match& and the other ta"es the #ennies if they do not

match. his is analogous to a soccer #enalty "ic"& where the goalie must dive to one

side or another !efore it is clear which way the "ic" will go& and the "ic"er cannot see

,22



which way the goalie will dive at the time of the "ic". n this case& the goalie wants a

match and the "ic"er wants a mismatch. a!le ,0., shows a game where the 3ow

#layer #refers to match:

a!le ,0.,. A 4odified 4atching 8ennies ame

=row s #ayoff& column s #ayoff>

3ow 8layer: Column 8layer:

1eft =heads> 3ight

=tails> o# =heads>

7ottom =tails>

=t can !e shown that this game is really e'uivalent to a matching #ennies game in which

there is always one #layer with a #enny gain and another with a #enny lossB see 'uestion,>. n each of the cells of the #ayoff ta!le& there is one #erson who would gain !y

altering the #lacement of their #enny unilaterally& as indicated !y the arrows. ?or

eam#le& if they were !oth going to choose Heads& then the column #layer would #refer

to switch to ails& as indicated !y the arrow #ointing to the right in the u##erleft !o of

the #ayoff ta!le. he arrows in each !o indicate the direction of a unilateral #ayoff+

increasing move& so there is no e'uili!rium in non+random strategies. =;on+random

strategies are commonly called #ure strategies !ecause they are not #ro!a!ility+weighted

mitures of other strategies.>

he game in a!le ,0., is +a#anced in the sense that each #ossi!le decision for

each #layer has the same set of #ossi!le #ayoffs& i.e. % and -2. n games with this ty#eof !alance& the ty#ical result is for su!9ects to #lay each decision with an a##roimately

e'ual #ro!a!ility =Ichs& ,/B oeree and Holt& 200,>. his tendency to choose each

decision with #ro!a!ility one half is not sur#rising given the sim#le intuition that one

must not !e #redicta!le in this game. ;evertheless& it is useful to review the

re#resentation of the ;ash e'uili!rium as an intersection of #layers !est res#onse

functions& as e#lained in Cha#ter 5. ?irst consider the 3ow #layer& who will choose

o# if Column is e#ected to choose 1eft. 6ince 3ow s #ayoffs are -2 and % in the to#

#art of a!le ,0., and are % and -2 in the !ottom #art& it is a##arent that 3ow would

choose o# as long as 1eft is thought to !e more li"ely than 3ight. his !est+res#onse

!ehavior is re#resented !y the solid line in ?igure ,0., that !egins in the to#left corner&

continuing along the to# until there is an a!ru#t dro# to the !ottom when the #ro!a!ilityof 3ight is 0.5. =8lease ignore the curved line for now.>

,2

-2& % ) ( %& -2

%& -2 & ' -2& %



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Probability of " op

.os est .esponse

+olumns est .esponse

.os Noisy .esponse


0 0., 0.2 0. 0./ 0.5 0.% 0.- 0. 0. ,

8ro!a!ility of 3ight

0ig$re '+.'. Best Response 0$nctions for a S!mmetric Matching Pennies Game:he "ffect of 8ois! Behavior

he column #layer s !est res#onse line is derived analogously and is shown !y

the thic" dashed line that starts in the lowerleft corner& rises to 0.5 and then crosses over

hori<ontally to the right side& !ecause Column would want to #lay 3ight whenever the

#ro!a!ility of o# is greater than 0.5. he intersection of these two lines is in the center

of the gra#h& at the #oint where each #ro!a!ility is 0.5& which is the ;ash e'uili!rium in

mied strategies.

II. 8ois! Best Responses

he #revious analysis is not altered if we allow a little noise in the #layers

res#onses to their !eliefs. ;oisy !ehavior of this ty#e was noticed !y #sychologists who

,2/



would show su!9ects two lights and as" which was !righter& or let them hear two sounds

and as" which was louder. $hen the signals =lights or sounds> were not close in

intensity& almost every!ody would indicate the correct answer& with any errors !eing

caused !y mista"es in recording decisions. As the two signals !ecame close in intensity&

then some #eo#le would guess incorrectly& e.g. !ecause of !ad hearing& random

variations in am!ient noise& or distraction and !oredom. As the intensity of the signals

a##roached e'uality& the #ro#ortions of guesses for each signal would a##roach ,2 in

the a!sence of measurement !ias. n other words& there was not a shar# !rea" where the

stronger signal was selected with certainty& !ut rather a smooth tendency to guess the

stronger signal more as its intensity increased. he #ro!a!ilistic nature of this !ehavior

is ca#tured in the #hrase: #ro!a!ilistic choice or noisy !est res#onse.

his #ro!a!ilistic+choice #ers#ective can !e a##lied to the matching #ennies

game. he intuitive idea is that 3ow will choose o# with high #ro!a!ility when the

e#ected #ayoff for o# is a lot larger than the e#ected #ayoff for 7ottom& !ut thatsome randomness will start to !ecome a##arent when the e#ected #ayoffs for the two

decisions are not that far a#art. o #roceed with this argument& we !egin !y calculating

these e#ected #ayoffs. 1et p denote 3ow s !eliefs a!out the #ro!a!ility of 3ight& so ,

p is the #ro!a!ility of 1eft. ?rom the to# row of a!le ,0.,& we see that if 3ow chooses

o#& then 3ow earns -2 with #ro!a!ility , p and % with #ro!a!ility p& so the e#ected

#ayoff is:

'o;9s 45pected Paoff for 8op J -2=, p> O %= p> J -2 % p.

6imilarly& !y #laying 7ottom& 3ow earns % with #ro!a!ility , p and -2 with #ro!a!ility

p& so the e#ected #ayoff is:

'o;9s 45pected Paoff for Bottom J %=, p> O -2= p> J % O % p.

t follows that the difference in these e#ected #ayoffs is: =-2 % p> =% O % p>& or

e'uivalently:

'o;9s 45pected Paoff (ifference 0for 8op Minus Bottom2 J % -2 p.

$hen p is near <ero& this difference is %& !ut as p a##roaches ,2 this difference goes

to <ero& in which case 3ow is indifferent and would !e willing to choose either decisionor to fli# a coin.

f 3ow were #erfectly rational =and res#onded to ar!itrarily small e#ected

#ayoff differences>& then o# would !e #layed whenever p is even a little less than ,2.

,25




he curved line in ?igure ,0., shows some de#arture from this rationality. ;otice that

the #ro!a!ility that 3ow #lays o# is close to , !ut not 'uite there when p is small& and

that the curved line deviates more from the !estres#onse line as p a##roaches ,2.

he curved noisy !est+res#onse line for 3ow intersects Column s dashed !est+

res#onse line in the center of ?igure ,0.,& so a relaation of the #erfect rationality

assum#tion for 3ow will not affect the e'uili!rium #rediction. 6imilarly& su##ose that

we allow some noise in Column s !est res#onse& which will smooth off the shar# corners

for Column s !est res#onse line& as shown !y the u#ward slo#ing dashed line on the

right side of ?igure ,0.2. his starts in the lower+left corner& rises with a smooth arc as

it levels off at 0.5 in the center of the gra#h !efore curving u#ward along the u##er+right

!oundary of the gra#h. 6ince this line will intersect 3ow s downward slo#ing noisy

res#onse line in the center of the gra#h& we see that the fifty+fifty #rediction is not

affected if we let each #layer s decision !e somewhat noisy.

,2%



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Probability of .i*,t

Probability of " op

,2-



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Probability of .i*,t Markets, Games, and Strategic Behavior Charles A. Holt

0ig$re '+.. Best Responses @Left SideA and 8ois! Best Responses @Right SideA

?inally& you should thin" a!out what would ha##en if the amount of randomness

in !ehavior were somehow reduced. his would corres#ond to a situation where each

#erson ta"es full advantage of even a slight tendency for the other to choose one

decision even slightly more often than the other. hese shar# res#onses to small

#ro!a!ility differences would cause the curved lines on the right to !ecome more li"e

the straight+line !est res#onse functions& i.e. the corners on in the curved lines would

!ecome shar#er. n any case& the intersection would remain in the center& so adding

noise has no effect on #redictions in the symmetric matching #ennies game.

III. he "ffects of Pa!off Im&a%ances As one would e#ect& the fact that the noisy !est res#onse lines always intersect in the

center of ?igure ,0.2 is due to the !alanced nature of the #ayoffs for this game =a!le

,0.,>. An un!alanced #ayoff structure is shown in a!le ,0.2& where the 3ow #layer s

#ayoff of -2 in the o#1eft !o has !een increased to %0. 3ecall that the game was

!alanced !efore this change& and the choice #ro#ortions should !e ,2 for each #layer.

he increase in 3ow s o#1eft #ayoff from -2 to %0 would ma"e o# a more

attractive choice for a wide range of !eliefs& i.e. 3ow will choose o# unless Column is

almost sure to choose 3ight. ntuitively& one would e#ect that this change would move

3ow s choice #ro#ortion for o# u# from the ,2 level that is o!served in the !alanced

game. his intuition is a##arent in the choice data for an e#eriment done with

Veconla! software& where the #ayoffs were in #ennies. *ach of the three sessionsinvolved ,0+,2 #layers& with 25 #eriods of random matching. he #ro#ortion of o#

choices was %-K for the %0 treatment game in a!le ,0.2.

a!le ,0.2. An Asymmetric 4atching 8ennies ame


3ow 8layer: Column 8layer:

1eft 3ight

o#

7ottom

,2

%0& % ) ( %& -2

%& -2 & ' -2& %



his intuitive own #ayoff effect of increasing 3ow s o#1eft #ayoff is not consistent

with the ;ash e'uili!rium #rediction. ?irst& notice that there is no e'uili!rium in non+

random strategies& as can !e seen from the arrows in a!le

,0.2& which go in a cloc"wise circle. o derive the mied e'uili!rium #rediction& let p

denote 3ow s !eliefs a!out the #ro!a!ility of 3ight& so , p is the #ro!a!ility of 1eft. hus

3ow s e#ected #ayoffs are:

'o;9s 45pected Paoff for 8op J %0=, p> O %= p> J %0 2/ p.

'o;9s 45pected Paoff for Bottom J %=, p> O -2= p> J % O % p.

t follows that the difference in these e#ected #ayoffs is: =%0 2/ p> =% O % p>& or

e'uivalently& 2/ %0 p. 3ow is indifferent if the e#ected #ayoff difference is <ero& i.e.

if # J 2/%0 J 0.. herefore& 3ow s !est res#onse line stays at the to# of the left #anel in ?igure ,0. as long as the #ro!a!ility of 3ight is less than 0.. he stri"ing

thing a!out this figure is that Column s !est res#onse line has not changed from the

symmetric caseB it rises from the 7ottom1eft corner and crosses over when the

#ro!a!ility of o# is ,2. his is !ecause Column s #ayoffs are eactly reflected =% and

-2 on the 1eft side& -2 and % on the right side>. n other words& the only way that

Column would !e willing to randomi<e is if 3ow chooses o# and 7ottom with e'ual

#ro!a!ility. n other words& the ;ash e'uili!rium for the asymmetric game in a!le ,0.2

re'uires that o# and 7ottom !e #layed with eactly the same #ro!a!ilities =,2 each> as

was the case for the case of !alanced #ayoffs in a!le ,0.,. he reason is that Column s

#ayoffs are the same in !oth games& and 3ow must essentially use a coin fli# in order to

"ee# Column indifferent.

,2



!ata

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Probability of " op


,0



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Pr obability of " op

0ig$re '+./. he /1+ reatment:

Best Responses @Left SideA and 8ois! Best Responses @Right SideA

he #revious treatment change made o# more attractive for 3ow !y raising 3ow s

o#1eft #ayoff from -2 to %0. n order to #roduce an im!alance that ma"es o# #ess

attractive& we reduce 3ow s o#1eft #ayoff in a!le ,0., from -2 to /0. hus 3ow s

e#ected #ayoff for o# is: /0=, p> O %= p> J /0 / p& and the e#ected #ayoff for

7ottom is: %=, p> O -2= p> J % O % p. hese are e'ual when p J //0& or 0.,. $hat

has ha##ened in ?igure ,0./ is that the reduction in 3ow s o#1eft #ayoff has #ushed

3ow s !est res#onse line to the !ottom of the figure& unless Column is e#ected to #lay

3ight with a #ro!a!ility that is less than 0.,. his downward shift in 3ow s !est

res#onse line is shown on the left side of ?igure ,0./. he result is that the !est

res#onse lines intersect where the #ro!a!ility of o# is ,2& which was the same #rediction o!tained from the left sides of ?igures ,0.2 and ,0.. hus a change in 3ow

s o#1eft #ayoff from /0 to -2 to %0 does not change the ;ash e'uili!rium #rediction

for the #ro!a!ility of o#. he mathematical reason for this result is that Column s

#ayoffs do not change& and 3ow must choose each decision with e'ual #ro!a!ility or

Column would not want to choose randomly.

,,



!ata40

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0 01 02 03 04 05 07 08 09 106




,2



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10 01 02 03 04 05 06 07 08 09Probability of .i*,t


0ig$re '+.3. he 3+ reatment:

Best Responses @Left SideA and 8ois! Best Responses @Right SideA

hese invariance #redictions are not !orne out in the data from the Veconla!

e#eriments re#orted here. *ach of the three sessions involved 25 #eriods for each

treatment& with the order of treatments !eing alternated. he #ercentage of o# choices

increased from %K to %-K when the 3ow s o#1eft #ayoff was increased from /0 to

%0. he Column #layers reacted to this change !y choosing 3ight only 2/K of the

time in the /0 treatment& and -/K of the time in the %0 treatment.

he 'ualitative effect of 3ow s own+#ayoff effect is ca#tured !y a noisyres#onse model

where the shar#+cornered !est res#onse functions on the left sides of ?igures ,0. and

,0./ are rounded off& as shown on the right sides of the figures. ;otice that theintersection of the curved lines im#lies that 3ow s #ro#ortion of o# choices will !e

!elow ,2 in the /0 treatment and a!ove ,2 in the %0 treatment& and the #redicted

change in the #ro#ortion of 3ight decisions will not !e as etreme as the movement

from 0., to 0. im#lied !y the ;ash #rediction. he actual data averages are shown !y

the !lac" dots on the right sides of ?igures ,0. and ,0./. hese data averages ehi!it

the strong own#ayoff effects that are #redicted !y the curved !est+res#onse lines& and

the #rediction is fairly accurate& es#ecially for the /0 treatment.

he 'uantitative accuracy of these #redictions is affected !y the amount of

curvature that is #ut into the noisy res#onse functions& and is ultimately a matter of

estimation. 6tandard estimation techni'ues are !ased on writing down a mathematical

function with a noise #arameter that determines the amount of curvature and thenchoosing the #arameter that #rovides the !est fit. he nature of such a function is

discussed net.

I. Pro&a&i%istic Choice @5ptiona%A

Anyone who has ta"en an introductory #sychology course will remem!er the little

stimulus+res#onse diagrams. 6tochastic or noisy res#onse models were develo#ed after

researchers noticed that res#onses could not always !e #redicted with certainty. he

wor" of a mathematical #sychologist& (uncan 1uce =,5>& suggested a way to model

noisy choices& i.e. !y assuming that res#onse #ro!a!ilities are increasing functions of the

strength of the stimulus. ?or eam#le& su##ose that a #erson must 9udge which of two

very faint sounds is loudest. he #ro!a!ility associated with one sound should !e an

increasing function of the deci!el level of that sound. 6ince #ro!a!ilities must sum to ,&

,




the choice #ro!a!ility associated with one sound should also !e a decreasing function of

the intensity of the other sound.

n economics& the stimulus intensity associated with a given res#onse =decision> might

!e thought of as the e#ected #ayoff of that decision. 6u##ose that there are two

decisions& (, and (2& with e#ected #ayoffs that we will re#resent !y E, and E2. ?or

eam#le& the decisions could !e 3ow s choice of o# or 7ottom& and the e#ected

#ayoffs would !e calculated using the e'uations given earlier& e.g. E, J %0=, p> O % p&

where p re#resents the #ro!a!ility which the 3ow #layer thin"s Column will choose

3ight. =hin" of E& the ree" letter #ronounced li"e #ie& as shorthand for #ayoff.>

Using 1uce s suggestion& the net ste# is to find an increasing function& i.e. one

with a gra#h that goes u#hill. =4athematically s#ea"ing& a function& f & is increasing if a

f0 E,> P f =E2> whenever E, P E2& !ut the intuitive idea is that the slo#e in a gra#h is from

lower+left to u##er+right.> 6everal eam#les of increasing functions are the linear

function: f =E,> J E,& or the e#onential function& f =E,> J e#=E,>. he linear functiono!viously has an u#hill slo#eB its gra#h is 9ust the /5degree line. he e#onential

function has a curved sha#e& li"e a hill that "ee#s getting stee#er& li"e a snow+ca##ed

4t. ?u9i in a Fa#anese wood!loc" #rint.

Ince we have found an increasing function& we might !e tem#ted to assume that

the #ro!a!ility of the decision is determined !y the function itself: 8r= (,> J f =E,> and

8r= (2> J f =E2>. he #ro!lem with this a##roach is that it does not ensure that the two

#ro!a!ilities sum to ,. his is easily fied !y a sim#le tric" with a fancy name:

normali<ation. Fust divide each function !y the sum of the functions:

=,0.,> 8r= (,> f=E,> and 8r= (2 > f

=E2 >.

f =E,> f =E2 > f =E E,> f = 2 >

f you are feeling a little unsure& try adding the two ratios in =,0.,> to show that 8r= (,>

O 8r= (2> J ,. ;ow let s see what this gives us for the linear case:

=,0.2> 8r= (,> E

, and 8r= (2 > E

2 .

E, E2 E E,

6u##ose E, J E2 J ,. hen each of the #ro!a!ilities in =,0.2> will e'ual ,=,O,> J ,2.

his result holds as long as the e#ected #ayoffs are e'ual& even if they are not !oth

e'ual to ,. his ma"es senseB if each decision is e'ually #rofita!le& then there is no

,/



reason to #refer one over the other& and the choice #ro!a!ilities should !e ,2 each.

;et notice that if E, J 2 and E2 J ,& the #ro!a!ility of choosing decision (, is higher

=2>& and this will !e the case whenever E, P E2.

Ine #otential limitation to the usefulness of the #ayoff ratios in =,0.2> is that

e#ected #ayoffs may !e negative if losses are #ossi!le. his #ro!lem can !e avoided if

we choose a function in =,0.,> that cannot have negative values& i.e. f =E,> P 0 even if E,

Q 0. his non+negativity is characteristic of the e#onential function& which can !e used

in =,0.,> to o!tain:

=,0.> 8r= (,> e#=E

,>

and 8r= (2 > e#=E

2>

.

e#=E,>e#=E2 > e#=E,>e#=E2 >

his avoids the #ossi!ility of negative #ro!a!ilities& and all of the other useful #ro#erties

of =,0.2> are #reserved. he #ro!a!ilities in =,0.> will sum to one& they will !e e'ual

when the e#ected #ayoffs are e'ual& and the decision with the higher e#ected #ayoff

will have a higher choice #ro!a!ility. he #ro!a!ilistic choice model that is !ased on

e#onential functions is "nown as the #ogit mode# . he curved lines in the right #arts of

the figures in this cha#ter were constructed using logit choice functions& !ut not 'uite

the ones in e'uation =,0.>. t is true that =,0.> a##lied to the e#ected #ayoffs for the

matching #ennies games will #roduce curved res#onse functions& !ut the lines will not

have as much curvature as those in the figures in this cha#ter. he lines drawn with

e'uation =,0.> have corners that are too shar# to e#lain the own #ayoff effects that we

are seeing in the matching #ennies games. Fust as we could ma"e it harder todistinguish !etween the width of two #ins !y ma"ing them each half as thic"& we can

add more noise or randomness into the choice #ro!a!ilities in =,0.> !y reducing all

e#ected #ayoffs !y a half or more. ntuitively s#ea"ing& dividing all e#ected #ayoffs

!y ,00 may in9ect more randomness& since dollars !ecome #ennies& and non+monetary

factors =!oredom& indifference& #layfulness> may have more influence. he right

#anels of the figures in this cha#ter were o!tained !y using the logit model with all

#ayoffs !eing divided !y ,0:

=,0./> 8r= (,> e#=E, ,0> & 8r= (2 > e#=E2 ,0> . e#=E, ,0>e#=E2 ,0> e#=E, ,0>e#=E2 ,0>

,5




At this #oint& you are #ro!a!ly wondering& why ,0& why not ,00N he degree to

which #ayoffs are diluted !y dividing !y larger and larger num!ers will determine the

degree of curvature in the noisy res#onse functions. hus we can thin" of the num!er in

the denominator of the e#ected #ayoff e#ressions as !eing an error #arameter that

determines the amount of randomness in the #redicted !ehavior. he #ogit error

parameter will !e called F& and it is used in the logit choice #ro!a!ilities:

=,0.5> 8r= (,> e#=E, F> & 8r= (2 > e#=E F2 > .

e#=E, F>e#=E F2 > e#=E F, >e#=E F2 >

he logit form in =,0.5> is flei!le& since the degree of curvature is ca#tured !y a

#arameter that can !e estimated. t also has the intuitive #ro#erty that an increase in the

#ayoffs will reduce the noise& i.e. will reduce the curvature of the noisy !est res#onsecurves in the figures =see 'uestion />.

"(tensions and 0$rther Reading

he use of #ro!a!ilistic choice functions in the analysis of games was #ioneered !y

4cDelvey and 8alfrey =,5>& and the intersections of noisy !est res#onse lines in

?igures ,0.2+,0./ corres#ond to the #redictions of a Huanta# response eHui#i+rium. =A

'uantal res#onse is essentially the same thing as a noisy !est res#onse.> he a##roach

ta"en in this cha#ter is !ased on oeree& Holt& and 8alfrey =200/& 2005>& who introduce

the idea of a regu#ar Huanta# response eHui#i+rium& and who discuss its em#irical

content and numerous a##lications. oeree& Holt& and 8alfrey =2000> use this a##roachto e#lain !ehavior #atterns in a num!er of matching #ennies games. heir analysis

also includes the effects of ris" aversion& which can e#lain the over+#rediction of own+

#ayoff effects for high #ayoffs that is seen on the right side of ?igure ,0.. 3is"

aversion introduces diminishing marginal utility that reduces the attractiveness of the

large %0 #ayoff for the row #layer& and this shifts that #erson s !est res#onse line down

so that the intersection is closer to the data average #oint.

All of the games considered in this cha#ter involved two decisions& !ut the same

#rinci#les can !e a##lied to games with more decisions =see oeree and Holt& ,B

Ca#ra& oeree& ome<& and Holt& ,& 200,& which will !e discussed in the net two

cha#ters>. If #articular interest is Com#osti =200>& who had University of Virginia

soccer #layers attem#t #enalty "ic"s into a net that was divided into three sectors !y

hanging stri#s of cloth. hus the "ic"er had three decisions corres#onding to the sector

of the "ic"& say -& M & and '. 6imilarly& the goalie had three decisions& corres#onding to

the direction of the defensive dive. n a symmetric treatment& the "ic"er received for

,%



each #oint scored& regardless of sector& and the goalie received each time a "ic" was

unsuccessful for any reason. he #enalty "ic"s were a##roimately evenly divided

!etween the two sides& , attem#ts to - and ,, attem#ts to '& with only % to the middle.

n an asymmetric treatment& the "ic"ers received % for each "ic" scored on the ' side&

and for each "ic" that scored to the M or - sides. he goalie s incentives did not

change = for each "ic" that did not score& regardless of direction>. his asymmetry in

the "ic"er s #ayoffs caused a strong own+#ayoff effect& with 25 attem#ts to '& 5 to -& and

none to the middle. his study ehi!its the advantages of an e#eriment& where

asymmetries can !e introduced in a manner that is not #ossi!le from an analysis of data

from naturally occurring soccer games.

he focus in this cha#ter has !een on games& !ut #ro!a!ilistic choice functions can !e

a##lied to sim#le decision #ro!lems. ?or eam#le& recall that a ris" neutral #erson

would ma"e eactly / safe choices in the lottery choice menu #resented in Cha#ter /&

and a #erson with s'uare root utility would ma"e % safe choices. hese #redictions #roduce lines with shar# corners& e.g. the dashed line in ?igure /.2& which loo"s a lot

li"e the curved lines in the figures in this cha#ter.

he actual data averages shown in that figure have smoothed corners that would result

from a #ro!a!ilistic choice function& and Holt and 1aury =200,> did find that such a

function #rovided a good e#lanation of the actual #attern of choice #ro#ortions.

<$estions,. n a matching #ennies game #layed with #ennies& one #erson loses a #enny and

the other wins a #enny& so the #ayoffs are , and ,. 6how that there is a sim#le

way to transform the game in a!le ,0., =with #ayoffs of % and -2> into this

form. =Hint: first divide all #ayoffs !y %& and then su!tract a constant from all

#ayoffs. his would !e e'uivalent to #aying in %+cent coins if they eisted& and

charging an entry fee to #lay.>

2. Consider a soccer #enalty "ic" situation where the "ic"er is e'ually s"illful at

"ic"ing to either side& !ut the goalie is !etter diving to one side. n #articular& the

"ic"er will always score if the goalie dives away from the "ic". f the goalie

dives to the side of the "ic"& the "ic" is always !loc"ed on the goalie s right !ut

is only !loc"ed with #ro!a!ility one half on the goalie s left. 3e#resent this as a

sim#le game& in which the goalie earns a #ayoff of O, for each !loc"ed "ic" and

, for each goal& and the "ic"er earns , for each !loc"ed "ic" and O, for each

,-




goal. A 0.5 chance of either outcome results in an e#ected #ayoff of 0.

(etermine the e'uili!rium #ro!a!ilities used in the ;ash e'uili!rium.

. 6how that the two choice #ro!a!ilities in =,0.5> sum to ,.

/. 6how that dou!ling all #ayoffs& i.e. !oth E, and E2& has the same effect as

reducing the noise #arameter F in =,0.5> !y half.

5. 6how that multi#lying all #ayoffs in =,0.2> !y 2 will not affect choice

#ro!a!ilities.

%. 6how that adding a constant amount& say 5& to all #ayoffs in =,0.5> will have no

effect on the choice #ro!a!ilities. Aint$ Use the fact that an e#onential

function of the sum is the #roduct of the e#onential functions of the two

com#onents of the #roduct: e#=EO 5> J e#=E>e#= 5>.

Cha#ter ,,. he raveler s (ilemma

*ach #layer s !est decision in a =single #lay> #risoner s dilemma is to defect& regardless

of what the other #erson is e#ected to do. he dilemma is that !oth could !e !etter off

if they resist these incentives and coo#erate. he traveler s dilemma is a game with a

richer set of decisionsB each #erson must claim a money amount& and the #ayment is the

minimum of the claims& #lus a =#ossi!ly small> #ayment incentive to !e the low

claimant. 7oth would !e !etter off ma"ing identical high claims& !ut each has an

incentive to undercut the other to o!tain the reward for having the lower claim. his

game is similar to a #risoner s dilemma in that the uni'ue e'uili!rium involves #ayoffsthat are lower than can !e achieved with coo#eration. 7ut here there is a more

interesting dimension& since the o#timal decision in the traveler s dilemma does de#end

on what the other #erson is e#ected to do. hus the game is more sensitive to

interactions of im#recise !eliefs and small variations in decisions. hese interactions

can cause data #atterns to !e 'uite far from ;ash #redictions& as will !e illustrated with a

set of iterated s#readsheet calculations. his is one of my favorite gamesG t can !e

#layed with the instructions in the A##endi or with the raveler s (ilemma #rogram on

ames 4enu of the Veconla! site. n addition& it is #ossi!le for students to #lay an

online demonstration version of the raveler s (ilemma& where the other decision is

ta"en from a data!ase of decisions made !y University of Virginia law students in a !ehavioral game theory class. his online simulation can !e accessed from:

htt#:veconla!.econ.virginia.edutddemo.htm

I. - acation 6ith an >nhapp! "nding

,



Ince u#on a time& two travelers were returning from a tro#ical vacation where they had

#urchased identical anti'ues that were #ac"ed in identical suitcases. $hen !oth !ags

were lost in transit& the #assengers were as"ed to #roduce recei#ts& an im#ossi!le tas"

since the anti'ues were #urchased with cash. he airline re#resentative calmly informed

the travelers that they should go into se#arate rooms and fill out claim sheets for the

contents of the suitcases. He assured them that !oth claims would !e honored if a> they

are e'ual& and !> they are no greater than the lia!ility limit of 200.00 #er !ag in the

a!sence of #roof of #urchase. here was also a minimum allowed claim of 0.00 to

cover the cost of the luggage and inconvenience. Ine of the travelers e#ressed some

frustration& since the value of the suitcase and anti'ue com!ined was somewhat a!ove

200.00. he other traveler& who was even more #essimistic& as"ed what would ha##en

if the claims were to differ. o this& the re#ly was: f the claims are une'ual& then we

will assume that the higher claim is un9ustly inflated& and we will reim!urse you !oth at

an amount that e'uals the minimum of the two claims.n addition& there will !e a 5.00 #enalty assessed against the higher claimant and a

5.00 reward added to the com#ensation of the lower claimant. he dilemma was that

each could o!tain reasona!le com#ensation if they made matching high claims of 200&

and resisted the tem#tation to come down to , to ca#ture the 5 reward for !eing

low. f one #erson e#ects the other one to as" for ,& the !est res#onse is ,& !ut

what if the other #erson is thin"ing similar thoughtsN his line of reasoning leads to an

unfortunate #ossi!ility that a cycle of antici#ated moves and counter+moves may cause

each traveler to #ut in very low claims. Unfortunately& there was no way for the two

travelers to communicate !etween rooms& and the stress of the tri# led each to !elieve

that there would !e no sharing of any une'ual com#ensation amounts received. he

unha##y ending has !oth ma"ing low claims& and the alternative ha##y ending involvestwo high claims. he actual outcome is an em#irical 'uestion that we will consider

!elow. his game was introduced !y 7asu =,/>& who viewed it as a dilemma for the

game theorist. $ith a very low #enalty rate& there is little ris" in ma"ing a high claim&

and yet each #erson has an incentive to undercut any common claim level. ?or eam#le&

su##ose that !oth are considering claims of 200B #erha#s they whis#ered this to each

other on the way to the se#arate rooms. Ince alone& each might reason that a deviation

to , would reduce the minimum !y ,& which is more than made u# for !y the 5

reward for !eing low. n fact& there is no !elief a!out the other s claim that would ma"e

one want to claim the u##er limit of 200. f one e#ects any lower claim& then it is

!etter to undercut that lower claim as well. 3easoning in this manner& we can rule out

all common claims as candidates for e'uili!rium& ece#t for the lowest #ossi!le claim of

0. 6imilar reasoning shows that no configuration with une'ual claims can !e an

e'uili!rium =Luestion ,>.

,




he uni'ue ;ash e'uili!rium for the traveler s dilemma has another interesting

#ro#erty: it can !e derived from an assum#tion that each #erson "nows that the other is

#erfectly rational. 3ecall that there is no !elief a!out the other s claim that would 9ustify

a claim of 200. 6ince they each "now that the other will never claim 200& then the

u##er !ound has shifted to ,& and there is no !elief that 9ustifies a claim of ,. o

see this& su##ose that claims must !e in integer dollar amounts and note that , is a

!est res#onse to the other s claim of 200 !ut it is not a !est res#onse to any lower

claim. 6ince 200 will not !e claimed !y any rational #erson& it follows that , is not

a !est res#onse to any !elief a!out the other s decision. 3easoning in this manner& we

can rule out all successively lower claims ece#t for the very lowest feasi!le claim.

=his argument !ased on common "nowledge of rationality is called rationali<a!ility&

and the minimum #ossi!le claim in this game is the uni'ue rationa#i)a+#e e'ulili!rium.

*conomists often #lace 9uicy& #ersuasive ad9ectives in front of the word e'uili!rium&

sometimes to no avail. his will turn out to !e one of those times.>he dilemma for the theorist is that the ;ash e'uili!rium is not sensitive to the si<e of

the #enaltyreward level& as long as this level is larger than the smallest #ossi!le amount

!y which a claim can !e reduced. ?or eam#le& the unilateral incentive to deviate from

any common claim in the traveler s dilemma is not affected !y if the #enaltyreward rate

is changed from 5 to /. f !oth were #lanning to choose a claim of 200& then one

#erson s deviation to , would reduce the minimum to ,& !ut would result in a

reward of / for the deviator. hus the #erson deviating would earn , O / instead

of the 200 that would !e o!tained if they !oth claim 200. he same argument can !e

used to show that there is an incentive to undercut any common claim& whether the

#enaltyreward rate is 2 or 200. hus the ;ash e'uili!rium is not sensitive to this

#enaltyreward rate as long as it eceeds ,& whereas one might e#ect o!served claimsto !e res#onsive to large changes in this #ayoff #arameter.

II. Data

he traveler s dilemma was analy<ed !y Ca#ra et al. =,>. his was an eciting

e#eriment. $e e#ected the #enaltyreward #arameter to have a strong effect on actual

claim choices& even though the uni'ue ;ash #rediction would !e inde#endent of

changes in this #arameter. he design involved two #arts& A and 7& with different

#enaltyreward #arameters. *ach session involved a!out ,0+,2 su!9ects& who were

randomly #aired at the start of each round& for a series of ,0 rounds. Claims were

re'uired to !e !etween 0.0 and 2.00. he #art A data averages for four of the

treatments are #lotted in ?igure ,,.,& where the hori<ontal ais shows the round num!er

and the #enaltyreward #arameter is denoted !y '. $ith a high #enaltyreward

#arameter of 0.0& the claims average a!out ,.20 in round , and fall to levels

,/0



@;er a*e +laim

a##roaching 0.0 in the final four rounds& as shown !y the thic" solid line at the !ottom

of the figure. he data for the 0.50 treatment start somewhat higher !ut also a##roach

the ;ash #rediction in the final rounds. n contrast& the round , averages for the 0.,0

and 0.05 treatments& #lotted as dashed lines& start at a!out a dollar a!ove the ;ash

#rediction and actually rise slightly& moving a;a from the ;ash #rediction. he data

for the intermediate treatments =0.20 and 0.25> are not shown& !ut they stayed in the

middle range =,.00 to ,.50> !elow the dashed lines and a!ove the solid lines& with

some more variation and crossing over.

3ound

0ig$re ''.'. Data for the rave%er9s Di%emma @Capra et a%. '444A

III. Learning and "(perience

;otice that the most salient feature of the traveler s dilemma data& the strong effect of

the #enaltyreward #arameter& is not #redicted !y the ;ash e'uili!rium. Ine might

,/,




dismiss this game on the grounds that it is somewhat artificial. here is some truth to

this& although many standard economic games do involve #ayoffs that de#end on the

minimum #rice& as is the case with 7ertrand #rice com#etition =Cha#ter ,5> or the net

cha#ter s minimum+effort coordination game. An alternative #ers#ective is that this

game involves an intentionally a!stract setting& which serves as a #aradigm for

#articular ty#es of strategic interactions. he traveler s dilemma is no more a!out lost

luggage than the #risoner s dilemma is a!out actual #risoners. f standard game theory

cannot #redict well in such sim#le situations& then some rethin"ing is needed. At a

minimum& it would !e nice to have an idea of when the ;ash e'uili!rium will !e useful

and when it will not. *ven !etter would !e a theoretical a##aratus that e#lains !oth the

convergence to ;ash #redictions in some treatments and the divergence in others. he

rest of this cha#ter #ertains to several #ossi!le a##roaches to this #ro!lem. $e !egin

with an intuitive discussion of learning. 7ehavior in an e#eriment with re#eated

random #airings may evolve from round to round as #eo#le learn what to e#ect. ?or eam#le& consider the data in a!le ,,., from a classroom e#eriment conducted at the

University of Virginia. Claims were re'uired to !e !etween 0 and 200 cents& and the

#enaltyreward #arameter was ,0 cents. here were 20 #artici#ants& who were divided

into ,0 #airs. *ach 2+#erson team had a handheld wireless 8(A with a touch+sensitive

color screen that showed the H41 dis#lays from the Veconla! software. t was a nice

s#ring day& and this class was conducted outside on the lawn& with students !eing seated

informally in a semi+circle. he ta!le shows the decisions for five of the teams during

the first four rounds. he round is listed on the left& and the average of all ,0 claims is

shown in the second column. he remaining columns show some of the team s own

decisions& which are listed net to the decision of the other team for that round =shown

in #arentheses>.?irst consider the round , decision for 6tacy;aomi& who claimed ,50. he

other team was lower& at ,5& so the earnings for 6tacy;aomi were the minimum& ,5&

minus the #enalty of ,0 cents& or ,25. 6u<6io !egan lower& at ,00& and encountered a

claim of ,. his team then cut their claim to 5& and encountered an even higher

claim of ,, in round 2. his caused them to raise their claim to ,25& and they finished

round ,0 =not shown> with a claim of ,%0.

a!le ,,.,. raveler s (ilemma (ata for a Classroom *#eriment with 3 J ,0 Dey:

Iwn Claim =Ither s Claim>

3ound Average

=,0 claims>

6u<6io D 6'uared Durt7ruce Fess*d 6tacy;aomi

, ,- ,00 =,> 00 =,5> , =,/0> , =,00> ,50 =,5>

,/2



2 ,,05 =,,>

0 =,,-> ,5 =,/0> 0 =,0> ,2- =200>

,// ,25 =,5> 0% =,,-> ,5 =,00> , =,> ,/ =200>

/ ,/2 ,,5 =,25> ,0 =,00> ,25 =,,5> , =,,5> ,50 =,./>

he team D 6'uared =Datie and Dari> !egan round , with a decision of 0 cents&

which is the ;ash e'uili!rium. hey had the lower claim and earned the minimum& 0&

#lus the ,0 cents for !eing low. After o!serving the other s claim of ,5 for that round&

they raised their claim to in round 2& and eventually to ,20 in round ,0. he #oint of

this eam#le is that the !est decision in a game is not necessarily the e'uili!rium

decisionB it ma"es sense to res#ond to and antici#ate the way that the others are actually #laying. Here the ;ash e'uili!rium decision of 0 #layed in the first four #eriods would

have earned 0 in each round& for a total of %0. D 6'uared actually earned /0 !y

ada#ting to o!served !ehavior and ta"ing more ris". A claim of 200 in all rounds would

have !een even more #rofita!le ='uestion />.

eams who were low relative to the other s claim tended to raise their claims&

and teams that were high tended to lower them. his 'ualitative ad9ustment rule&

however& does not e#lain why 6u<6io s claim was lowered even after it had the lower

claim in round ,. his reduction seems to !e in antici#ation of the #ossi!ility that other

claims might fall. n the final round& the claims ranged from ,,0 to 200& with an average

of ,/% and a lot of dis#ersion.

Ine way to descri!e the outcome is that #eo#le have different !eliefs !ased ondifferent e#eriences& !ut !y round ,0 most e#ect claims to !e a!out ,50 on average&

with a lot of varia!ility. f claims had converged to a narrow !and& say with all at ,50&

then the undercutting logic of game theory might have caused them to decline as all see"

to get !elow ,50. 7ut the varia!ility in claims did not go away& #erha#s !ecause #eo#le

had different e#eriences and different reactions to those e#eriences. his varia!ility

made it harder to figure out the !est res#onse to others claims. n fact& claims did not

diminish over timeB if anything there was a slight u#ward trend. he highest earnings

were o!tained !y the Fess*d team& which had a relatively high average claim of ,--.

he #art A treatment for this session was followed !y ,0 more rounds with a

higher #ayoff #arameter =50 cents>. his created a strong incentive to !e the low

claimant& and claims did decline to the ;ash e'uili!rium level of 0 after the first

several rounds. hus we see that convergence to a ;ash e'uili!rium seems to de#end on

the magnitudes of incentives& not 9ust on whether one decision is slightly !etter than

another. An analysis that is sensitive to the magnitudes of #ayoff differences is

,/




considered in the A##endi to this cha#ter. he goal is to develo# a theoretical

e#lanation of the o!served convergence to the ;ash #rediction in the high+ ' treatments

and the divergence in the low+ ' treatments. he calculations in the a##endi are set u#

in terms of a s#readsheet& so the re'uired mathematics involves sim#le alge!ra and is no

higher than that of the other cha#ters of this !oo".

I. "(tensions and 0$rther Reading

he main result of the traveler s dilemma e#eriment is that the most salient as#ect of

the data& sensitivity to the si<e of the #enalty+reward rate& is not e#lained !y the ;ash

e'uili!rium& which is unaffected !y changes in this rate. (ecisions in the e#eriment

seem to !e affected !y the magnitudes of #ayoff differences& even though these

magnitudes do not affect the 'ualitative =is it greater than or less than> com#arisons used

to find a ;ash e'uili!rium. An im#ortant lesson to !e learned is that it may !e verycostly to use a ;ash e'uili!rium strategy if others are not using their e'uili!rium

strategies.

his lesson can !e illustrated with an outcome o!served for the online version of the

traveler s dilemma mentioned in the introductory #aragra#h of this cha#ter& which has

!een #layed !y over ,&000 #eo#le to date. his demo involves 5 rounds with a

#enaltyreward rate of ,0 cents& and with claims re'uired to !e in the 0 to 200 range.

he other decisions are claims saved in a data!ase from an e#eriment involving

Virginia law students from a !ehavioral game theory class. After finishing 5 rounds& the

#erson #laying the demo is told how their total earnings com#are with the total earned

!y the law student who faced the same se'uence of other claims that they themselves

saw. he average claims from this law class were fairly high& in the ,0 range& sosomeone who stays low =closer to the ;ash e'uili!rium of 0> will ty#ically earn less

than the law student earnings !enchmar". n fact& this was the case when a ;o!el+#ri<e

winning economist #layed the game and maintained claims near ,0 in the lower+middle

#art of the rangeB he earned a!out 25K less than the law student who had faced the same

se'uence of other claims.

he e#eriments discussed thusfar in this cha#ter involved re#eated random matching&

so that #eo#le can learn from e#erience& even though #eo#le are not interacting with

the same #erson in each round. oeree and Holt =200,> #rovide e#erimental evidence

for traveler s dilemma games #layed only once. here is no o##ortunity to learn from

#ast e#erience in a one+shot gameB each #erson must !ase their claim decision on

introspection a!out what the other is li"ely to do& a!out what the other thin"s they will

do& etc. he average claims are strongly influenced !y the si<e of the #enaltyreward&

even though changes in this #arameter have no effect on the uni'ue ;ash e'uili!rium.

If course& it is not reasona!le to e#ect data to conform to a ;ash e'uili!rium in a one+

,//



shot game with no #ast e#erience& since this e'uili!rium =in #ure strategies> im#lies

that the other s claim is somehow "nown. he develo#ment of models of intros#ection

is an im#ortant and relatively #oorly understood to#ic in game theory. 6ee oeree and

Holt =200/> for a formal model of intros#ection.

here are several other games with a similar #ayoff that de#ends on the minimum of all

decisions. Ine such game& discussed in the net cha#ter& has the wea"est+lin" #ro#erty

that the out#ut is determined !y the minimum of the individual effort levels. 6imilarly&

the sho##ing !ehavior of uninformed consumers may ma"e firms #rofits sensitive to

whether or not their #rice is the minimum in the mar"et. Ca#ra et al. =2002> #rovide

e#erimental data for a #rice com#etition game with meet+or+release clauses& which

release the !uyer from the contract if a lower #rice offer is found and the original seller

refuses to match. f all consumers are informed and all sellers are #roducing the same

#roduct& then each would rather match the other instead of losing all !usiness as

informed consumers switch. A unilateral #rice cut& however& may fail to #ic" u# some !usiness for a grou# of uninformed consumers. n any case& if one firm sets a lower

#rice than the other& then it will o!tain the larger mar"et share& and the high+#rice firm

will have to match the other s #rice to get any sales at all. his is li"e the traveler s

dilemma in that earnings are determined !y the minimum #rice& with a #enalty for

having a higher #rice. n #articular& each firm earns an amount that e'uals the minimum

#rice times their sales 'uantity& !ut the firm that had the lower #rice initially will have

the larger mar"et share !y virtue of #ic"ing u# sales to the informed sho##ers.

he Ca#ra et al. =2002> #rice+com#etition game also has a uni'ue ;ash

e'uili!rium #rice at marginal cost& since at any higher #rice there is a unilateral

incentive to cut #rice !y a very small amount and #ic" u# the informed sho##ers. he

;ash e'uili!rium is inde#endent of the num!er of informed !uyers who res#ond to evensmall #rice differences. (es#ite this inde#endence #ro#erty& it is intuitively #lausi!le

that a large fraction of !uyers who are uninformed a!out #rice differences would #rovide

sellers with some #ower to raise #rices. his intuition was confirmed !y the results of

the e#eriments re#orted !y the authors. As with the traveler s dilemma& data averages

were strongly affected !y a #arameter that determines the #ayoff differential for !eing

low& even though this #arameter does not affect the uni'ue ;ash e'uili!rium at the

lowest #ossi!le decision.

-ppendi(: Bo$nded Rationa%it! in the rave%er9s Di%emma: -

SpreadsheetBased -na%!sis of the "#$i%i&ri$m "ffects of 8ois! Behavior

3ecall from Cha#ter ,0 that stochastic res#onse functions can !e used to ca#ture the

idea that the magnitudes of #ayoff differences matter. he form of these functions is

!ased on the intuitive idea that a #erson is much more li"ely to !e a!le to determine

,/5




which of two sounds is louder when the deci!el levels are not close together. n these

games& the e#ected #ayoff is the stimulus analogous to the deci!el level& so let s !egin

!y calculating e#ected #ayoffs for each decision. o "ee# the calculations from

!ecoming tedious& su##ose that there are only , #ossi!le decisions in even ten+cent

amounts: 0& 0& ,00& 200. A #erson s !eliefs will !e re#resented !y , #ro!a!ilities

that sum to ,.

hese !elief #ro!a!ilities can !e used to calculate e#ected #ayoffs for each

decision& and the logit functions introduced in the #revious cha#ter can then !e used to

determine the choice #ro!a!ilities that result from the initial !eliefs. he intuitive idea

is that a decision is more li"ely if its e#ected #ayoff is higher. n #articular& choice

#ro!a!ilities are assumed to !e increasing =e#onential> functions of e#ected #ayoffs&

normali<ed so that all #ro!a!ilities sum to ,. 6ome notation will !e useful for the

calculation of e#ected #ayoffs. 1et

P i !e the #ro!a!ility associated with claim i& so !eliefs are characteri<ed !y P 0& P 0& . P 200. ?irst consider a claim of 0& which will tie with #ro!a!ility P 0 and will !e low

with #ro!a!ility , P 0. he e#ected #ayoff is:

e#ected #ayoff for 0 J 0 P 0 =0 '>=, P 0 >.

;otice that the first term covers the case of a tie and the second covers the case of

having the low claim. ?or a claim of 0& we have to consider the chance of having the

higher claim and incurring the #enalty of '& so the e#ected #ayoff is:

e#ected #ayoff for 0 J 0 P 0 O =0 '>=, P 0 P 0 > O =0 ' P > 0.

As !efore& the first term is for the #ossi!ility of a tie at 0& and the second is for the

#ossi!ility that the other claim is a!ove 0& which occurs with #ro!a!ility , P 0 P 0.

he third term now reflects the #ossi!ility of having the higher claim. he #ayoff

structure can !e clarified !y considering a higher claim& say ,50.

e#ected #ayoff for ,50 J ,50 P ,50

=,,.,> O =,50 '>, P 0 P 0 ... P ,50

O =0 '> P 0 =0 ' P > 0 ....=,/0 ' P > ,/0.

n order from to# to !ottom& the #arts on the right side of =,,.,> corres#ond to the cases of

a tie& of having the lower claim& and of having the higher claim.

,/%



his section shows you how to do these calculations in an *cel s#readsheet that

ma"es it #ossi!le to re#eat the calculations iteratively in order to o!tain #redictions for

how #eo#le might actually !ehave in a traveler s dilemma game when we do not assume

#erfect rationality. he logic of this s#readsheet is to !egin with a column of

#ro!a!ilities for each claim =0& 0& 200>. iven these #ro!a!ilities& the s#readsheet

will calculate a column of e#ected #ayoffs& one for each claim. 1et the e#ected #ayoff

for claim i !e denoted !y Eie& where i J 0& 0& ,00& etc. hen there is a column of

e#onential functions of e#ected #ayoffs& e#=EeF>& where the e#ected #ayoffs have

!een divided !y an error #arameter F. $e want choice #ro!a!ilities to !e increasing in

e#ected #ayoffs& !ut these e#onential functions cannot !e used as #ro!a!ilities unless

they are normali<ed to ensure that they sum to ,. hus the column of e#onential

functions is summed to get the normali<ing element in the denominator of the logit

#ro!a!ility formula: e#=EeF>Gi= e#=EeiF>>& which corres#onds to e'uation =,0.5>

from the #revious cha#ter.he !est way to read this section is to o#en u# a s#readsheet. he instructions will !e

#rovided for *cel& !ut analogous instructions would wor" in other s#readsheet

#rograms. a!le ,,.2 is laid out li"e a s#readsheet& with columns la!eled A& 7& & and

rows la!eled ,& 2& .

a!le ,,.2. *cel 6#readsheet for raveler s (ilemma 1ogit 3es#onses

- B C D " 0 G 7 I K J

' F J '+

' J '+

/

3

*

6 ' J' P P 602 P0 '2 Ee

45p0 Ee ? F 2 P

) )+ + +2 0 -0 0 , 0 , -0 -0 0 20 0.,/

4 0 0 ,00 0 0 , 0 -0 -0 ,0% 0.0%-

'+ ,00 0 ,,0 0 0 , 0 -0 -0 ,0% 0.0%-

'' ,,0 ,00 ,20 0 0 , 0 -0 -0 ,0% 0.0%-

' ,20 ,,0 ,0 0 0 , 0 -0 -0 ,0% 0.0%-

'/ ,0 ,20 ,/0 0 0 , 0 -0 -0 ,0% 0.0%-'3 ,/0 ,0 ,50 0 0 , 0 -0 -0 ,0% 0.0%-

'* ,50 ,/0 ,%0 0 0 , 0 -0 -0 ,0% 0.0%-

'1 ,%0 ,50 ,-0 0 0 , 0 -0 -0 ,0% 0.0%-

') ,-0 ,%0 ,0 0 0 , 0 -0 -0 ,0% 0.0%-

,/-



Gi= ep=

E

ei<F>>


'2 ,0 ,-0 ,0 0 0 , 0 -0 -0 ,0% 0.0%-

'4 ,0 ,0 200 0 0 , 0 -0 -0 ,0% 0.0%-

+ 200 ,0 2,0 0 0 , 0 -0 -0 ,0% 0.0%-

' ' 0 '1'3+ '

a!le ,,.. Column Dey for 6#readsheet in a!le ,,.2

=formulas for row should !e co#ied down to row 20>

Column

Varia!le ;otation ?ormula for 3ow A Claim -) '+

7 8ayoff if 1ower ' -2 B

C 8ayoff if Higher J ' -2 B

( 8ro!a!ility P

'

* 8roduct P D2F-2

? Cumulative P 602 0)D2

8roduct 2 P0 '2 D2FB2

H Cumulative 8roduct 2 7) G2

*#ected 8ayoff Ee "2 @'02AFC2 7)

F *#onential of #ayoffe5p0 Ee ? F 2 e(p@I2;B'A

Cell F2, 6um of e#onentials Gi0 e5p0 Eei ? F 2

[email protected]+A

D 8ro!a!ilitye5p0 Ee ? F 2? K2;K'

Step 1$ Parameter Specification. t is convenient to have the error #arameter& F& !e at a

focal location in the u##er left corner& along with the #enaltyreward #arameter. 8ut a

value of '+ for the error #arameter in cell B'& and #ut '+ for the #enaltyreward

#arameter in cell B. hese num!ers can later !e changed to e#eriment with different

amounts of noise for different treatments.

Step $ C#aim :a#ues. 8ut a )+ in cell -)& so that the formula in -2 can !e:

s$m@-)'+A& which will yield a value of 0. hen this formula can !e co#ied to cells

A to A20 !y clic"ing on the lower right corner of the A !o and dragging it down.

=Mou could later go !ac" and e#and the ta!le to allow for all #enny amounts from 0 to200& !ut the smaller ta!le will do for now.>

,/



Step 3$ Initia# :a#ues. 7efore !eginning to fill in the other cells& #ut values of 0 in cells

?- and H-B these <eros are used to start cumulative sums& in a manner that will !e

e#lained !elow.

Step &$ 6ormu#as. ;otice that the formula that you used in ste# 2 is shown at the to# of

the far right column of a!le ,,.. he formulas for row cells of other columns are

also shown in the right column. *nter these in cells B2& C2& J2& ma"ing sure to #lace

the sym!ols in the #laces indicated. he dollar sym!ol forces the reference to stay

fied even when the formula is co#ied to another location. ?or eam#le& the references

to the error #arameter will !e 7,& with two signs& since !oth the row and location of

the reference to this #arameter must stay fied. A reference to cell 7& however& would

"ee# the column fied at 7 and allow one to co#y the formula from row to other rows.

8lease use the sym!ols only where indicated& and not elsewhere. ?inally& co#y all of

these formulas down from row to rows +20& ece#t in column D where rows +20should !e filled with <eros as shown in a!le ,,.2.

Step "$ Sums. he num!ers in column D will not ma"e sense until you #ut the formula

for the sum of e#onentials into cell K'B this formula is given in the net+to+last row of

a!le ,,.. Add a formula to sum the #ro!a!ilities and #ayoffs !y co#ying the formula

in cell K' to cells D'& "'& and J'.

At this #oint& the num!ers in your s#readsheet should match those in a!le ,,.2.

he initial #ro!a!ility column reflected a !elief that the other #erson would choose 0

with certainty. herefore the e#ected #ayoff in column is 0 if one matches this

claim. iven these !eliefs& any higher claim will result in a ,0+cent #enalty& and the

#ayoff will !e other s claim of 0& which is the minimum& minus ,0& or -0 as shown in

column . his column will #rovide e#ected #ayoffs for each decision as the initial

!elief column ( is changed& so let s loo" at the structure of the formula in column :

e#ected #ayoff for claim of 0 *

=,,.2> =,?>RC

H-.

his formula has three #arts that corres#ond to the three #arts of e'uation =,,.,>& i.e.

de#ending on whether the other s claim is e'ual to& higher& or lower than one s own

claim. $e will discuss these three elements in turn.

,/




Case of a 8ie$ n e'uation =,,.2>& the first term& "2& was calculated as the #ro!a!ility in

D2 that the other chooses 0 times the claim itself in -2. hus this first term covers the

case of a tie.

Case of Aaving the -o;er C#aim$ he second term in =,,.2> involves C2& which is the

claim #lus the reward& '& so this term #ertains to the case where one s claim is the lower

one. he =,02> term is the #ro!a!ility that the other s claim is higher& so 02 is the

#ro!a!ility that the other s claim is less than or e'ual to one s own claim of 0. 6ince

there is no chance that the other is less than 0& 02 is calculated as 0)& which has !een

set to 0& #lus the #ro!a!ility that the other s claim is eactly 0& i.e. C2. As this formula

is co#ied down the column& we get a sum of #ro!a!ilities& which can !e thought of as

the cumulative less+than+ore'ual+to #ro!a!ilities. 3etracing our ste#s& one minus the

less+than+or+e'ual+to #ro!a!ilities in column ? will !e the #ro!a!ility that the other s

claim is higher& so the second term in =,,.,> has the =, 02> !eing multi#lied !y theclaim #lus the reward.

Case of Aaving the Aigher C#aim$ Here we have to consider each of the claims that are

lower than one s own claim. Column 7 has claims with the #enalty su!tracted& and

these #ayoffs are multi#lied !y the #ro!a!ility of that claim to yield the num!ers in

column . hen column H calculates a cumulative lessthan+or+e'ual+to sum of the

elements in . his sum is necessary since& for any given claim& there may !e many

lower claims that the other can ma"e& with associated #ayoffs that must !e multi#lied !y

#ro!a!ilities and summed to get an e#ected #ayoff.

;ow that the e#ected #ayoffs are calculated in column & they are #ut into e#onentialfunctions in column F after dividing !y the error #arameter in cell B.

hese are then summed in cell K'& and the e#onentials are divided !y the sum to get

the choice #ro!a!ilities in column D. ?or eam#le& in a!le ,,.2 the #ro!a!ility

associated with a claim of 0 is a!out 0.,/& as shown in cell J2. he other

#ro!a!ilities are a!out a third as high. hese other #ro!a!ilities are all e'ual& since the

other #erson is e#ected to choose a claim of 0 for sure& and therefore all higher claims

lead to earnings of -0& as com#ared with the 0 that could !e earned !y matching the

other s e#ected claim. his reduction in e#ected #ayoff is not so severe as to #revent

deviations from ha##ening& at least for the error #arameter of ,0 in cell 7, that is !eing

used here. ry a lower error #arameter& say 5& to !e sure that it results in a higher #ro!a!ility for the claim of

0& with less chance of an error in terms of ma"ing a claim that is higher than

0.

,50



;et consider cell "'& which shows the e#ected value of the other #erson s claim for

the initial !eliefs. f you still have a #ro!a!ility of , in cell D2 for the lowest claim&

then the num!er in "' should !e 0. As you change the #ro!a!ilities in column (

=ma"ing sure that they sum to ,>& the e#ected claim in "' will change.

?inally& we can use the choice #ro!a!ilities in column D to calculate the resulting

e#ected claims. o ma"e this calculation& we essentially need to get the formula in "'

over to the right of the D column. At this #oint& you should save your s#readsheet =in

case something goes wrong> and then #erform the final two ste#s:

Step * . Highlight the shaded !loc" of cells in a!le ,,.2& i.e. from "1 to J'. hen

co#y and #aste this !loc" with the cursor in cell L1& which will essentially re#licate

columns * to D& #utting them in columns 1 to 3. ;ow you should see that the average

claim in cell L' is a!out ,& way a!ove the average claim on 0 !ased on the initial

!eliefs.

Step 7. he net ste# is to #erform another iteration. Mou could mar" the shaded !loc"

in a!le ,,.2 again =or s"i# this ste# if it is still on the cli#!oard> and then #lace the

cursor in cell S1 and co#y the !loc". he new average claim in cell S' should !e ,%%.

n two more iterations this average will converge to ,- =in cell -G'>& and this average

will not change in su!se'uent iterations.

At this #oint& there will !e a vector of #ro!a!ilities in column A with the

following interesting #ro#erty: if these re#resent initial !eliefs& then the stochastic !est

res#onses to these !eliefs will yield essentially the same #ro!a!ilities& i.e. the !eliefs will

!e confirmed. his is the notion of e'uili!rium that was introduced in cha#ter ,0. *ven

more interesting is the fact that the level of convergence& ,-& is a##roimately e'ual to

the average claim for the ' J ,0 treatment in ?igure ,,.,.

;ow try changing the #ayoff #arameter from ,0 to 50 to !e sure that the average claims

converge to 0& and chec" to !e sure that these convergence levels are not too sensitive

to changes in initial !eliefs& e.g. if you #ut a , in the !ottom =-+> element of column (

instead of in the -2 element. n this sense& the model of logit stochastic res#onses can

e#lain why !ehavior converges to the ;ash #rediction in some treatments =50 and 0>&

and why data go the other side of the set of feasi!le decisions in other treatments =5 and

,0>.

n e'uili!rium& !eliefs are confirmed& and the result is called a logit e'uili!rium&which was used !y Ca#ra et al. =,> to evaluate data from the e#eriment. Iur

s#readsheet calculations in this section have !een !ased on an error rate of ,0& which is

close to the value of . that was estimated from the actual data& with a standard error of

0.5.

,5,




<$estions

,. 6how that une'ual claims cannot constitute a ;ash e'uili!rium in a traveler s

dilemma.

2. $hat is the ;ash e'uili!rium for the traveler s dilemma game where there is a 5

#enalty and a 5 reward& and claims must !e !etween 50 and 50N =A

negative claim means that the traveler #ays the airline& not the reverse.>

. Consider a traveler s dilemma with F #layers who each lose identical items& and

the airline re'uests claim forms to !e filled out with the understanding that

claims will !e !etween 0 and 200. f all claims are not e'ual& there is a 5

#enalty if one s claim is not the lowest and a 5 reward rate for the #erson with

the lowest claim. 6#eculate on what the effect on average claims of increasing F from 2 to / #layers in a setu# with re#eated random matchings.

/. Calculate what the earnings would have !een for the D s'uared team if they had

chosen a claim of 200 in each of the first four rounds of the game summari<ed in

ta!le ,,.,.

5. Use the s#readsheet constructed in the A##endi to show that the choices

converge to the ;ash level of 0 when the error rate is reduced to , for the ' J

,0 treatment.

Cha#ter ,2. Coordination amesA ma9or role of management is to coordinate decisions so that !etter outcomes can !e

achieved. he game considered in this cha#ter highlights the need for such eternal

management& since otherwise #layers may get stuc" in a situation where no!ody eerts

much effort !ecause others are not e#ected to wor" hard either. n these games& the

#roductivity of each #erson s effort de#ends on that of others& and there can !e multi#le

;ash e'uili!ria& each at a different common effort level. 7ehavior is sensitive to factors

li"e changes in the effort cost or the si<e of the grou#& even though these have no effect

on the set of ;ash e'uili!ria. he e#eriment can !e run with the Veconla!

Coordination ame #rogram that can !e found on the ames menu. Alternatively& the

matri game with - #ossi!le effort decisions discussed !elow is im#lemented !y thedefault #arameters for the large matri game #rogram& the ;H ; 4atri ame #rogram.

he instructions in the a##endi are for the matri+game version.

,52



I. he Minim$m "ffort Game hat9s 5ne I Can P%a!NE

4ost #roductive #rocesses involve s#eciali<ed activities& where distinct individuals or

teams assem!le se#arate com#onents that are later com!ined into a final #roduct. f this

#roduct re'uires one of each of the com#onents& then the num!er of units finally sold is

sensitive to !ottlenec"s in #roduction. ?or eam#le& a marine #roducts com#any that

#roduces ,00 hulls and 0 engines will only !e a!le to mar"et 0 !oats. hus there is a

!ottlenec" caused !y the division with the lowest out#ut. 6tudents =and #rofessors too>

have little trou!le understanding the incentives of this ty#e of minimum+effort game& as

one student s comment in the section title indicates.

he minimum+effort game was originally discussed !y 3ousseau in the contet of a stag

hunt& where a grou# of hunters form a large circle and wait for the stag to try to esca#e.

he chances of "illing the stag de#end on the watchfulness and effort eerted !y the

encircling hunters. f the stag o!serves a hunter to !e na##ing or hunting for smaller

game instead& then the stag will attem#t an esca#e through that sector. f the stag is a!leto 9udge the wea"est lin" in the circle& then the chances of esca#e de#end on the

minimum of the hunters efforts. he other as#ect of the #ayoffs is that effort is

individually costly& e.g.& in terms of giving u# the chance of !agging a hare or ta"ing a

rest.

he game in a!le ,2., is a 22 version of a minimum effort game. ?irst& consider the

lower+left corner where each #erson has a low effort& and #ayoffs are -0 each. f the

3ow #layer increases effort& while the Column #layer maintains low efforts& then the

relevant 3ow #ayoff is reduced to %0& as can !e seen from the to#+left !o. hin" of it

this way: the unilateral increase in effort will not increase the minimum& !ut the etra

cost reduces 3ow s #ayoff form -0 to %0& so the cost of this etra unit of effort is ,0.

a!le ,2.,. A 4inimum *ffort ame =3ow s #ayoff& Column s #ayoff>

Column 8layer:

3ow 8layer: 1ow *ffort =,> High

*ffort =2>

High *ffort =2>

1ow *ffort =,>

;ow su##ose that the initial situation is a high effort for 3ow and a low effort for

Column& as in the u##er+left !o. An increase in Column s effort will raise the

minimum& which raises 3ow #ayoff from %0 to 0. hus a unit increase in the minimum

,5

%0& -0 ( ) 0& 0

-0& -0 ' & -0&

%0



3 5421 10 5030 70 11090 130

20 60 80 10040 120 12030 7050 90 110 110 110

6040 80 100 10010010050 70 90 9090 90 9060 80 80 808080 8070 70 70 707070 70


effort will raise #ayoffs !y 20& holding one s own effort constant. hese o!servations

can !e used to devise a mathematical formula for #ayoffs that will !e useful in devising

more com#le games. 1et the low effort !e , and the high effort !e 2& as indicated !y

the num!ers in #arentheses net to the row and column la!els in the #ayoff ta!le. hen

the #ayoffs in this ta!le are determined !y the formula: %0 O =20 times the minimum

effort> W =,0 times one s own effort>. 1et M denote the minimum effort and 4 denote

one s own effort& so that this formula is:

=,2.,> own #ayoff %0 20 M ,0 . 4

?or eam#le& the u##er right #ayoffs are determined !y noting that !oth efforts are 2& so

the minimum = M > is 2& and the formula yields: %0 O =20>R2 =,0>2 J 0.

he formula in =,2.,> was used to construct the #ayoffs in a!le ,2.2& for the

case of efforts that can range from , to -. ;otice that the four num!ers in the !ottom+left corner of the ta!le corres#ond to the 3ow #ayoffs in a!le ,2., a!ove. $ith a larger

num!er of #ossi!le effort levels& we see the dramatic nature of the #otential gains from

coordination on high+effort outcomes. At a common effort of -& each #erson earns ,0&

which is almost twice the amount earned at the lowest effort level. 4ovements along

the diagonal from the lower+left to the u##er+right corner show the !enefits from

coordinated increases in effortB a one unit increase in the minimum effort raises #ayoff

!y 20& minus the cost of ,0 for the increased effort& so each diagonal #ayoff is ,0 larger

than the one lower on the diagonal.

7esides the gains from coordination& there is an additional feature of a!le ,2.2

that is related to ris". $hen the row #layer chooses the lowest effort =in the !ottom

row>& the #ayoff is -0 for sure& !ut the highest effort may yield #ayoffs that range from

,0 to ,0& de#ending on the column #layer s choice. his is the strategic dilemma in

this coordination game: there is a large incentive to coordinate on high efforts& !ut the

higher effort decisions are ris"y.

a!le ,2.2. he Van Huyc" et al. =,0> 4inimum *ffort ame

$ith 3ow 8layer s 8ayoffs (etermined !y the 4inimum of Ithers *fforts

Column s *ffort =or 4inimum of Ither *fforts>

%-

-

%

,5/



3ow s 5

*ffort /

2

,

An additional element of ris" is introduced when more than two #eo#le are

involved. 6u##ose that #ayoffs are still determined !y the formula in =,2.,>& where M is

the minimum of all #layers efforts. he result is still the #ayoff ta!le ,2.2& where the

#ayoff num!ers #ertain to the row #layer as !efore& !ut where the columns corres#ond to

the minimum of the other #layer s efforts. ?or eam#le& the #ayoffs are all -0 in the

!ottom row !ecause 3ow s effort of , is the minimum regardless of which column is

determined !y the minimum of the others efforts. ;otice that the incor#oration of larger num!ers of #layers into this minimumeffort game does not alter the essential strategic

dilemma& i.e. that high efforts involve high #otential gains !ut more ris". At a dee#er

level& however& there is more ris" with more #layers& since the minimum of a large

num!er of inde#endently selected efforts is li"ely to !e small when there is some

variation from one #erson to another. his is analogous to having a large num!er of

hunters s#read out in a large circle& which gives the stag more of a chance to find a

sector where one of the hunters is a!sent or na##ing.

II. 8ash "#$i%i&ria, 8$m&ers "ffects, and "(perimenta% "vidence

hese intuitive considerations =the gains from coordination and the ris"s of un+

matched high efforts> are not factors in the structure of the ;ash e'uili!ria in this game.

Consider a!le ,2.,& for eam#le. f the other #erson is going to eert a low effort& the

!est res#onse is a low effort that saves on effort cost. hus the lower+left outcome& low

efforts for each& is a ;ash e'uili!rium. 7ut if the other #layer is e#ected to choose a

high effort& then the !est res#onse is a high effort& since the gain of 20 for the increased

minimum eceeds the cost of ,0 for the additional unit of effort. hus the high+effort

outcome in the u##er+right corner of a!le ,2., is also a ;ash e'uili!rium. his is an

im#ortant feature of the coordination game: there are multi#le e'uili!ria& with one that is

#referred to the other=s>.

he #resence of multi#le e'uili!ria is a feature that differentiates this game froma #risoner s dilemma& where all #layers may #refer the high+#ayoff outcome that results

from coo#erative !ehavior. his high+#ayoff outcome is not a ;ash e'uil!rium in a

#risoner s dilemma since each #erson has a unilateral incentive to defect. he

e'uili!rium structure for the game in a!le ,2., is not affected !y adding additional

,55




#layers& each choosing !etween efforts of , and 2& and with the row #layer s #ayoffs

determined !y the column that corres#onds to the minimum of the others efforts. n this

case& adding more #layers does not alter the fact that there are two e'uili!ria =in non+

random strategies>: all choose low efforts or all choose high efforts. Fust restricting

attention to the ;ash e'uili!ria would mean ignoring the intuition that adding more

#layers would seem to ma"e the choice of a high effort ris"ier& since it is more li"ely that

one of the others will choose low effort and #ull the minimum down.

he #ro!lem of multi#le e'uili!ria is more dramatic with more #ossi!le efforts&

since any common effort is a ;ash e'uili!rium in a!le ,2.2. o see this& #ic" any

column and notice that the row #layer s #ayoffs are highest on the shaded diagonal. As

!efore& this structure is inde#endent of changes in the num!er of #layers& since such

changes do not alter the #ayoff ta!le. hese considerations were the !asis of an

e#eriment conducted !y Van Huyc" et al. =,0>& who used the #ayoffs from a!le ,2.2

for small grou#s =si<e 2> and large grou#s =si<e ,/+,%>. he large grou#s #layed thesame game ten consecutive times with the same grou#& and the lowest effort was

announced after each round to ena!le all to calculate their #ayoffs =in #ennies>. *ven

though a ma9ority of individuals selected high efforts of % or - in the first round& the

minimum effort was no higher than / in the first round for any large grou#. $ith a

minimum of /& higher efforts were wasted& and effort reductions followed in su!se'uent

rounds. he minimum fell to the lowest level of , in all large grou#s& and almost all

decisions in the final round were at the lowest level.

his e#eriment is im#ortant since #reviously it had !een a common #ractice in

theoretical analysis to assume that individuals could coordinate on the !est ;ash

e'uili!rium when there was general agreement a!out which one is !est& as is the case for

a!le ,2.2. n contrast& the su!9ects in the e#eriment managed to end u# in thee'uili!rium that is ;orst for all concerned. $ith grou#s of si<e 2& individuals were a!le

to coordinate on the highest effort& ece#t when #airings were randomly reconfigured in

each round. $ith random matching& the outcomes were varia!le& with average efforts in

the middle range. n any case& it is clear that grou# si<e had a large im#act on the

outcomes& even though changes in the num!ers of #layers had no effect on the set of

e'uili!ria.

he coordination failures for large grou#s ca#tured the attention of

macroeconomists& who had long s#eculated a!out the #ossi!ility that whole economies

could !ecome mired in low+#roductivity states& where #eo#le do not engage in high

levels of mar"et activity !ecause no one else does. he macroeconomic im#lications of

coordination games are discussed in 7ryant =,>& Coo#er and Fohn =,>& and 3omer

=,%>& for eam#le.

,5%



III. "ffort?Cost "ffects

;et consider what ha##ens when the cost of effort is altered. ?or eam#le& su##ose

that the effort cost of ,0 used to construct a!le ,2., is raised to ,& so that the #ayoffs

=for an effort of * and a minimum effort of M > will !e determined !y: %0 O 20 M + , 4 .

n this case& a one unit increase in each #erson s effort raises #ayoffs !y 20 minus the

cost of ,& so the #ayoffs in the u##er+right !o of a!le ,2. are only , cent higher than

the #ayoffs in the lower+left !o. ;otice that this increase in effort cost did not change

the fact that there are two ;ash e'uili!ria& and that !oth #layers #refer the high+effort

e'uili!rium. However& sim#le intuition suggests that effort levels in this game may !e

affected !y effort costs. ?rom the row #layer s #ers#ective& the to# row offers a #ossi!le

gain of only one cent and a #ossi!le loss of , cents& as com#ared with the !ottom row.

a!le ,2.. A 4inimum+*ffort ame $ith High *ffort Cost=3ow s #ayoff& Column s #ayoff>

Column 8layer:

3ow 8layer: 1ow *ffort =,> High *ffort =2>

High *ffort =2>

1ow *ffort =,>

oeree and Holt =,& 200,> re#ort e#eriments in which the cost of effort is varied

!etween treatments& using the #ayoff formula:

=,2.2> own #ayoff J M c4 &

where M is the minimum of the efforts& 4 is one s own effort& and c is a cost #arameter

that is varied !etween treatments. *ffort decisions were restricted to !e any amount

!etween =and including> ,.,0 and ,.-0. As !efore& any common effort is a ;ash

e'uili!rium in this game& as long as the cost #arameter& c& is !etween 0 and ,. his is

!ecause a unilateral decrease in effort !y one unit will reduce the minimum !y ,& !ut it

will only reduce the cost !y an amount that is less than ,. herefore& a unit decrease in

effort will reduce one s #ayoff !y , c. Conversely& a unilateral increase in effort !y one

unit a!ove some common level will not raise the minimum& !ut the #ayoff will fall !y c.

*ven though deviations from a common effort are un#rofita!le when c is greater than 0

and less than ,& the magnitude of c determines the relative cost of errors in either

,5-

/2& %, %2& %2

%,& %, %,& /2




direction. A large value of c& say 0.& ma"es increases in effort more costly& and a small

value of c ma"es decreases more costly.

?igure ,2., shows the results for sessions that consisted of ,0+,2 su!9ects who

were randomly #aired for a series of ,0 rounds. here were three sessions with a low

effort cost #arameter of 0.25& and the averages !y round for these sessions are #lotted as

thin dashed lines. he thic" dashed line is the average over all three sessions of this

treatment. 6imilarly& the thin solid lines #lot round+!yround averages for the three

sessions with a high effort cost #arameter of 0.-5& and the thic" solid line shows the

average for this treatment.

*fforts in the first round average in the range from ,.5 to ,.50& with no se#aration

!etween treatments. 6uch se#aration arises after several rounds& and average efforts in

the final round are ,.%0 for the low+cost treatment& versus ,.25 for the high+cost

treatment. hus we see a strong cost effect& even though any common effort is a ;ash

e'uili!rium.Ine session in each treatment seemed to a##roach the !oundary& which raises the

issue of whether !ehavior will loc" on one of the etremes. his #attern was o!served

in a Veconla! classroom e#eriment in which efforts went to ,.-0 !y the ,0 th #eriod.

his "ind of etreme !ehavior is not universal& however. oeree and Holt =,> re#ort

a #air of sessions that were run for 20 rounds. $ith an effort cost of 25 cents& the

decisions converged to a!out ,.55& and with an effort cost of -5 cents the decisions

leveled off at a!out ,.. 7oth of these outcomes seem to fit the #attern seen in ?igure

,2.,.

he s#readsheet+!ased analysis of noisy !ehavior for the ravelers (ilemma&

#resented in the a##endi to the #revious cha#ter& can !e ada#ted to the coordination

game. he ste#s for constructing this new s#readsheet are #rovided in the A##endi atthe end of this cha#ter. As !efore& the #ur#ose of this analysis is to show how some

randomness in individual decisions can result in data #atterns =for num!ers and effort+

cost effects> that are consistent with those o!served in the e#eriments& even though

these data #atterns are not #redicted on the !ase of an analysis of the ;ash e'uili!ria for

the game.

,5



@;er a*e :# ort

3ound

,5



0ig$re '.'. -verage "fforts for the Goeree and 7o%t @'444A Coordination "(periment:

hick Lines are -verages 5ver hree Sessions for "ach Cost reatment

I. "(tensions

oeree and Holt =,> also considered the effects of raising the num!er

of #layers #er grou# from 2 to & holding effort cost constant& which resulted in ashar# reduction of effort levels. *ffort+cost effects were o!served as well in games

in which #ayoffs were determined !y the median of the three efforts. 6ome

coordination games may !e #layed only onceB oeree and Holt =200,> re#ort

strong effort+cost effects in such games as well. A theoretical analysis of these

effects =in the contet of a logit e'uili!rium> can !e found in Anderson& oeree&

and Holt =200,!>.

here is an interesting literature on factors that facilitate coordination on good

outcomes in matri games& e.g. 6efton =,>& 6trau! =,5>& and Ichs =,5>.

n #articular& notions of ris" dominance and #otential #rovide good #redictions

a!out !ehavior when there are multi#le e'uili!ria. Anderson& oeree& and Holt

=200,!> introduce the more general notion of stochastic potentia# and use it to

e#lain results of coordination game e#eriments. 4odels of learning and

evolution have also !een widely used in the study of coordination games =see

references in Ichs& ,5>.

-ppendi(: -n -na%!sis of 8ois! Behavior in the Coordination Game

he traveler s dilemma and the coordination game have similar #ayoff structures&

so it is straightforward to modify the s#readsheet in a!le ,,.2 so that it a##lies to

the coordination game. Here are the ste#s:

Step 1. 6ave the traveler s dilemma s#readsheet under a different name& e.g.

cg.ls& !efore deleting all information in column 1 and farther to the right.

Step . he coordination e#eriment has fewer #ossi!le decisions& when

restricted to ,0+cent increments: ,,0& ,20& ,-0. herefore& delete all material in

rows -+,0& and in rows ,+20& leaving row 2, as it is. Mou will have to enter the

#ossi!le effort levels in column A: ''+ in -''& '+ in -'& etc.

Step 3. here is no #enaltyreward #arameter in the coordination game& so enter a

+ in cell B. ;et #ut C in cell -/& and enter a value of +.)* in cell B/.

Step &. he starting values for the cumulative column sums will have to !e

moved& so #ut values of + in cells 0'+ and 7'+.



Step ". o set the initial #ro!a!ilities& change the entry in cell D'' to '& and leave

the other entries in that column to !e 0.

Step *. ;et we have to su!tract the effort cost from the e#ected #ayoff formula

in cell I''. he cost #er unit effort in cell 7 must !e multi#lied !y the effort incolumn A to calculate the total effort cost. hus you should append the term B/F-'' to the end of the formula that is already in cell I''. hen this

modified formula should !e co#ied into rows ,2 to ,- in this column.

Step 7. At this #oint& the num!ers you have should !e essentially the same as

those in a!le ,2./ =which have !een truncated>& with a new vector of

#ro!a!ilities in column D. o #erform the first iteration& mar" the !loc" from

"'+ to J'& co#y& and #lace the cursor in cell 1,0 !efore #asting. his same

#rocedure can !e re#eated ,0 times& so that the average effort after ,0 iterations

will !e found on the far right side of the s#readsheet& in cell B'.

a!le ,2./. *cel 6#readsheet for Coordination ame 1ogit 3es#onses

- B C D " 0 G 7

I

K J

' F J '+

' J +

/ C J +.)*

3

*

6 P P 602 P Ee

45p0 Ee ? F 2 P

)

2

4

'+,00 + +

'' ,,0 ,,0 ,,0 , ,,0 , ,,0 ,,0 2-.5 ,5.%/ 0.50

' ,20 ,20 ,20 0 0 , 0 ,,0 20 -. 0.250

'/ ,0 ,0 ,0 0 0 , 0 ,,0 ,2.5 ./ 0.,,

'3 ,/0 ,/0 ,/0 0 0 , 0 ,,0 5 ,.%/ 0.05%

'* ,50 ,50 ,50 0 0 , 0 ,,0 +2.5 0.-- 0.02%'1 ,%0 ,%0 ,%0 0 0 , 0 ,,0 +,0 0.% 0.0,2

') ,-0 ,-0 ,-0 0 0 , 0 ,,0 +,-.5 0.,- 0.00%



'2

'4

+

'' ,,0 34.34'

'

he e#ected effort after one iteration& found in 12,& is a!out ,,. his

rises to ,25 !y the fifth iteration. $hen the initial #ro!a!ilities in column ( are

changed to #ut a #ro!a!ility of , for the highest effort and 0 for all other efforts&

the iterations converge to a!out ,25 !y the ,0 th iteration. $hen the cost of effort

is reduced from 0.-5 to 0.25& the average claims converge to a!out ,55 after

several iterations. hese average efforts are 'uite close to the levels for the two

treatment averages in ?igure ,2., in round ,0. As noted in the #revious cha#ter&

the convergence of iterated stochastic res#onses indicates an e'uili!rium in which

a given set of !eliefs a!out others effort levels will #roduce a matching

distri!ution of effort choice #ro!a!ilities. he resulting effort averages are 'uite

good #redictors of !ehavior in the minimum+effort e#eriment when the error rate

is set to ,0. *ven though any common effort is a ;ash e'uili!rium& intuition

suggests that lower effort costs may #roduce higher efforts. hese s#readsheet

calculations of the logit e'uili!rium are consistent with this intuitionB they e#lain

why efforts are inversely correlated with effort costs. A lower error rate will #ush

effort levels to one or another of the etremes& i.e. to ,,0 or to ,-0 ='uestion >.

<$estions

,. Consider the game in the ta!le !elow& and find all ;ash e'uili!ria in #ure

strategies. s this a coordination gameN

Column 8layer:

3ow 8layer: 1eft 4iddle ;on+;ash 3ight

=2%K> =K> =%K>

=0K>

o#

=%K>

7ottom

=2K>

200& 50 0& /5 ,0& 0 20& 250

0& 250 ,0&

,00

0& 0 50& /0



2. he num!ers under each decision la!el in the a!ove ta!le show the

#ercentage of #eo#le who chose that strategy when the game was #layed

only once =oeree and Holt& 200,>. $hy is the column #layer s most

commonly used strategy la!eled as ;on+;ash in the ta!leN =;ote: it was

not given this la!el in the e#eriment.> Con9ecture why the ;on+;ash

decision is selected so fre'uently !y column #layersN

. Use the s#readsheet for the minimum+effort coordination game to fill in

average efforts for the following a!le. Use initial !elief #ro!a!ilities of

, =or 0.,25> for each of the elements in column ( !etween rows ,, to

,-. hen discuss the effects of the error #arameter on these #redictions.

Average *ffort 8redictions

F J , F J 5 F J ,0 F J 20

High Cost

=c J 0.-5>

1ow Cost

=c J 0.25>

/. Consider the game shown !elow& which gives the Column 8layer a safe

decision& 6. 3ow s #ayoffs de#end on a #arameter that changes from

one treatment to the other. oeree and Holt =200,> re#ort that themagnitude of affects the fre'uency with which the 3ow 8layer chooses

o#& so the issue is whether this effect is #redicted in theory. ?ind all

;ash e'uili!ria in #ure and mied strategies for J 0& and for J /00&

under the assum#tion that each #erson is ris" neutral.

Column 8layer:

3ow 8layer: 1 3 6

o#

7ottom

0& 0 0& 0 & /0

0& 0 ,0& ,0 0 & /0



5. =Advanced> ?ind the mied+strategy e'uili!rium for the game shown in

'uestion ,. =Hint: n this e'uili!rium& column randomi<es !etween 1eft

and 4iddle. he #ro!a!ilities associated with ;on+;ash and 3ight are

<ero& so consider the truncated 22 game involving only 1eft and 4iddle

for column and o# and 7ottom for row. ?inally& chec" to !e sure that

column is not tem#ted to deviate to either of the two strategies not used.>



8art V. 4ar"et *#eriments

he games in this #art re#resent many of the standard mar"et models that are

used in economic theory. n a monopo#& there is a single seller& whose

#roduction decision determines the #rice at which the out#ut can !e sold. hismodel can !e generali<ed !y letting firms choose 'uantities inde#endently& where

the aggregate 'uantity determines the mar"et #rice. his Cournot setu# is widely

used in economic theory& and it has the intuitive #ro#erty that the e'uili!rium

#rice outcomes are decreasing in the total num!er of sellers. he e#eriment

discussed in Cha#ter , im#lements a setu# with linear demand and constant cost&

which #ermits an analysis of the effects of changing the num!er of com#etitors&

including mono#oly. he Cournot model may !e a##ro#riate when firms

#recommit to #roduction decisions& !ut in many situations it may !e more realistic

to model firms as choosing #rices inde#endently.

4any im#ortant as#ects of a mar"et economy #ertain to mar"ets of intermediate

goods along a su##ly chain. Cha#ter ,/ considers e#eriments !ased on a very

sim#le case of a mono#oly manufacturer selling to a mono#oly retailer. ?rictions

and distortions = the !ullwhi# effect > that may arise along longer su##ly chains

are also discussed.

Cha#ters ,5 and ,% #ertain to #rice com#etition& im#erfections& and the effects of

alternative trading institutions when !uyers are not simulated. 7uyer and seller

activities are essentially symmetric in the dou+#e auction, ece#t that !uyers tend

to !id the #rice u#& and sellers tend to undercut each other s #rices. All !ids and

as"s are #u!lic information& as is the transaction #rice that result when someone

acce#ts the #rice #ro#osed !y someone on the other side of the mar"et. his

strategic symmetry is not #resent in the postedoffer auction& where sellers #ost #rices inde#endently& and !uyers are given a chance to #urchase at the #osted

#rices =further !argaining and discounting is not #ermitted>. he #osted offer

mar"et resem!les a retail mar"et& with sellers #roducing to order and selling at

catalogue or list #rices. n contrast& the dou!le auction more closely resem!les a

com#etitive o#en outcry mar"et. Collusion and the eercise of mar"et #ower

cause larger distortions in mar"ets with #osted #rices& although these effects may

!e diminished if sellers are a!le to offer secret discounts.

he effects of asymmetric information a!out #roduct 'uality are considered in

Cha#ter ,-. $hen sellers can select a 'uality grade and a #rice& the outcomes

may !e 'uite efficient under full information condition& !ut 'ualities fall to low

levels when !uyers are una!le to o!serve the 'uality grade #rior to #urchase.4any mar"ets are for commodities or assets that #rovide a stream of !enefits

over time. ?or eam#le& a share of stoc" may #ay dividends over time.

4ar"ets for assets are com#licated !y the fact that ownershi# #rovides two

#otential sources of value& i.e. the !enefits o!tained each #eriod =services&



dividends& etc.> and the ca#ital gain =or loss> in the value of the asset. Asset

values may !e affected !y mar"et fundamentals li"e dividends and the

o##ortunity cost of money =the interest that could !e earned in a safe account>. n

addition& values can !e affected !y e#ectations a!out future value& and such

e#ectations+driven values can cause trading #rices to deviate from levels

determined !y mar"et fundamentals. he final cha#ter in this section summari<es

some e#erimental wor" on such asset mar"ets where #rice !u!!les and crashes

are often o!served.



Cha#ter ,. 4ono#oly and Cournot 4ar"ets

7y restricting #roduction 'uantity& a mono#olist can ty#ically o!tain a higher

#rice& and #rofit maimi<ation involves finding a !alance !etween the desire to

charge a high #rice and the desire to maintain a reasona!le sales 'uantity.6u!9ects in la!oratory e#eriments with simulated !uyers are a!le to ad9ust

'uantity so that #rices are at or near the #rofit+maimi<ing mono#oly level. his

ty#e of e#eriment can !e done with the Veconla! Cournot #rogram !y setting the

num!er of sellers to !e ,. his seller selects a 'uantity that determines the

resulting #rice. he setu# allows #rice to !e su!9ect to random shoc"s& which

adds interest and realism.

he analysis of mono#oly #rovides a natural !ridge to the most widely used

model of the interaction of several sellers =oligo#oly>. he "ey !ehavioral

assum#tion of this model is that each seller ta"es the others 'uantity choices as

given and fied& and then !ehaves as a mono#olist maimi<ing #rofits for the

resulting residual demand. he #o#ularity of the Cournot model is& in #art& due to

the intuitive #rediction that the mar"et #rice will decrease from mono#oly levels

toward com#etitive levels as the num!er of sellers is increased& and this tendency

can !e verified with the we!+!ased Cournot #rogram& or with the hand+run

version that is given in the a##endi.

I. Monopo%!

A mono#olist is defined as !eing a sole seller in a mar"et& !ut the general model

of mono#oly is central to the analysis of antitrust issues !ecause it can !e a##lied

more widely. ?or eam#le& su##ose that all sellers in a mar"et are somehow a!leto collude and set a #rice that maimi<es the total #rofit& which is then divided

among them. n this case& the mono#oly model would !e relevant& either for

#roviding a #rediction of #rice and 'uantity& or as a !enchmar" from which to

measure the success of the cartel.

he mono#oly model is also a##lied to the case of one large firm and a num!er

of small fringe firms that !ehave com#etitively =e#anding out#ut as long as the

#rice that they can get is a!ove the cost of each additional unit of out#ut>. he

!ehavior of the firms in the com#etitive fringe can !e re#resented !y a su##ly

function& which shows the 'uantity #rovided !y these firms in total as a function

of the #rice. $hether or not a grou# of small firms can !e counted u#on to

!ehave com#etitively is a !ehavioral issue that might !e investigated with ane#eriment. 1et this fringe su##ly function !e re#resented !y 6= P >& which is

increasing in #rice if marginal costs are increasing for the fringe firms. hen the

residua# demand is defined to !e the mar"et demand& (= P >& minus the fringe

su##ly& so the residual demand is 3= P > J (= P > 6= P >. his residual demand



function indicates a relationshi# !etween #rice and the sales 'uantity that is not

ta"en !y the fringe firmsB this is the 'uantity that can !e sold !y the large

dominant firm. n this situation& it may !e a##ro#riate to treat the dominant firm

as a mono#olist facing a demand function < J 3= P >.

?rom the mono#olist s #oint of view& the demand =or residual demand> function

reveals the amount that can !e sold for each #ossi!le #rice& with high #rices

generally resulting in lower sales 'uantities. t is useful to invert this demand

relationshi# and thin" of #rice as a function of 'uantity& i.e. selling a larger

'uantity will reduce #rice. ?or eam#le& a linear inverse demand function would

have the form: P J A 7<& where A is the vertical interce#t of demand in a gra#h

with #rice on the vertical ais& and 7 is the slo#e& with 7 P 0. he e#eriments

to !e discussed in this cha#ter all have a linear inverse demand& which for

sim#licity& will !e referred to as the demand function.

he left side of ?igure ,., shows the results of a la!oratory e#eriment in which

each #erson had the role of a mono#oly seller in a mar"et with a constant cost of

, #er unit and a linear demand curve: P J , <& where P is #rice and < is the

'uantity selected !y the mono#olist. 6ince the slo#e is minus one& eachadditional unit of out#ut raises the cost !y , and reduces the #rice !y ,. he

vertical ais in the figure is the average of the 'uantity choices made !y the

#artici#ants in the e#eriment. t is a##arent that the #artici#ants 'uic"ly settle on

a 'uantity of a!out %& which is the #rofit+maimi<ing choice& as will !e verified

net.

0ig$re '/.'. Monopo%! ers$s D$opo%!

he demand curve used in the e#eriment is also shown in the to# two rows of

a!le ,.,. ;otice that as #rice in the second row decreases from ,2 to ,, to ,0&



the sales 'uantity increases from , to 2 to . he total revenue& P<& is shown in

the third row& and the total cost& which e'uals 'uantity& is given in the fourth row.

8lease ta"e a minute to fill in the missing elements in these rows& and to su!tract

cost from revenue to o!tain the #rofit num!ers that should !e entered in the fifth

row. (oing this& you should !e a!le to verify that #rofit is maimi<ed with a

'uantity of % and a #rice of -.

a!le ,., 4ono#oly with 1inear (emand and Constant Cost

Luantity , 2 / 5 % - ,0 ,, ,2

8rice ,2 ,, ,0 - % 5 / 2 ,

3 ,2 22 0

C , 2

8rofit ,, 20 2-

43 ,2 ,0

4C , , ,

*ven though the #rofit calculations are straightforward& it is instructive to

consider the mono#olist s decision as 'uantity is increased from , to 2& and then

to & while "ee#ing an eye on the effects of these increases on revenue and cost at

the margin. he first unit of out#ut #roduced yields a revenue of ,2& so the

marginal revenue of this unit is ,2& as shown in the 43 row at the left. An

increase from L J , to L J 2 raises total revenue from ,2 to 22& which is an

increase of ,0& as shown in the 43 row. hese additional revenues for the first

and second units are greater than the cost increases of , #er unit& so the increases

were 9ustified. ;ow consider an increase to an out#ut of . his raises revenuefrom 22 to 0& an increase of & and this marginal revenue is again greater than the

marginal cost of ,. ;otice that #rofit is going u# as long as the marginal revenue

is greater than the marginal cost& a #rocess that continues until the out#ut reaches

the o#timal level of %& as you can verify.

Ine thing to notice a!out the 43 row of ta!le ,., is that each unit increase in

'uantity& which reduces #rice !y ,& will reduce marginal revenue !y 2& since

marginal revenue goes from ,2 to ,0 to & etc. his fact is illustrated in ?igure

,.2& where the demand line is the outer thic" line& and the marginal revenue line

is the thic" dashed line. he marginal revenue line has a slo#e that is twice as

negative as the demand line. he 43 line intersects the hori<ontal marginal cost

line at a 'uantity of %. hus the gra#h illustrates what you will see when you fillout the ta!le& i.e. that marginal revenue is greater than marginal cost for each

additional unit& the ,st& 2nd& rd& /th& 5th& and %th& !ut the marginal revenue of

the -th unit is !elow marginal cost& so that unit should not !e added. =he

marginal revenues in the ta!le will not match the num!ers on the gra#h eactly&



since the marginal revenues in the ta!le are for going from one unit u# to the net.

?or eam#le& the marginal revenue in the sith column of the ta!le will !e 2&

which is the increase in revenue from going from 5 to % units. hin" of this as

5.5& and the marginal revenue for 5.5 in the figure is eactly 2.>

n addition to the ta!le and the gra#h& it is useful to redo the same derivation of

the mono#oly 'uantity using sim#le calculus. =A !rief review of the needed

calculus formulas is #rovided in the a##endi to this cha#ter.> 6ince demand is:

P J , <& it follows that total revenue& P<& is =, <>< & which is a 'uadratic

function of out#ut: ,< <2. 4arginal revenue is the derivative of total revenue&

which is , 2<& which is a line that starts at a value of , when < J 0 and

declines !y 2 for each unit increase in 'uantity& as shown in ?igure ,.2. 6ince

marginal cost is ,& it follows that marginal revenue e'uals marginal cost when ,

J , 2<& ie. when L J %& which is the mono#oly out#ut for this mar"et.

0 , 2 / 5 % - ,0 ,, ,2 ,

'uantity

0ig$re '/.. Monopo%! Profit Ma(imiOation

he case against mono#oly can !e illustrated in ?igure ,.2& where the mono#oly

#rice is - and the marginal cost is only ,. hus !uyers with valuations !etween

, and - would !e willing to #ay amounts that cover cost& !ut the mono#oly

does not satisfy this demand& in order to "ee# #rice and earnings high. As shown

in Cha#ter 2& the loss in !uyers value can !e measured as the area under the

demand curve& and the net loss is o!tained !y loo"ing at the area !elow demand&to the right of % and a!ove the marginal cost line in the figure. his triangular

area& !ounded !y light gray lines& is a measure of the welfare cost of mono#oly& as



com#ared with the com#etitive outcome where #rice is e'ual to marginal cost

=,> and the 'uantity is ,2.

t is im#ortant to mention that the convergence of the 'uantity to the mono#oly

#rediction on the left side of ?igure ,., was o!served in a contet where the

demand side of the mar"et was simulated. his "ind of e#eriment is #ro!a!ly

a##ro#riate if one is thin"ing of a mar"et with a very large num!er of consumers&

none of whom have any significant si<e or #ower to !argain for reductions from

the mono#oly #rice levels.

II. Co$rnot D$opo%!

;et consider what would ha##en if a second firm were to enter the mar"et that

was considered in the #revious section. n #articular& su##ose that !oth firms

have constant marginal costs of ,& and that they each select an out#ut 'uantity&

with the #rice then !eing determined !y the sum of their 'uantities& using the to#

two rows of a!le ,.,. his duo#oly structure was the !asis for the second #art

of the e#eriment summari<ed in ?igure ,.,. ;otice that the 'uantity #er seller starts out at the mono#oly level of % in round & which results in a total 'uantity of

,2 and forces #rice down to , =J , + ,2>. $hen #rice is ,& which e'uals the

cost #er unit& it follows that earnings are <ero. he average 'uantity is o!served to

fall in round ,0B which is not sur#rising following a round with <ero #rofit. he

incentive to cut out#ut can !e seen from the gra#h in ?igure ,.2. 6u##ose that

one seller =the entrant> "new the other would #roduce a 'uantity of %. f the

entrant were to #roduce 0& the #rice would stay at the mono#oly level of -. f the

entrant were to #roduce ,& the #rice would fall to %& etc. hese #rice'uantity

#oints for the entrant are shown as the large dots la!eled 3esidual (emand in

?igure ,.2. he marginal revenue for this residual demand curve has a slo#e that

is twice as stee#& as shown !y the thin dotted line that crosses 4C at a 'uantity of . his crossing determines an out#ut of for the entrant& since the incum!ent

seller is #roducing %. n summary& when one firm #roduces %& the !est res#onse

of the other is to choose a 'uantity of . his suggests why the 'uantities& which

start at an average of % for each firm in #eriod & !egin to decline in su!se'uent

#eriods& as shown on the right side of ?igure ,.,. he out#uts fall to an average

of / for each seller& which suggests that this is the e'uili!rium& in the sense that if

one seller is choosing /& the !est res#onse of the other is to choose / also.

n order to show that the Cournot e'uili!rium is in fact / units #er seller& we need

to run through some other !est res#onse calculations& which are shown in

a!le ,.. his ta!le shows a firm s #rofit for the eam#le under consideration

for each if its own out#ut decisions =increasing from !ottom to to#> and for each

out#ut decision of the other firm =increasing from left to right>. ?irst consider the

column la!eled 0 on the left side& i.e. the column that is relevant when the other

firm s out#ut is 0. f the other firm #roduces nothing& then this is the mono#oly



case& and the #rofits in this column are 9ust co#ied from the mono#oly #rofit row

of a!le ,.,: a #rofit of ,, for an out#ut of ,& 20 for an out#ut of 2& etc. ;ote

that the highest #rofit in this column is % in the u##er left corner& at the

mono#oly out#ut of %. his #rofit has !een highlighted !y #utting a dar" !order

around the !o. o !e sure that you understand the ta!le& you should fill in the

two missing num!ers in the right column. n #articular& if !oth have out#uts of %&

then the total out#ut is ,2& the #rice is SSSS& the total revenue for the firm is SSS&

the total cost is SSSS& and then the #rofit is 0 =#lease verify>. 6ome of the other

!est res#onse #ayoffs are also indicated !y !oes with dar" !orders. 3ecall that

the !est res#onse to another firm s out#ut of % is to #roduce =as seen in ?igure

,.2>& and this is why the !o with a #ayoff of in the far+right column has a dar"

!order.

a!le ,. A 6eller s Iwn 8rofit 4atri and 7est 3es#onses

Dey: Columns for Ither 6eller s Iut#uts& and 3ows for Iwn Iut#uts

0 , 2 / 5 %% /1 /+ 2/ , ,2 %

5 5 /+ * + ,5 ,0

/ 2 2 2/ + '1FF '

2- 2/ 2, ,R ,5 ' 4

2 20 , ,% ,/ ,2 ,0

, ,, ,0 - % 5

A ;ash e'uili!rium in this duo#oly mar"et is a #air of out#uts& such that each

seller s out#ut is the !est res#onse to that of the other. *ven though is a !est

res#onse to %& the #air =% for one& for the other> is not a ;ash e'uili!rium=#ro!lem ,>. he #ayoff of ,%& mar"ed with a dou!le asteris" in the ta!le& is the

location of a ;ash e'uili!rium& since it indicates that an out#ut of / is a !est

res#onse to an out#ut of /& so if each firm were to #roduce at this level& there

would !e no incentive for either to change unilaterally. If course& if they could

coordinate on 9oint out#ut reductions to each& the total out#ut would !e % and

the industry #rofit would !e maimi<ed at the mono#oly level of %& or , each.

his 9oint maimum is indicated !y the single asteris" at the #ayoff of , in the

ta!le. his 9oint maimum is not a ;ash e'uili!rium& since each seller has an

incentive to e#and out#ut =#ro!lem 2>.

he fact that the ;ash e'uili!rium does not maimi<e 9oint #rofit raises an

interesting !ehavioral 'uestion& i.e. why couldn t the su!9ects in the e#eriment

somehow coordinate on 'uantity restrictions to raise their 9oint earningsN he

answer is given in the legend on the right side of ?igure ,.,& which indicates that

the matchings were random for the duo#oly #hase. hus each seller was matched



with a randomly selected other seller in each round& and this switching would

ma"e it difficult to coordinate 'uantity restrictions. n e#eriments with fied

matchings& such coordination is often o!served& #articularly with only two sellers.

Holt =,5> re#orted #atterns where sellers sometimes wal"ed the 'uantity down

in unison& e.g. !oth duo#olists reducing 'uantity from - to % in one #eriod& and

then to 5 in the net& etc. his "ind of tacit collusion occurred even though sellers

could not communicate e#licitly. here were also cases where one seller

#roduced a very large 'uantity& driving #rice to 0& followed !y a large 'uantity

reduction in an effort to send a threat and then a conciliatory message and there!y

induce the other seller to coo#erate. 6uch tacit collusion is less common with

more than 2 sellers. 8art of the #ro!lem in these 'uantity+choice models is that

when one seller cuts out#ut& the other has a unilateral incentive to e#and out#ut&

so when one shows restraint& the other has greater tem#tation.

he ;ash e'uili!rium 9ust identified is also called a Cournot e'uili!irum& after

the ?rench mathematician who #rovided an e'uili!rium analysis of duo#oly and

oligo#oly models in ,. Although the Cournot e'uili!rium is symmetric in this

case& there can !e asymmetric e'uili!ria as well. A further analysis of a!le ,.indicates that there is at least one other ;ash e'uili!rium& also with a total

'uantity of & and hence an average of /. Can you find this asymmetric

e'uili!rium =#ro!lem >N

As was the case for mono#oly& it is useful to illustrate the duo#oly e'uili!rium

with a gra#h. f one firm is #roducing an out#ut of /& then the other can #roduce

an out#ut of , =total 'uantity J 5> and o!tain a #rice of =J , + 5>& as shown !y

one of the residual demand dots in ?igure ,.. hin" of the vertical ais as

having shifted to the right at the other firm s 'uantity of /& as indicated !y the

vertical dotted line& and the residual demand dots yield a demand curve with a

slo#e of minus ,. As !efore& marginal revenue will have a slo#e that is twice as

negative& as indicated !y the heavy dashed line in the figure. his line crossesmarginal cost 9ust a!ove the 'uantity of & which re#resents a 'uantity of / for

this firm& since the other is already #roducing /. hus the 'uantity of / is a !est

res#onse to the other s 'uantity of /. As seen in the figure& the #rice in this

duo#oly e'uili!rium is 5& which is lower than the mono#oly #rice of - for this

mar"et.



01234567

89

10111213

3 4 5 7 8 9101112130 1 2 6

price eman!

.esi!ual .

+

.esi!ual eman!

'uantity

0ig$re '/./. Co$rnot D$opo%!

III. Co$rnot 5%igipo%!

he classroom e#eriment shown in ?igure ,./ was done with fied matchings&

for duo#oly mar"ets =left side> and then for three+firm mar"ets =right side>.

(es#ite the fied nature of the matchings& #artici#ants were not a!le to coordinate

on out#ut reductions !elow the duo#oly #rediction of / #er seller. n the trio#oly

treatment& the out#uts converge to on average. ;otice that out#uts of for each

of three firms translates into an industry out#ut of & as com#ared with for the

duo#oly case =/ each> and % for the mono#oly case. hus we see that increases inthe num!er of sellers raise the total 'uantity and reduce #rice toward com#etitive

levels.

t is #ossi!le to redraw ?igure ,. to show that if two firms each #roduce

units each& then the remaining firm would also want to #roduce units =#ro!lem

/>. nstead& we will use a sim#le calculus derivation. 6u##ose that there are two

firms which #roduce units each& so their total is 2 . he remaining firm must

choose a 'uantity L to maimi<e its #rofit. he residual demand for the

remaining firm is: A =2 > 7 7<& where A J , and 7 J , in our eam#le.

4ulti#lying the residual demand !y L& we o!tain the total revenue function for

the remaining firm: A< =2 > 7 < 7<2& which has two terms that are linear in <

and one term that is 'uadratic. he derivative of this total revenue e#ressionwith res#ect to < is: A 2 7 27<. =he derivatives of the linear terms are 9ust

the coefficients of those terms& and the derivative of the 'uadratic term is ta"en !y

moving the 2 in the e#onent down to !e multi#lied !y the 7.> his derivative is

the e#ression for the marginal revenue that is associated with the residual



demand function =the thic" dashed line in ?igure ,.>. 7y e'uating marginal

revenue to marginal cost& one o!tains a single e'uation& A 2 7 27< J ,. his

e'uation could !e solved for L& which would show the !est res#onse out#ut L for

any given level of the other firms common out#uts& units each. Fust as there

was a symmetric e'uili!rium in the duo#oly case& there will !e one here& with X J

L& and with this su!stitution& the marginal+revenue+e'uals+marginal+cost

condition !ecomes: A 2<7 27< J ,& which can !e solved for L J =A ,>/7.

$ith A J , and 7 J ,& this reduces to ,2/ J & which is the Cournot e'uili!rium

for the trio#oly case.

0ig$re '/.3. - C%assroom 5%igopo%! "(periment 6ith 0i(ed Matchings

I. "(tensions

he Cournot model is #erha#s the most widely used model in theoretical wor" in

ndustrial Irgani<ation. ts #o#ularity is !ased& to some etent& on the #rediction

that the e'uili!rium #rice will !e a decreasing function of the num!er of sellers in

a symmetric model. his #rediction is su##orted !y evidence from la!oratory

e#eriments that im#lement the Cournot assum#tion that firms select 'uantities

inde#endently =e.g. Holt& ,5>.

he o!vious shortcoming of the Cournot model is the s#ecific way in which #rice

is determined. he im#licit assum#tion is that firms ma"e 'uantity decisionsinde#endently& and then #rice is cut so that all #roduction can !e sold. ?or

eam#le& you might thin" of a situation in which the 'uantities have !een

#roduced& so the short run su##ly curve is vertical at the total 'uantity& and #rice

is determined !y the intersection of this vertical su##ly function with the mar"et



demand function. n other words& the #rice+com#etition #hase is etremely

com#etitive. As we will see in later cha#ters& there is some game+theoretic and

e#erimental evidence to su##ort this view. *ven so& a Cournot e'uili!rium

would not !e a##ro#riate if it is #rice that firms set inde#endently& and then

#roduce to fill the orders that arrive. nde#endent #rice choice may !e an

a##ro#riate assum#tion if firms mail out catalogues or #ost !uy now #rices on the

internet& with the a!ility to #roduce 'uic"ly to fill orders. 6ome of the richer

models of #rice com#etition with discounts will !e discussed in the coming

cha#ters.

-ppendi(: 5ptiona% <$ick Ca%c$%$s Revie6

he derivative of a linear function is 9ust its slo#e. 6o for the demand function:

P J , < in ?igure ,.2& the slo#e is , =each unit increase in 'uantity decreases

#rice !y ,>. o find the slo#e using calculus& we need a formula: the derivative

of 7< with res#ect to < is 9ust 7& for any value of the slo#e #arameter 7. hus

the derivative of = ,>< is ,. he slo#e of the demand curve is the derivative of , < and we "now that the derivative of the second #art is ,& which is the correct

answer& so the derivative of , must !e 0. n fact& the derivative of any constant is

<ero. o see this& note that the derivative is the slo#e of a function& and if you

gra#h a function with a constant height& then the function will have a slo#e of 0 in

the same manner that a ta!le to# has no slo#e. ?or eam#le& consider the

derivative of a more general linear demand function: A 7L. he interce#t& A& is

9ust a constant =it does not de#end on L& which is varia!le& !ut rather it stays the

same>. 6o the derivative of A is 0& and the derivative of 7L is 7& and therefore

the derivative of A 7L is 0 7 J 7. Here we have used the fact that the

derivative of the sum of two functions is the sum of the derivatives. o

summari<e =ignore rules / and 5 for the moment>:

,> Constant 0$nction: dAdL J 0.

he derivative of a constant li"e A is 9ust 0.

2> Linear 0$nction: d=DL>d< J D.

he derivative of a constant times a varia!le& which has a constant slo#e& is 9ust

the constant slo#e #arameter& i.e. the derivative of D < with res#ect of < is 9ust D.

> S$m of 0$nctions: he derivative of the sum of two functions is the sum of

the derivatives.

/> <$adratic 0$nction: d=D <2>d< J 2D <.

he derivative of a 'uadratic function is o!tained !y moving the 2 in the e#onent

down& so the derivative of D <2 is 9ust 2D <.



5> Po6er 0$nction: d=D <>d< J D <+,.

he derivative of a varia!le raised to the #ower 5 is o!tained !y moving the 5

down and reducing the #ower !y ,& so the derivative of D < is 9ust D <2& the

derivative of D </ is /D <& and in general& the derivative of D < with res#ect to L

is D <

+,

.

?or eam#le& the mono#olist !eing discussed has a constant marginal cost of ,

#er unit& and since there is no fied cost& the total cost of for #roducing < units is

9ust < dollars. hin" of this total cost function as !eing the #roduct of , and L&

so the derivative of ,< is 9ust , using rule 2 a!ove. f there had !een a fied cost

of 6 & then the total cost would !e ? O <. ;ote that ? is 9ust a constant& so its

derivative is 0 =rule ,>& and the derivative of this total cost function is the

derivative of the first #art =0> #lus the derivative of the second #art =,>& so

marginal cost is again e'ual to ,.

;et consider a case where demand is P J A 7<& with has a vertical interce#t of

A and a slo#e of 7. he total revenue function is o!tained !y multi#lying !y < toget: the total revenue function: A< 7<2& which has a linear term with slo#e A

and a 'uadratic term with a coefficient of 7. $e "now that the derivative of the

linear #art is 9ust A =rule 2>. he fourth rule indicates how to ta"e the derivative

of 7<2& you 9ust move the 2 in the e#onent down& so the derivative of this #art is

27<. $e add these two derivatives together to determine that the derivative of

the total revenue function is: A 27<& so marginal revenue has the same vertical

interce#t as demand& !ut the slo#e is twice as negative. his is consistent with the

calculations in a!le ,.,& where each unit increase in 'uantity reduces #rice !y

, and reduces marginal revenue !y 2.

<$estions

,. Use a!le ,. to show that out#uts of % and do not constitute a ;ash

e'uili!rium to the duo#oly model that is the !asis for that ta!le.

2. Use a!le ,. to show that out#uts of for each firm do not constitute a

Cournot ;ash e'uili!rium.

. ?ind a ;ash e'uili!rium in a!le ,. with the #ro#erty the total 'uantity

is !ut one seller #roduces more than the other. herefore& you must

s#ecify what the two out#uts are& and you must show that neither seller has a unilateral incentive to deviate.

/. 3edraw ?igure ,. for the trio#oly case& #utting the vertical dotted line

at a 'uantity of % =three for each of two other firms> and show that the



residual marginal revenue for the remaining firm would cross marginal

cost in a manner that ma"e the out#ut of a !est res#onse for that firm.

5. =re'uires calculus> 6u##ose that demand is linear: P J A 7<& and

marginal cost is constant at C. Use the a##roach ta"en at the end of

6ection to find the Cournot trio#oly e'uili!rium in terms of the

#arameters A& 7& and C. =o chec" your answer& set C J , and com#are

with the calculations in the reading.>

%. =re'uires calculus> ?ind the Cournot duo#oly e'uili!rium for the model

given in #ro!lem 5.

-. =re'uires calculus> ?ind a symmetric Cournot e'uili!rium for the model

in 'uestion 5 with the num!er of firms set to F instead of . o chec"

your answer& set F J and com#are.

. =re'uires calculus> 6how that the #rice is a decreasing function of the

num!er of firms& ;& in the Cournot e'uili!rium model for 'uestion -.

. =for the A##endi> ?ind the derivatives with res#ect to < of: a> <, !> 2<&

c> 5 O ,0L& d> 5& and e> A 7< where A and 7 are #ositive #arameters.

,0. =for the A##endi> ?ind the derivatives with res#ect to of : a> ,0 & !>,0 O & c> 2y2& d> y2 O & and e> , / J 2.

Cha#ter ,/. Vertical 4ar"et 3elationshi#s

A com#le economy is characteri<ed !y considera!le s#eciali<ation along the

su##ly chainB with connections !etween manufacturers of intermediate #roducts&

manufacturers of final #roducts& wholesalers& and retailers. here is some

theoretical and em#irical evidence that frictions and mar"et im#erfections may !e

induced !y these vertical su##ly relationshi#s. his cha#ter !egins !yconsidering the very s#ecific case of a vertical mono#oly& i.e. interaction !etween

an u#stream mono#oly wholesaler selling to a firm that has a local mono#oly in a

downstream retail mar"et. n theory& the vertical alignment of two mono#olists

can generate retail #rices that are even higher than those that would result with a

single integrated firm& i.e. a merger of the u#stream and downstream firms. hese

effects can !e investigated with the Veconla! Vertical 4ono#oly #rogram or with

the instructions for a hand+run version that can !e found in 7adasyan et al.

=200/>& which is the !asis for most of the discussion in this cha#ter. he addition

of more layers in the su##ly chain raises the 'uestion of the etent to which

demand shoc"s at the retail level may cause even larger swings in orders and

inventories at the u#stream levels of the su##ly chain& a #henomenon that is

"nown as the !ullwhi# effect. he Veconla! 6u##ly Chain #rogram& listed under

the 4ar"ets menu& can !e used to im#lement this setu#.



I. Do$&%e Margina%iOation

6ince Adam 6mith& economists have !een "nown for their o##osition to

mono#oly. 4ono#oli<ation =carefully defined> is a crime in U.6. antitrust law&

and hori<ontal mergers are commonly challenged !y the antitrust authorities if the

effect is to create a mono#oly. n contrast& there is much more leniency shown

toward vertical mergers& i.e. !etween firms that do !usiness with each other along

a su##ly chain. Ine motivation for this relative tolerance of vertical mergers is

that two mono#olies may !e worse than one& at least when these two mono#olies

are arrayed vertically. he intuition is that each mono#olist restricts out#ut to the

#oint where marginal revenue e'uals marginal cost& a #rocess that reduces

#roduction !elow com#etitive levels and results in higher #rices. $hen this

marginal restriction is made !y one firm in the su##ly chain& it raises the #rice to a

downstream firm& which in turn raises #rice again at the retail level& and the

resulting dou!le marginali<ation may !e worse than the out#ut restriction that

would result if !oth firms were merged into a single& vertically integrated

mono#olist.

he effects of dou!le marginali<ation can !e illustrated with a sim#le la!oratorye#eriment !ased on a linear demand function: P J ,2 <& which is the demand at

the retail level. here is an u#stream manufacturer with a constant marginal cost

of #roduction of 2. he retailer #urchases units and then #ac"ages them& there!y

incurring an additional cost of 2 #er unit. hus there is a total cost

=manufacturing #lus retail> of / #er unit. his would !e the marginal cost of a

single vertically integrated firm that manufactures and sells the good in a retail

mono#oly mar"et.

?irst consider the mono#oly #ro!lem for this integrated firm& who will find the

out#ut that e'uates the marginal cost of / with the marginal revenue& as e#lained

in the #revious cha#ter. $ith a demand of P J ,2 <& and an associated total

revenue of ,2< <2& the marginal revenue will !e ,2 2<& since the marginalrevenue is twice as stee# as the demand curve. *'uating this to the marginal cost

of / and solving for < yields a mono#oly out#ut of / and an associated #rice of

. hese calculations are illustrated in ?igure ,/.,. he demand curve has a

vertical interce#t of ,2& with a slo#e of ,& so it has a hori<ontal interce#t of ,2.

he marginal revenue line =43> also has a vertical interce#t of ,2& !ut its slo#e is

twice as stee#& the hori<ontal interce#t is %. his 43 line intersects the hori<ontal

4C line at a 'uantity of /& which leads to a #rice of & as can !e seen !y moving

from the intersection #oint vertically u# to the demand curve.



0ig$re '3.'. - C%assroom ertica% Monopo%! "(periment

;et& consider the case in which the two firms are not vertically integrated. *ach

firm will !e e'uating marginal revenue and marginal cost. t turns out that in

these "inds of #ro!lems& it is easiest to start at the end of the su##ly chain =retail>

and wor" !ac"wards to consider the manufacturing level last. ?or any wholesale

#rice& = & charged !y the manufacturer& the marginal cost to the retailer will !e =

O 2& since each unit sold at retail must !e #urchased at a wholesale #rice = and

then #rocessed at a cost of 2. hus the downstream retailer is a mono#olist who

will e'uate this marginal cost& = O 2& to the marginal revenue& ,2 2<& to o!tain a

single e'uation: = O 2 J ,2 2< that can !e solved for the retail order 'uantity L

as a function of the wholesale #rice: L J 5 = 2. ;et consider the u#stream

firm s o#timal decision. he retail order is the demand function faced !y the

wholesaler& i.e. for each wholesale #rice it determines how much is sold to the

retailer. Fust as we draw the inverse of retail demand with #rice on the vertical

ais& it is useful to gra#h the inverse of the wholesale demand with the wholesale

#rice on the vertical ais. o do this& invert the wholesale demand function& L J

5 = 2& to o!tain the wholesale #rice as a function of <: $ J ,0 2<. his

inverse wholesale demand function is gra#hed on the left side of ?igure ,/., as

the thin line with a vertical interce#t of ,0 and a hori<ontal interce#t of 5. he

associated marginal revenue line has the same vertical interce#t !ut a slo#e that is

twice as stee#& as shown !y the line that has a hori<ontal interce#t of 2.5. he

wholesaler will want to increase out#ut until this marginal revenue is e'ual to thewholesale marginal cost of 2& as shown in the figure& where the intersection

occurs at a 'uantity of 2& which determines a wholesale #rice of $ J ,0 2< J ,0

/ J %. $hen the wholesale #rice is %& the marginal cost to the retailer is % O 2 J &

as shown !y the hori<ontal line at in the figure. his marginal cost line



intersects the retail marginal revenue line at a 'uantity of 2& which determines a

retail #rice of ,0. o summari<e& for the mar"et in ?igure ,/., with a mono#oly

out#ut of / and a #rice of & the #resence of two vertically stac"ed mono#olists

reduces the out#ut to 2 and raises the retail #rice to ,0. hus the out#ut

restriction is greater for two mono#olists than would !e the case for one vertically

integrated firm.

II. Some "(perimenta% "vidence

he first 5 rounds of the e#eriment shown in ?igure ,/., were done with #airs

of students& with one retailer and one wholesaler in each #air. he wholesale

#rice #rediction =%> and the retail #rice #rediction =,0> are indicated !y the

hori<ontal lines& and the average #rices are indicated !y the dots converging to

those lines. ;otice that wholesale #rices are a little low relative to theoretical

#redictionsB this could !e due to the fact that the !uyer& the retailer& is a #artici#ant

in the e#eriment& not a simulated !uyer as was the case for the retail demand in

this e#eriment and in the mono#oly e#eriments discussed in the #reviouscha#ter. n #articular& the downstream !uyer may res#ond to a wholesale #rice

that is #erceived to !e unfairly high !y cutting !ac" on #urchases& even if those

#urchases might result in more #rofit for the downstream seller.

he center #art of ?igure ,/., shows 5 rounds in which each of the ,2

#artici#ants from the first #art was #ut into the mar"et as a vertically integrated

mono#olist facing the retail demand determined !y P J ,2 <. he average #rices

converge to the mono#oly level of that is indicated !y the hori<ontal line for

#eriods %+,0. his reduction in retail #rice& from a!out ,0 to a!out & resulted

in an increase in sales 'uantity& and less of a mono#oly out#ut restriction& as

#redicted !y theory. n addition& the #rofits of the integrated firm are larger& since

!y definition& #rofit is maimi<ed at the mono#oly level. his increase in #rofita!ility associated with vertical integration can !e verified !y calculating the

#rofits for each firm se#arately =#ro!lems , and 2>.

Vertical integration may not always !e feasi!le or desira!le& or even cost efficient

in all cases. An alternative& which wor"s in theory and is o!served in #ractice& is

for the u#stream firm to re'uire the retailer to #ay a fied franchise fee in order to

!e a!le to sell the #roduct at all. n #articular& the wholesale firm chooses a

wholesale #rice and a franchise fee. he retailer then may re9ect the arrangement&

in which case !oth earn 0& or acce#t and #lace an order for a s#ecified num!er of

units. he idea !ehind the franchise fee is to lower the wholesale #rice from the

% charged when the firms are not integrated& to a level of 2 that reflects the true

wholesale marginal cost. he retailer s marginal cost would then !e the new low

wholesale #rice of 2 #lus the retail marginal cost of 2& which e'uals /& the true

social cost of #roducing and retailing each unit. $hen the retailer faces this

marginal cost of /& it will !ehave as an integrated mono#olist& choosing an



out#ut of /& charging the mono#oly #rice of & and earning the mono#oly #rofit

of R/ /R/ J ,%. f this were the whole story& the wholesaler would have <ero

#rofits& since the units were sold at a #rice that e'uals marginal cost. he

wholesaler can recover some of this mono#oly #rofit !y charging a franchise fee.

A fee of would s#lit the mono#oly #rofit& leaving for each. n theory& the

wholesaler could demand ,5. of the #rofit& leaving one #enny for the retailer&

under the assum#tion that the retailer would #refer a #enny to a #ayoff of <ero

that results from re9ecting the franchise contract. ntuition and other e#erimental

evidence& however& suggest that such aggressive franchise fees would !e re9ected

=see the Cha#ter 2 on 7argaining>. n effect& retailers will !e li"ely to re9ect

contract offers that are viewed as !eing unfair& and such re9ections may even have

the effect of inducing the wholesaler to ma"e a more moderate demand in a

su!se'uent #eriod.

he right side of ?igure ,/., shows average #rices and franchise fees for #eriods

,,+,5 of the e#eriment& in which the #artici#ants were again divided into

wholesalerretailer #airs. he average franchise fees are the light dots that lie

!etween % and . he wholesalers were not a!le to demand a high share of themono#oly #rofit in this e#eriment. nstead& it is a##arent that they did not lower

the #rice all of the way to their wholesale cost level of 2. Average #rices do fall

!elow the wholesale #rice levels in the first 5 rounds& !ut they only fall to a level

of a!out . hus fairness considerations seem to !e #reventing the franchise fee

treatment from solving the vertical mono#oly #ro!lem.

II. "(tensions: he B$%%6hip "ffect

(urham =2000> uses e#eriments to com#are #rice+setting !ehavior !y an

u#stream mono#olist who selects a wholesale #rice and announces it to one or more downstream sellers. here are two treatments& one with a single

downstream firm& and another with three downstream firms. 6he finds that the

#resence of downstream com#etition leads to outcomes similar to those under

vertical integration. n effect& the com#etition at the downstream level ta"es out

one of the sources of mono#oly marginali<ation& so the #rice and 'uantity

outcomes a##roach those that would result from a single mono#olist. Another

way of loo"ing at this is to note that if there is enough com#etition downstream&

then the #rice downstream will !e driven down to marginal cost& and the total

out#ut will !e a #oint on the retail demand curve& not a #oint on the retail

marginal revenue curve. hen the u#stream firm can essentially !ehave as an

integrated mono#olist who can select a #oint on the retail demand curve that

maimi<es total #rofit.

o summari<e& the dou!le marginali<ation #ro!lem associated with two

mono#olies may !e alleviated to some etent !y a vertical merger or !y the



introduction of com#etition downstream. $ithout these "inds of corrections& this

analysis of the #revious section would a##ly to an even greater etent with a

longer su##ly chain& e.g. with a mono#oly manufacturer selling to a wholesaler&

who sells to a distri!utor& who sells to a retailer& where the firms at each stage are

mono#olists. n this contet& the effects of successive marginali<ations get

com#ounded& unless these effects were somehow diluted !y com#etition !etween

sellers at each level.

A second source of inefficiency in a long su##ly chain is the #ossi!ility that

orders and inventories are not well coordinated& and that information is not

transmitted efficiently from one level to the net. ?or eam#le& 8rocter Y

am!le found that dia#er orders !y distri!utors to !e too varia!le relative to

consumer demand& and Hewett+8ac"ard found that #rinter orders made !y retail

sellers are much more varia!le than consumer demand itself. hese and other

eam#les are discussed in 1ee et al. =,-a& ,-!>.

here is a long tradition in !usiness schools of #utting 4.7.A. students into a

su##ly chain simulation& which is "nown as the 7eer ame. here are four

vertical levels: manufacturing& wholesale& distri!ution& and retail =?orrester&,%,>. he #artici#ants in these classroom games must ty#ically fill #urchase

orders from inventory& and then #lace new orders to the level a!ove in the su##ly

chain. here is a cost of carrying unsold inventory& and there is also a cost

associated with not !eing a!le to fill an order from !elow& i.e. the lost #rofit #er

unit on sales. he setu# ty#ically involves having a sta!le retail demand for

several rounds !efore it is su!9ected to an une#ected and unannounced increase

that #ersists in later #eriods. he effect of this demand increase is to cause larger

and larger fluctuations in orders #laced u#stream& which is "nown as the !ullwhi#

effect.

n classroom e#eriments with the 7eer ame& the u#stream sellers tend to

attri!ute these large fluctuations to eogenous demand shifts& des#ite the fact thatmost of the fluctuation is due to the reactions of those lower in the su##ly chain

=6terman& ,>. 6ee 1ee& 8admana!han& and $hang =,-a> for a theoretical

analysis of 'uasi+rational !ehavior that may cause orders to fluctuate more than

sales& and that may cause this distortion to !e greater as one moves u# the su##ly

chain. ?or more recent e#eriments& see Holt and 4oore =200/a>.



<$estions

,. ?ind the #redicted #rofit for a mono#oly seller in the mar"et descri!ed in section

.

2. ?ind the #rofits for wholesaler and retailer for the mar"et descri!ed in section

when the wholesale #rice is % and the retail #rice is ,0& as #redicted. 6how that

#rofits for these two firms are& in total& less than that of a vertically integrated

mono#olist& as calculated in the #revious 'uestion.

. Consider a mar"et with an inverse demand function that is linear: P J /% 2<.

he cost at the wholesale level is 0 for each unit #roduced. At the retail level&

there a cost of % associated with retailing each unit #urchased at wholesale. hus

the average cost for !oth levels com!ined is also %. ?ind the o#timal out#ut and

retail #rice for a vertically integrated mono#olist& either using a gra#h or calculus.

n either case& you should illustrate your answer with a gra#h.

/. f you answered 'uestion correctly& your answers should im#ly that the firm s

total revenue is 2%0& the total cost is %0& and the #rofit for the integrated

mono#olist is 200. ;ow consider the case in which the u#stream and downstream

firms are se#arate& and the u#stream seller chooses a wholesale #rice& = & which isannounced to the downstream firm on a ta"e+it+or+leave+it !asis =no further

negotiation>. hus the marginal cost downstream is % O = . *'uate this with

marginal revenue =with a gra#h or with calculus> to determine the o#timal

'uantity for the downstream firm& as a function of = . hen use this function to

find the o#timal level of $ for the u#stream seller =using a gra#h or calculus>& and

illustrate your answer with a gra#h in either case. Hint: if you answered this

correctly& the 'uantity should !e half of the mono#oly 'uantity that you found in

your answer to 'uestion .

5. ;ow su##ose that the u#stream mono#olist for the setu# in 'uestion / can charge

a franchise fee. f the downstream seller is #erfectly rational and #refers a small

#rofit to none at all& what is the highest fee that the u#stream seller can chargeN$hat is the !est =#rofit+maimi<ing> com!ination of wholesale #rice and

franchise fee from the #oint of view of the u#stream sellerN



Cha#ter ,5. 4ar"et nstitutions and 8ower

raders in a dou!le auction can see all transactions #rices and the current

!idas" s#read& as is the case with trading on the ;ew Mor" 6toc" *change. he

dou!le auction is an etremely com#etitive institution& given the tem#tation for traders to im#rove their offers over time in order to ma"e trades at the margin. n

contrast& mar"ets with #osted #rices =7ertrand or 8osted+Iffer> allow sellers to

#re+commit to fied& ta"e+it+or+leave+it #rices that cannot !e ad9usted during a

trading #eriod. he focus of this cha#ter is #rice and efficiency outcomes of

mar"ets with #osted #rices& in #articular when sellers #ossess market po;er &

which is& roughly s#ea"ing& the a!ility of a firm to raise #rice #rofita!ly a!ove

com#etitive levels. n most cases& the mar"et efficiency is higher in the dou!le

auction& since the #rice flei!ility !uilt into that institution tends to !ring

outcomes closer to a com#etitive =su##ly e'uals demand> outcome. *fficiency is

also enhanced !y the incor#oration of more #rice flei!ility into mar"ets with

#osted #rices& e.g.& !y allowing sellers to offer secret discounts to s#ecific !uyers.hese various mar"et institutions can !e im#lemented !y hand with the

instructions in the a##endi to (avis and Holt =,>& or with the V econla!

dou!le+auction& #osted+offer& or 7ertrand #rograms& which #rovide flei!le setu#

o#tions and automatic data and gra#hical summaries.

I. Introd$ction

he common #erce#tion that la!oratory mar"ets yield efficient com#etitive

outcomes is sur#rising given the em#hasis on mar"et im#erfections that #ervades

theoretical wor" in industrial organi<ation. his a##arent contradiction isresolved !y considering the effects of trading institutions: As discussed in

Cha#ter 2& com#etitive outcomes are ty#ical in Zdou!le auctionZ mar"ets& with

rules similar to those used in many centrali<ed financial echanges. 7ut most

mar"ets of interest to industrial organi<ation economists have different

institutional characteristicsB sellers #ost #rices and !uyers must either !uy at those

#rices or engage in costly search and negotiation to o!tain discounts. Unli"e

more com#etitive dou!le auctions& the #erformance of mar"ets with #osted #rices

can !e seriously im#eded !y the #resence of mar"et #ower& #rice+fiing

cons#iracies& and cyclical demand shoc"s.

n many mar"ets& it is common for #rices to !e set !y the traders on the thin side

of the mar"et. 6ellers& for eam#le& ty#ically #ost #rices in retail mar"ets. ?or this reason& theoretical =7ertrand> models are often structured around an

assum#tion that #rices are listed simultaneously at the !eginning of each #eriod.

Using la!oratory e#eriments& it is #ossi!le to ma"e controlled com#arisons

!etween mar"ets with #osted #rices and more symmetric institutions such as the



Zdou!le auction&Z where !oth !uyers and sellers #ost !ids and as"s in an

interactive setting that resem!les a centrali<ed stoc" mar"et. 1a!oratory dou!le

auctions yield efficient& com#etitive outcomes in a sur#risingly wide variety of

settings& sometimes even in a mono#oly =6mith& ,%2& ,,B (avis and Holt&

,>. n contrast& #rices in mar"ets with #osted #rices are often a!ove

com#etitive levels =8lott& ,B (avis and Holt& ,>.

?igures ,5., and ,5.2 show a matched #air of mar"ets& one dou!le auction and

one #osted+offer auction& which were run under research conditions with

#ayments e'ualing ,5 of earnings. he values and costs were arrayed in a

manner that #redicted earnings #er #erson were a##roimately e'ual. n each

case& the !uyer values were reduced after #eriod 5 to shift demand down !y /&

there!y lowering the com#etitive #rice #rediction !y 2& as shown on the left

sides of each figure. ?or eam#le& the two !uyers with the highest value units& at

,5 in #eriod 5& had these values reduced to ,, in #eriod %& whereas costs stayed

the same in all #eriods.

0ig$re '*.'. Price Se#$ence for a Do$&%e -$ction

?irst& consider the dou!le auction& in ?igure ,5.,. he vertical lines to the right

of the su##ly and demand curves indicate the starting #oint of each new trading

#eriod& and the dots re#resent transactions #rices in the order in which they wereo!served in each #eriod. 3ecall that each trader in a dou!le auction may ma"e a

#rice offer =!ids for !uyers and as"s for sellers> or may acce#t the !est offer on

the other side of the mar"et. hus traders ty#ically see a se'uence of declining



as" #rices andor increasing !id #rices& until the !id+as" s#read narrows and one

#erson acce#ts another s #ro#osal. he com#uteri<ed mar"et ma"er ensured that

a #erson ma"ing a successful !id or as" would o!tain the !est availa!le terms.

?or eam#le& su##ose that the highest !id were 5 and there were as"ing #rices of

,0 and & and that a !uyer decided to acce#t !y !idding . f another as" were

to come in at - #rior to the time at which the !uyer confirmed the acce#tance !id

of & then the !uyer would o!tain the unit at the #rice of -.

(es#ite the considera!le #rice variation in the first #eriod& the dou!le auction

mar"et achieved #er cent of the maimum #ossi!le sur#lus& and this efficiency

measure averaged #er cent in the first 5 rounds& with 'uantities of ,/ that e'ual

the com#etitive #rediction. All traders where told that some #ayoff #arameters

may have changed in round %. !ut sellers who o!served their own costs had not

changed had no idea whether !uyer values or others costs had gone u# or down.

ransactions #rices !egan in round % at the old com#etitive level of & !ut fell to

% !y the end of the #eriod. his is a ty#ical #attern& where the high+value and

low+cost units trade early in a #eriod at levels close to those in the #revious

#eriods& !ut late trades for units near the margin are forced to !e closer to thesu##ly+demand intersection #rice. he transaction 'uantity was at the

com#etitive #rediction of ,0 in all rounds of the second treatment. As early+

#eriod #rices fell toward the com#etitive level after round %& the #rice averages

were a!out e'ual to the new #rediction of %& and efficiencies were at a!out %

#ercent.

3ecall from the discussion in Cha#ter 2 that !uyers do not #ost !ids in a #osted+

offer mar"et. 6ellers #ost #rices inde#endently at the start of each #eriod& along

with the maimum num!ers of units that are offered for sale. hen !uyers are

selected in a random order to ma"e desired #urchases. he #eriod ends when all

!uyers have finished sho##ing or when all sellers are out of stoc". As shown in

?igure ,5.2& #rices in the #osted offer mar"et also !egan in the first #eriod withconsidera!le variation& although all #rices were far from the com#etitive

#rediction. 8rices in #eriods 2+/ seemed to converge to a level at a!out , a!ove

the com#etitive #rediction. 7oth the average 'uantity =,2> and the efficiency =%

#ercent> were well !elow the com#etitive levels o!served in the matched dou!le

auction. 8rices fell slowly after the demand shift in #eriod %& !ut they never 'uite

reached the new com#etitive #rediction& and efficiencies averaged % #ercent in

the last 5 #eriods. he lowest efficiencies were o!served in the first two #eriods

of each treatment& illustrating the more sluggish ad9ustment of the #osted offer

mar"et. $hat is ha##ening here is that sellers are not a!le to see !uyer !ids or to

learn from them& and instead& they must loo" at sales data to ma"e inferences

a!out how #rice should !e altered. 7y the final #eriod of each treatment&

efficiencies had clim!ed a!ove 0 #ercent& and the transactions 'uantity had

reached the relevant com#etitive #rediction.



0ig$re '*.. Price Se#$ence for a Posted?5ffer -$ction

he results shown in ?igures ,5., and ,5.2 are fairly ty#ical. Com#ared with

dou!le auctions& la!oratory #osted+offer mar"ets converge to com#etitive

#redictions more slowly =Detcham& 6mith and $illiams& ,/> and less

com#letely =8lott& ,%& ,>. *ven in non+mono#oli<ed designs with stationary

su##ly and demand functions& traders in a #osted+offer mar"et generally forego

a!out ,0K of the #ossi!le gains from trade& whereas traders in a dou!le auction

routinely o!tain 5+K of the total sur#lus in such designs =(avis and Holt&

,& cha#ters and />. he sluggish #rice res#onses shown in ?igure ,5.2 areeven more a##arent in mar"ets with se'uences of demand shifts that create a

!oom and !ust cycle. ?irst consider a case where the su##ly curve stayed

stationary as demand shifted u# re#eatedly in a se'uence of #eriods& creating an

u#ward momentum in e#ectations. his !oom was followed !y a se'uence of

downward demand shifts that reduced the com#etitive #rice #rediction

incrementally until it returned to the initial level. 8rices determined !y dou!le

auction trading trac" the #redicted #rice increases and decreases fairly accurately&

with high efficiencies. his is !ecause the demand shifts are conveyed !y the

intensity of !uyer !idding !ehavior during each #eriod& so that sellers could learn

a!out new mar"et conditions as they started ma"ing sales. n contrast& #rices in

#osted #rice mar"ets are selected !efore any sho##ing !egins& so sellers cannots#ot changes in mar"et conditions& !ut rather& must try to ma"e inferences from

sales 'uantities. $hen #osted+offer mar"ets were su!9ected to the same se'uence

of demand increases and decreases mentioned a!ove& the actual trading #rices

lagged !ehind com#etitive #redictions in the u#swing& and #rices continued to rise



even after demand started shifting downward. hen #rices fell too slowly& relative

to the declining com#etitive #redictions =where su##ly e'uals demand>. he

result was that #rices stayed too high on the downswing #art of the cycle& and

these high #rices caused transactions 'uantities to fall dramatically& essentially

drying u# the mar"et for several #eriods.

(avis and Holt =,%> re#licate these sluggish ad9ustment #atterns on #osted+

#rice mar"ets su!9ected to a series of demand increases& followed !y a series of

demand decreases. *ven in mar"ets with #osted #rices& sellers do receive some

useful information from !uyer !ehavior. ?or eam#le& sellers might !e a!le to

o!serve ecess demand on the u#swing& as !uyers see" to ma"e #urchases !ut are

not accommodated !y the #roduction 'uantities. (avis and Holt ran a second

treatment with the !oom!ust cycle of demand shifts& with an added feature that

let sellers o!serve ecess demand !y frustrated !uyers at the #rices that they set.

his ecess demand information im#roved #rice res#onsiveness and raised mar"et

efficiency& !ut not to the high levels o!served in dou!le auctions. 6imilar

increases in efficiency were o!served when sellers were allowed to offer a single

#rice reduction = clearance sale > after the #osted #rices had !een su!mitted andsome sales had !een made at those #rices.

II. he "(ercise of Se%%er Market Po6er 6itho$t "(p%icit Co%%$sion

Ine of the ma9or factors considered in the antitrust analysis of mergers !etween

firms in the same mar"et is the #ossi!ility that a merged firm may !e a!le to raise

#rice& to the detriment of !uyers. If course& any seller may raise a #rice

unilaterally& and so the real issue is the etent to which #rice can !e raised

#rofita!ly. 6uch a #rice increase is more li"ely to raise #rofits if others in the

mar"et are not in a #osition to a!sor! increases in sales at lower #rices& so the

ca#acities of other sellers may constrain a firm s mar"et #ower& and a merger thatreduces others ca#acities may create mar"et #ower.

*ven if a firm loses some sales as a result of a #rice increase& this may not !e

very detrimental to the firm s #rofit if these marginal units are low+#rofit units&

i.e. if the costs of these units are close to the initial #rice. ?or eam#le& consider

the su##ly and demand structure on the left side of ?igure ,5.. he su##ly

function has a flat s#ot at a #rice of 2.%0& where it crosses demand& and this

com#etitive #rice is indicated !y the thin hori<ontal line !elow the dots that

re#resent transactions #rices for the first / rounds of trading& which was done with

#osted+offer rules. he failure of #rice to converge to the com#etitive #rediction

was due to the fact that two of the sellers had a num!er of units with costs at

2.%0& so they had little incentive to cut #rice and sell these units& es#ecially at

#rices 9ust a!ove this level. he narrow =, unit> ga# !etween demand and su##ly

at #rices 9ust a!ove the com#etitive level means that a seller who refuses to sell 2



of these low+#rofit marginal units would shift su##ly to the left !y 2 units and

raise the su##lydemand intersection to the net highest su##ly ste#& at 2.0.

0ig$re '*./ - Posted?Price Market 6ith Se%%er Market Po6er

o summari<e& mar"et #ower to raise #rice a!ove com#etitive levels can eist

when com#etitors ca#acities are limited& and when a firm s marginal units are

selling at low #er+unit #rofits. n addition& mar"et #ower can !e influenced !y

the nature of the trading institution. Vernon 6mith =,,> investigated an etreme

case of mar"et #ower& with a single seller& and found that even mono#olists in a

dou!le auction are sometimes not a!le to raise #rices a!ove com#etitive levels&

whereas #osted+offer mono#olists are ty#ically a!le to find and enforce mono#oly

#rice levels. he difference is that a seller in a #osted offer auction sets a single&

ta"e+it+or+leave+it #rice& so sellers are not tem#ted to cut #rice late in the #eriod in

order to sell marginal units. his tem#tation is #resent in a dou!le auction. n

#articular& a mono#olist who cuts #rice late in a trading #eriod to sell marginal

units will have a harder time selling units at nearmono#oly levels at the start of

the net #eriod& and this !uyer resistance may cause #rices in a dou!le auction

mono#oly to !e lower than the mono#oly #rediction.

his effect of the trading institution is also a##arent from e#eriments conducted

!y Holt& 1angan& and Villamil =,%> with the design from ?igure ,5.& using 5

sellers and 5 !uyers. As noted a!ove& two of the sellers had higher ca#acities that

included a num!er of low+#rofit units with costs at the com#etitive level of 2.%0.n contrast& the large vertical difference !etween demand and the com#etitive

#rice for all included units means that !uyers had a strong incentive to !uy these

high+value units& even if #rice were increased a!ove com#etitive levels. hus the

design was intended to create an asymmetry !etween !uyers who were eager to



!uy& and sellers with little incentive to sell units at the margin& and hence with a

large incentive to try to raise #rices a!ove com#etitive levels. (es#ite this

asymmetry& #rices converged to com#etitive levels in a!out half of the sessions.

6ome modest #rice increases a!ove com#etitive levels were o!served in the other

sessions& indicating that sellers could sometimes eercise mar"et #ower even in a

dou!le auction.

(avis and $illiams =,,> re#licated the Holt& 1angan& and Villamil results for

dou!le auctions with the mar"et #ower design in ?igure ,5.& and the resulting

#rices were slightly a!ove com#etitive levels. n addition& they ran a new series

of sessions using #osted+offer trading& which generated somewhat higher #rice

se'uences li"e those o!served in the figure.

6u#ra+com#etitive #rices in #osted+offer mar"ets are not sur#rising& since

e#erimental economists have long "nown that #rices in #osted+offer auctions

tend to converge to com#etitive levels from a!ove& if at all. his raises the

'uestion of whether these high #rices can !e e#lained !y game+theoretic

calculations. t is straightforward to s#ecify a game+theoretic definition of mar"et

#ower& !ased on the incentive of one seller to raise #rice a!ove a commoncom#etitive level =Holt& ,>. n other words& mar"et #ower is said to eist

when the com#etitive e'uili!rium is not a ;ash e'uili!rium.

Although it is ty#ically easy to chec" for the #rofita!ility of a unilateral deviation

from a com#etitive outcome& it may !e more difficult to identify the ;ash

e'uili!rium for a mar"et with #osted #rices. he easiest case is where firms do

not have constraints on what they can #roduce& a case commonly referred to as

7ertrand com#etition. ?or eam#le& if each firm has a common& constant

marginal cost of C& then no common #rice a!ove C would !e a ;ash e'uili!rium&

since each firm would have an incentive to cut #rice slightly and ca#ture all

mar"et sales. he 7ertrand #rediction for a #rice com#etition game p#aed once

is for a very harsh ty#e of com#etition that drives #rice to marginal cost levels&even with only two or three firms.

*ven with a re#eated series of mar"et #eriods& the one+shot ;ash #redictions may

!e relevant if random matchings are used to ma"e the mar"et interactions have a

one+shot nature& where no!ody can #unish or reward other s #ricing decisions in

su!se'uent #eriods. 4ost mar"et interactions are re#eated& !ut if the num!er of

mar"et #eriods is fied and "nown& then the one+shot ;ash #rediction a##lies in

the final #eriod& and a #rocess of !ac"ward induction can !e used to argue that

#rices in all #eriods would e'ual this one+shot ;ash #rediction. n most mar"ets&

however& there is no well+defined final #eriod& and in this case& there is a

#ossi!ility that a "ind of tacit coo#eration might develo#. n #articular& a seller s

#rice restraint in one #eriod might send a message that causes others to follow

suite in su!se'uent #eriods& and sellers #rice cuts might !e deterred !y the threat

of retaliatory #rice cuts !y others. here are many ways that such tacit

coo#eration might develo#& as su##orted !y #unishment and reward strategies&



and the multi#licity of #ossi!le arrangements ty#ically ma"es a theoretical

analysis im#ossi!le without strong sim#lifying assum#tions. Here is where

e#eriments can hel#.

?igure ,5./ shows the results of a session with #osted #rices in which

#artici#ants were matched in grou#s of & with no ca#acity constraints. (emand

was simulated& with aggregate 'uantity !eing determined !y the function& < J ,2

P . n the first ,0 rounds& all sellers had a constant marginal cost of ,& so the

7ertrand #rediction is ,. he +#erson grou#s were fied& and this is #ro!a!ly a

factor in the a!ility of sellers in one of the si trio#oly grou#s to maintain #rices

at a!out & well a!ove the 7ertrand #rediction. he #rice dots in the figure

re#resent averages over all +#erson grou#s for each #eriod& and these averages

mas" the fact that the 5 other trio#oly grou#s were #ricing much nearer to the

7ertrand #rediction of ,. he marginal cost was reduced to 0 for the final ,0

#eriods& and matchings were reconfigured randomly after each #eriod for this

second treatment. he random matching is #ro!a!ly a factor in the somewhat

shar#er convergence of average #rices to the 7ertrand #rediction in the final

#eriods.o summari<e& we have identified several factors that may facilitate #ricing at

su#ra+com#etitive levels: ,> the #rice fiity and asymmetry of the #osted+offer

institution& relative to the dou!le auction& 2> the etent to which mar"et

interactions are re#eated& and > the structure of the seller costs and ca#acities&

which may ena!le one seller to raise #rice without losing significant sales to

others. he net section e#ands on the third factor& !y #roviding a more detailed

analysis of #rice com#etition with ca#acity constraints.

0ig$re '*.3. - Posted?Price Market 6ith 8o Capacit! Constraints and

8o Market Po6er



III. Market Po6er and "#$i%i&ri$m Price Distri&$tions

$hen sellers all offer the same homogeneous #roduct& !uyers with good #rice

information will floc" to the seller with the lowest #rice. his can create a #rice

insta!ility& which may lead to randomi<ed #rice choices. ?irst consider a s#ecific

eam#le in which demand is inelastic at units for all #rices u# to a limit #rice of : & i.e. the demand curve is flat at a height of : for less than units and !ecomes

vertical at a 'uantity of for #rices !elow : . here are two sellers& each with a

ca#acity to #roduce 2 units at <ero cost. hus the mar"et su##ly is vertical at a

'uantity of / units& and su##ly and demand intersect at a #rice of 0 and a

'uantity of ='uestion 2>. he sellers are identical& so each can only e#ect to

sell to half of the mar"et& ,.5 units on average& if they offer the same #rice. f the

#rices are different& the seller with the lower #rice sells 2 units and the other only

sells , unit =as long as #rice is no greater than : >. here is no ;ash e'uili!rium at

any common #rice !etween 0 and V& since each seller could increase e#ected

sales from ,.5 to 2 !y decreasing #rice slightly. ;or is a common #rice of 0 a ;ash e'uili!rium& since earnings are <ero and either seller would have an

incentive to raise #rice and earn a #ositive amount on the one+unit residual

demand ='uestion >. hus there is no e'uili!rium in #ure =non+random>

strategies& and therefore& one would not e#ect to see sta!le #rice #atterns.

!n 4dge;orth Cc#e

Ine #ossi!ility& first considered !y *dgeworth& is the idea that #rices would

cycle& with each firm undercutting the other s #rice in a downward s#iral. At

some #oint #rices go so low that one seller may raise #rice. n the eam#lediscussed a!ove& su##ose that one seller has a #rice of that is a #enny a!ove a

level p !etween 0 and : . he other seller could cut #rice to p and sell 2 units&

there!y earning 2#& or raise #rice to : & sell the single residual demand unit& and

earn : . hus cutting #rice would !e !etter if 2 p P : & or if p P : 2. 3aising #rice

would !e !etter if p Q : 2. his reasoning suggests that #rices might fall in the

range from : down to : 2& at which #oint one seller would raise #rice to : and the

cycle would !egin again. he #ro!lem with this argument is that if one "nows

that the other will cut #rice !y one cent& then the !est reaction is to cut !y 2 cents&

!ut then the other would want to cut !y cents& etc. hus the declining #hase of

#rices is li"ely to !e s#oradic and somewhat un#redicta!le. his raises the

#ossi!ility that #rices will !e random& i.e. that there will !e a ;ash e'uili!rium in

mied strategies of the ty#e considered in Cha#ter 5. n this section& we

consider such an e'uili!rium& in which #rices can vary continuously on some

interval. 7ut first we must introduce some notation for a distri!ution of #rices.



Price (istri+utions

$ith a continuous distri!ution of #rices& the #ro!a!ility that another seller s #rice

is less than or e'ual to any given amount p will !e an increasing function of p&

and we will denote this #ro!a!ility !y ?= p>. ?or eam#le& su##ose that #rice is

e'ually li"ely to !e any dollar amount !etween 0 and ,0.00. his #rice

distri!ution could !e generated !y the throw of a ,0+sided die& with sides mar"ed

,& 2& ... ,0& and using the outcome to determine #rice. hen the #ro!a!ilities

would !e given !y the num!ers in the second row of a!le ,5.,. ?or each value

of p in the to# row& the associated num!er in the second row is the #ro!a!ility that

the other seller s #rice is less than or e'ual to p. ?or a #rice of p J ,0& all #rices

that might !e chosen !y the other seller are less than or e'ual to ,0 !y

assum#tion& so the value of ?= p> in the second row of the far+right column is ,.

6ince all #rices are e'ually li"ely& half of them will !e less than or e'ual to the

mid#oint of 5& so ?= p> J 0.5 when p J 5. he other num!ers in the ta!le are

calculated in a similar manner. A mathematical formula for this uniform

distri!ution of #rices would !e: ?= p> J p,0& as can !e verified !y directcalculation ='uestion ,>.

a!le ,5., A Uniform (istri!ution of 8rices

p 0 , 2 / 5 % - ,0

?= p> 0 0., 0.2 0. 0./ 0.5 0.% 0.- 0. 0. ,

Uniform #rice distri!utions& such as the one shown in a!le ,5.,& were

encountered in the earlier cha#ter on se'uential search and will come u#

fre'uently in some of the su!se'uent cha#ters on auctions.

he function& p,0& shown in a!le ,5., is increasing in p& !ut it increases at auniform rate& since all #rices in the range from 0 to ,0 are e'ually li"ely. Ither

distri!utions may not !e uniform& e.g. some central ranges of #rices may !e more

li"ely than etreme high or low #rices. n such cases& ?= p> would still !e an

increasing function& !ut the rate of increase would !e faster in a range where

#rices are more li"ely to !e selected. n a mied+strategy e'uili!rium& the

distri!ution of #rices& ?= p>& must !e determined in a manner to ma"e each seller

indifferent over the range of #rices& since no seller would !e willing to choose

#rice randomly unless all #rices in the range of randomi<ation yield the same

e#ected #ayoff. As was the case in Cha#ter 5& we will calculate the e'uili!rium

under the assum#tion that #layers are ris" neutral& so that indifference meanse'ual e#ected #ayoffs =with ris" averse #layers& we would need to force

indifference in terms of e#ected utilities>.

Mi5ed Strategies for the (uopo# 45amp#e



Again& the discussion #ertains to a duo#oly eam#le with <ero costs and

ca#acities of 2 units for each seller. his su##ly is vertical at a 'uantity of /. As

!efore& assume that demand is inelastic at units for all #rices !elow : . hus the

com#etitive #rice is 0& which is not a ;ash e'uili!rium& as e#lained #reviously.

7egin !y considering a #rice& p& in the range from 0 to : . he #ro!a!ility that the

other seller s #rice is less than or e'ual to p is ?= p>& which is assumed to !e

increasing in p. $hen the other seller s #rice is lower& a firm sells a single unit&

with earnings e'ual to p. $hen the other s #rice is higher& one s own sales are 2

units& with earnings of 2 p. 6ince #rices are continuously distri!uted& we will

ignore the #ossi!ility of a tie& and hence& ?= p> is the #ro!a!ility associated with

having the high #rice and earning the #ayoff of p& and , ?= p> is the #ro!a!ility

associated with having the low #rice and earning the #ayoff of 2 p. hus the

e#ected earnings are: ?= p> p O , ?= p>2 p. his e#ected #ayoff must !e

constant for each #rice in the range over which a firm randomi<es& to ensure that

the firm is indifferent !etween #rices in this range. o determine this constant&

note that setting the highest #rice& : & will result in sales of only , unit& since the

other seller will have a lower #rice for sure. hus we set the e#ected #ayoff e#ression e'ual to the constant : :

=,5.,> ?= > O , p p ?= >2 J . p

:

o summari<e& e'uation =,5.,> ensures that the e#ected #ayoff for all #rices is a

constant and e'uals the earnings level for charging the !uyer #rice limit of : .

his e'uation can !e solved for the e'uili!rium #rice distri!ution ?= p>:

=,5.2> ?= > p 2 p : p

for : 2 I p I : . ;otice that e'uation =,5.2> im#lies that ?= p> J , when p J : .

n words& the #ro!a!ility that the other seller s #rice is less than or e'ual to : is ,.

he lower limit of the #rice distri!ution& p J : 2& is the value of # for which the

right side of =,5.2> is 0. ;ote that this is also the lower !ound of the *dgeworth

cycle discussed a!ove. he fact that the lower !ound is half of the !uyer

reservation value : is due to the assum#tion that the high+#rice firm only sells

half of what the low+#rice firm sells ='uestion />.

I. -n "(periment on the "ffects of Market Po6er

here may !e several reasons for o!serving #rices that are a!ove com#etitive

levels in a design li"e that considered in the #revious section. ?or eam#le& with



only two sellers& a ty#e of tacit collusion may !e #ossi!le& es#ecially if the sellers

interact re#eatedly. Another #ossi!le reason is that demand is inelastic and that

the ecess demand is only one unit at #rices a!ove the com#etitive level of 0 in

this eam#le. A final reason is that earnings would !e <ero at the com#etitive

outcome& which might #roduce erratic !ehavior. hese ty#es of arguments led

(avis and Holt =,/> to consider a design with two treatments& each with the

same aggregate su##ly and demand functions& !ut with a reallocation of units that

creates mar"et #ower. n #articular& a reallocation of ca#acity from one seller to

others changed the ;ash e'uili!rium #rice from the com#etitive #rice =7ertrand

result> to higher =randomi<ed> #rices over the range s#anned !y the *dgeworth

cycle.

Consider (esign , on the left side of ?igure ,5.5. As indicated !y the seller

num!ers !elow each unit& sellers 6,+6 each have units& and 6/ and 65 each

have a single& low+cost unit. he demand curve has a vertical interce#t of r and

intersects su##ly at a range of #rices from the highest com#etitive #rice& pc& to the

level of the highest cost ste#& c. he demand is simulated with the high+value

units !eing #urchased first. his demand #rocess ensures that a unilateral #riceincrease a!ove a common #rice pc would leave the high+value units to !e

#urchased !y the other sellers& whose ca#acity totals to units. hus a unilateral

increase from a common com#etitive #rice will result in no sales& and hence& will

!e un#rofita!le. t follows that no seller has mar"et #ower in this design.

4ar"et #ower is created !y giving seller 6)s two high+cost units to 6, and 62& as

shown in (esign 2 on the right side of the figure. ;ow each of the large sellers&

6, and 62& has / units. f one of these sellers were to raise #rice unilaterally to

the demand interce#t& r & one of these / units would sell since the other / sellers

only have enough ca#acity to sell - of the units that are demanded at #rices

a!ove the com#etitive level. 7y ma"ing the demand interce#t high relative to the

high+cost ste#& we ma"e such deviations #rofita!le for the two large sellers&there!y creating mar"et #ower. n this case& it is #ossi!le to calculate the #rice

distri!utions in the mied+strategy e'uili!rium& !y e'uating sellers) e#ected

#ayoffs to a constant =since a seller would only !e willing to randomi<e if

e#ected #ayoffs are inde#endent of #rice on some range>. hese calculations

#arallel those in the #revious section& !ut the analysis is more tedious& given the

asymmetries in sellers cost structures =see (avis and Holt& ,/& for details>. ?or

(esign 2& the range of randomi<ation is shown as the dar"ened region on the

vertical ais. ;ote that this design change holds constant the num!er of sellers

and the aggregate su##ly and demand arrays& so that #rice differences can !e

attri!uted to the creation of mar"et #ower and not to other factors such as a small

num!er of sellers or a low ecess su##ly at su#ra+com#etitive #rices.



0ig$re '*.*. Capacit! -%%ocations for a 8o?Po6er;Po6er Design @So$rce: Davis and 7o%t, '443A

?igure ,5.% shows the results of a session in which 0 #eriods with the 8ower

design were followed !y 0 #eriods of the ;o+8ower design. he #eriod num!ers

and treatments are shown at the !ottom of the figure. he su##ly and demandfunctions are re#roduced on the left side =on a different scale from figure ,5.5> to

indicate the relative #ositions of the "ey #rice #rediction: the com#etitive #rice is

0. (emand was determined !y a #assive& #rice+ta"ing simulated !uyer& and

sellers were told the num!er of #eriods and all as#ects of the demand and su##ly

structure. n each #eriod& S1)s #osted #rice is #lotted as a !o& S)s #rice is

#lotted as a cross& and the other three small sellers) #rices are #lotted as dots.

n all !ut one of the first nine #eriods of this session& seller 62 =cross> stays at the

demand interce#t #rice& 5& selling the residual demand of , unit. his action

lured the other sellers u#& and seller 6, =!o> reached the demand interce#t #rice

in #eriod . hen 6,)s #rice cut in #eriod ,0 started a general #rice decline&which was sto##ed as cross goes !ac" u# to 5 for four #eriods. he final ,2

#eriods of this treatment contain two relatively tight #rice cycles.



0ig$re '*.1. Price Series for a Po6er;8o?Po6er Se#$ence @So$rce: Davis and 7o%t, '443A

Dey: 6,)s #rices are indicated !y !oes& 62)s #rices !y crosses& and the other #rices !y

dots.

Units are reallocated after #eriod 0 to ta"e away mar"et #ower =a movement

from (esign 2 to (esign , from ?igure ,5.5>. he initial high #rice !y 62 =cross>

results in no sales& and the aggressive com#etition that follows drives #rices to the

com#etitive =and now ;ash> levels. his mar"et efficiently e#loits all gains

from trade in the a!sence of mar"et #ower. 4ar"et #ower also resulted in large

#rice increases in five other sessions& holding constant the num!er of sellers and

the aggregate su##ly and demand structure. n fact& the #rice+increasing effect of mar"et #ower was considera!ly greater than the difference !etween the

;ash7ertrand #rice in the ;o+8ower treatment and the mean of the mied #rice

distri!ution in the 8ower treatment. 6ince e#licit communications !etween

sellers were not #ermitted& it is a##ro#riate to use the term tacit collusion to

descri!e this a!ility of sellers to raise #rices a!ove the levels determined !y a

;ash e'uili!rium. o summari<e& mar"et #ower has a dou!le im#act: it raises the

#redicted mean #rice& and it facilitates tacit collusion that raises #rices a!ove the

;ash #rediction.

. "(tensions4uch of the more recent effects of mar"et #ower can !e understood !y

reconsidering the narrowness of the vertical ga# !etween demand and su##ly to

the left of the su##lydemand intersection =see ?igure ,5.5>. his narrow ga#

im#lies that a seller who refuses to sell units with these relatively high costs will



not forego much in the way of earnings. his seller might #rofit from such

withholding if the #rice increase on low cost units sold would more than

com#ensate for the lost earnings on these marginal units. 6u##ose that the

demandsu##ly ga# is large during a !oom #eriod& which ma"es marginal units

more #rofita!le and limits the eercise of mar"et #ower. n contrast& the ga#

might fall during a contraction& there!y ena!ling sellers to raise #rices #rofita!ly.

hus the effects of mar"et #ower in some mar"ets may !e counter+cyclical

=3eynolds and $ilson& ,- and $ilson& ,->. A similar consideration may

arise in the analysis of mergers that may create synergies which lower costs. f

these cost reductions occur on units near the margin& i.e. near the su##lydemand

intersection& then the cost reduction on the marginal unit may ma"e the eercise

of mar"et #ower less #rofita!le. (avis and $ilson =,> discuss this case& and

also some contrary cases where cost+reducing synergies of a merger may create

#ower where none eists #reviously.

A second as#ect of mar"et #ower is that unilateral #rice increases are more li"ely

to !e successful if com#etitors are not a!le to move in and e#and their

#roduction. od!y =,-> e#lores the eercise of mar"et #ower in situationswhere #roducers must ac'uire #ollution #ermits to cover the !y#roducts of certain

#roduction activities. hen the ac'uisition of other sellers #ermits may limit their

ca#acities& and hence ena!le a firm to raise #rice #rofita!ly.

<$estions

,. 6how that the formula& ?= p> J p,0& #roduces the num!ers shown in the

second row of ta!le ,%.,. How would the formula have to !e changed if

#rices were uniform on the interval from 0 to 20& e.g. with half of the

#rices !elow ,0& ,/ !elow 5& etc.N2. Use the information given in the duo#oly eam#le from 6ection to

construct the su##ly and demand gra#h& and verify that they intersect at a

'uantity of and a #rice of 0.

. ?or the duo#oly eam#le in section & show that #rices of 0 for !oth

sellers do not consitute a ;ash e'uili!rium& i.e. show that a unilateral

#rice increase !y either seller will raise earnings.

/. Consider a modification of the duo#oly eam#le in section & where

demand is vertical at % units for all #rices !elow : & and each seller has a

ca#acity of 5 units at <ero cost. $hat is the range of #rices over which an

*dgeworth cycle would occurN ?ind the mied+strategy ;ash e'uili!rium

distri!ution of #rices& and com#are the range of randomi<ation with the

range of the *dgeworth cycle.

5. Answer 'uestion / for the case in which each seller s costs are constant at

, #er unit =with : P ,>.



%. Construct an eam#le with at least sellers& in which a merger of two of

these sellers reduces the costs of marginal units and ma"es the eercise of

mar"et #ower more #rofita!le. llustrate your answer with a gra#h that

shows each seller s (& and the way in which costs are reduced.

-. Construct an eam#le =again with at least sellers> of a merger that

destroys mar"et #ower !y reducing costs on marginal units. (raw a gra#h

that shows demand and seller costs and (s for each unit.



Cha#ter ,%. Collusion and 8rice Com#etition *ver since Adam 6mith& economists have !elieved that sellers often cons#ire to

raise #rice. 6uch collusion involves trust and coordination& and therefore& the #lan

may fall a#art if some sellers defect. n fact& 6mith)s oft+'uoted warning a!outthe li"elihood of #rice fiing is immediately 'ualified: n a free trade an effectual

com!ination cannot !e esta!lished !ut !y the unanimous consent of every single

trader& and it cannot last longer than every single trader continues of the same

mind. he ma9ority of a cor#oration can enact a !ye+law with #ro#er #enalties&

which will limit the com#etition more effectually and more dura!ly than any

voluntary com!ination whatever. =6mith& ,--%& #. ,//>. 8rice+fiing is illegal in

the U.6. and most other develo#ed economies& and hence it is difficult to study.

4oreover& cons#irators will try to "ee# their activities secret from those who have

to #ay the high #rices. $ithout good data on #artici#ants and their costs& it is

difficult to evaluate the nature and success of collusion& and the causes of

!rea"downs in #ricing disci#line. 1a!oratory e#eriments are not ham#ered !ythese data #ro!lems& since controlled o##ortunities for #rice fiing can !e

allowed& holding constant other structural and institutional elements that may

facilitate su#ra+com#etitive #ricing.

I. Co%%$sion in Posted 5ffer Markets

saac and 8lott =,,> allowed sellers in dou!le auctions to go to a corner of the

room and discuss #rices !etween trading #eriods. hese cons#iracies were not

very effective in actually raising transactions #rices in the fast+#aced com#etition

of a dou!le auction& where sellers are faced with the tem#tation to cut #rices in

order to sell marginal units late in the trading #eriod. saac& 3amey& and $illiams

=,/> su!se'uently showed that #rice+fiing activities can !e more effective in

#ostedoffer auctions& since sellers are not #ermitted to lower #rices once they are

#osted. 6ome #articularly interesting #atterns of #rice collusion are re#orted in

(avis and Holt =,>. here were three !uyers and three sellers in each session.

At the !eginning of each #eriod& the !uyers were ta"en from the room under the

guise of assigning different redem#tion values to them. $hile !uyers were out of

the room& sellers were allowed to #ush their chairs !ac" from their visually

isolated cu!icles so that they could see each other and discuss #rice. hey were

not #ermitted to discuss their own #roduction costs or to divide u# earnings. hen

they returned to their com#uters as the !uyers came !ac" into the room. At thattime& sellers would enter their #osted #rices inde#endently& without further

discussion. he !uyers were not aware of the seller #rice discussions.

he structure of the values and costs #roduced the su##ly and demand functions

shown on the left side of ?igure ,%.,. he intersection of these functions



determines the com#etitive #rice& which is shown as a hori<ontal thin line. f all

three sellers could set a #rice that maimi<ed total earnings for the three of them&

then they would each sell a single unit at a #rice indicated !y the thic" hori<ontal

line in the gra#h.

0ig$re '1.'. Prices for a Session 6ith 8o Co%%$sion @Lo6er Se#$ence of Circ%esA

and for a Second Session 6ith Co%%$sion @>pper Se#$ence of S#$aresA So$rce:

Davis and 7o%t @'441A

?irst consider a session in which !uyers were ta"en from the room for value

assignments& as in all sessions& !ut sellers were not #ermitted to fi #rices. he

#rices for this session are shown as the lower se'uence in ?igure ,%.,. he

circles are the list #rices& and the units sold are shown as small dots which create a

short hori<ontal line that !egins in the circle and may etend to the right if more

than two units are sold at this #rice. he #rices for each #eriod are se#arated !y

vertical lines& and the #rices for the three sellers are shown in order from left to

middle to right& for sellers 6,& 62& and 6 res#ectively. ;otice that the low+#rice

firm sells the most units in the first two rounds& and that the other #rices fall

'uic"ly. 8rices are roughly centered around the com#etitive #rice in the later

#eriods. he com#etitive nature of the mar"et =without collusion> was an

intentional design feature.

he u##er se'uence in ?igure ,%., shows a #attern of #rices& mar"ed withs'uares& that develo#ed when sellers were a!le to collude while !uyers were out

of the room. Attem#ts to fi a #rice resulted in high !ut varia!le #rices in the

early #eriods. 6ellers agreed on a common #rice in the 5th round& !ut all !uyers

made #urchases for seller 6,& which created an earnings dis#arity. 6eller 6, then



suggested that they ta"e turns having the low #rice& and that he& 6,& !e allowed to

go firstG his agreement was ado#ted& and 6, made all sales in #eriod %. he low

#rice #osition was rotated from seller to seller in su!se'uent rounds& much as the

famous #hases of the moon #rice fiing cons#iracy involving electrical e'ui#ment

in the sities. ;otice that there was some e#erimentation with #rices a!ove and

!elow the 9oint+#rofit+maimi<ing collusive level& !ut that #rices stayed at

a##roimately this level in most #eriods. his rotation scheme was 'uite

inefficient& since each seller had a low+cost unit that would not sell when it was

not their turn to have the low #rice. n other sessions with collusion& earnings

were enhanced !y agreeing to choose a common #rice and to limit sales to one

unit each& there!y avoiding the inefficiencies caused !y the rotation.

0ig$re '1.. Prices for a Second Posted 5ffer Session 6ith Co%%$sion

So$rce: Davis and 7o%t @'441A

8rices for a second session with collusion are shown in ?igure ,%.2. Again there

is some #rice varia!ility until sellers agree on a common #rice& this time in the

fourth #eriod. he failure of seller 62 to ma"e a sale at this #rice forces them to

deal with the allocation issue& which is solved more efficiently !y an agreementthat each will limit sales to , unit. his agreement !rea"s down several #eriods

later as 62& whose #rice is always listed in the center& lists a #rice !elow the

others. A high #rice is reesta!lished in the final % #eriods& !ut #rices remained

slightly !elow the 9oint#rofit+maimi<ing mono#oly level. n two of these final



#eriods& the sellers agreed to raise #rice slightly and hold sales to one unit each&

!ut in each case 62 sold two units& leaving 6 with nothing. hese defections

were covered u# !y 62& who did not admit the etra sale& !ut claimed in the

su!se'uent meeting that this is economics& and that there is less sold at a higher

#rice. In !oth occasions& the others went along with this e#lanation and agreed

to lower #rice slightly.

II. Co%%$sion 6ith Secret Disco$nts

4ost mar"ets of interest to industrial organi<ation economists cannot !e

classified as continuous dou!le auctions =where all #rice activity is #u!lic> or as

#osted offer mar"ets =which do not #ermit discounts and sales>. his raises the

issue of how effective #rice collusion would !e in mar"ets with a richer array of

#ricing strategies and information conditions. n #articular& mar"ets for #roducer

goods or ma9or consumer #urchases ty#ically differ from the ty#ical #osted+offer

institution in that sellers can offer #rivate discounts from the ZlistZ #rices. he

effectiveness of cons#iracies in such mar"ets is im#ortant for antitrust #olicy&since many of the famous #rice+fiing cases& li"e the electrical e'ui#ment !idding

cons#iracy discussed a!ove& involve #roducer goods. 4oreover& sales contracts

and !usiness #ractices that may deter discounts have !een the target of antitrust

litigation& as in the ?ederal rade Commission)s 4th# case& where the ?C

alleged that certain !est#rice #olicies deterred sellers from offering selective

discounts. he anticom#etitive nature of these !est+#rice #ractices is su##orted

!y results of some e#eriments run !y rether and 8lott =,/>& who used a

mar"et structure that was styled after the main characteristics of the mar"et for

lead+!ased anti+"noc" gasoline additives that was litigated in the *thyl case. Holt

and 6cheffman =,-> #rovide a theoretical analysis of !est+#rice #olicies and of

the e#erimental data re#orted !y rether and 8lott.n order to evaluate the effects of discount o##ortunities for the mar"et structure

considered in ?igures ,%., and ,%.2& (avis and Holt =,%> ran a third series of

sessions& in which sellers could collude as !efore& !ut when !uyers returned and

saw the sellers #osted #rices& they could re'uest discounts. he way this wor"ed

was that a !uyer whose turn it was to sho# could either #ress a !uy !utton for a

#articular seller or a re'uest discount !utton. he seller would then ty#e in a

#rice& which could !e e'ual to the list #rice =no discount> or lower. he seller

res#onse was not o!served !y other sellers& so the discounts were given secretly.

6ellers were free to discount selective to some !uyers and not others& and to hold

discounts until later in a #eriod.

8rices were much lower in the collusion sessions with o##ortunities for

discounting than in the collusion sessions with no such o##ortunities. n one

session& one of the sellers got so mad at the others that she refused to s#ea" with

them in the discussion #eriod !etween #eriods. *ven in grou#s that maintained an



active #rice+fiing discussion& the results were often sur#rising. Consider& for

eam#le& the #rice se'uence in ?igure ,%.. As !efore& the small s'uares indicate

list #rices& and the small dots indicate actual sales& often at levels well !elow

#osted list #rice. n #eriod & for eam#le& all sellers offer the same list #rice& !ut

6, sold two units at a dee# discount. n #eriods - and & seller 2 !egan secret

discounting& as indicated !y the dots !elow the middle #rice s'uare. hese

discounts caused 6, to have no sales in these #eriods& and 6, res#onded with a

shar# discount in #eriod and a lower list #rice in #eriod ,0. After this #oint&

discounts were #ervasive& and the outcome was relatively com#etitive. his

com#etitive outcome is similar to the results of several other sessions with

discounting.

0ig$re '1./. Prices for a Session 6ith Co%%$sion and Secret Disco$nts

So$rce: Davis and 7o%t @'441A

Ine factor that ham#ered the a!ility of sellers to maintain high #rices in the face

of secret discounts is the ina!ility to identify who is cheating on the agreement.

Antitrust investigations have found that many successful #rice+fiing

cons#iracies& es#ecially those with many #artici#ants& involve industries where atrade association re#orts relia!le sales information for each seller =Hay and Delly&

,-/>. his o!servation motivated the (avis and Holt =,> treatment where

sellers were given re#orts of each #erson s sales 'uantity at the end of each round.

All other #rocedures were identical& and secret discounts were #ermitted. *ven



with discounting& this e5 post sales information #ermitted sellers to raise #rices

a!out half way !etween the com#etitive #rice and the collusive #rice =the two

hori<ontal lines in the figures>.

III. "(tensions

o summari<e& the effects of mar"et #ower and e#licit collusion are much moresevere in mar"ets where sellers #ost #rices on a ta"e+it+or+leave+it !asis. he

o##ortunities to offer #rice reductions during the trading #eriod ma"es it difficult

to coordinate collusive #rice increases and to eercise #ower in the a!sence of

collusion. hese results are consistent with the antitrust hostility to industry

#ractices that are seen as limiting sellers o#tions to offer selective discounts.

here have !een a num!er of follow+u# studies on the effects of #rice collusion.

saac and $al"er =,5> found that the effects of collusion in sealed+!id auctions

are similar to those in #osted+#rice auctions. his is not sur#rising& since a sealed

!id auction is similar to a #osted+offer auction& ece#t that only a single unit or

#ri<e is ty#ically involved. Collusion in sealed !id auctions is an im#ortant to#ic&since many #rice+fiing cons#iracies have occurred in such auctions& and since

many com#anies are relying more and more on auctions to #rocure su##lies. he

design of the rrigation 3eduction Auction discussed in Cha#ter 22 was !ased to a

large etent on efforts to thwart collusion& and for that reason& the auction allowed

!idders to ad9ust their !ids in a series of rounds& with no fied final round.

Another interesting issue is the etent to which the #attern of !ids might !e used

to infer collusion =(avis and $ilson& ,>. he idea here is that there may !e

less correlation !etween losing !ids and costs when the losing !idders have

agreed in advance to !id high and lose =8orter and Tona& ,>. Cason =,->

investigates the #ossi!ility that the ;A6(AL dealers convention of relying on

even eighths may have facilitated collusion.here have !een allegations that com#etitors who #ost #rices on com#uter

networ"s may !e a!le to signal threats and coo#erative intentions. ?or eam#le&

some !idders in early ?CC !andwidth auctions used decimal #laces to attach <i#

codes to !ids in an attem#t to deter rivals !y threatening to !id on licenses that

rivals were trying to o!tain. 6imilarly& some airlines attached aggressive letter

com!inations =e.g. ?U> to tic"et #rices #osted on the Airline ariff 8u!lishing

=A8> com#uteri<ed #rice system. Cason =,5> re#orts that such non!inding

= chea# tal" > communications can raise #rices& !ut only tem#orarily. 6ee Holt

and (avis =,0> for a similar result for a mar"et where sellers could #ost non+

!inding intended #rices !efore #osting actual #rices. hese non+!inding #rice

announcements would corres#ond to the #osting of intended future #rices on the

A8& which are visi!le to com#etitors !efore such #rices are actually availa!le for

consumers. Cason and (avis =,5> find that such #rice signaling o##ortunities



had more of an effect in a multi+mar"et setting& !ut even here the effects of #urely

non!inding #rice announcements were 'uite limited.



Cha#ter ,-. 8roduct Luality& Asymmetric

nformation& and 4ar"et ?ailure

$hen !uyers can o!serve !oth #rice and #roduct 'uality #rior to #urchase& thereis #ressure on sellers to #rovide good 'ualities at reasona!le #rices. 7ut when

'uality is not o!served& a seller may !e tem#ted to cut 'uality& and the result will

!e disa##ointed !uyers who are hesitant to #ay the high #rices needed to cover the

costs to the seller of #roviding a high 'uality. $hen !uyers e#ect to encounter

low+'uality items& or lemons& the resulting downward #ressure on #rice may force

sellers to cut !oth #rice and 'uality. he Zlemons mar"etZ terminology is due to

eorge A"erlof =,-0>& who e#lained how the #ressure of com#etition may

cause 'uality to deteriorate to such low levels that the mar"et may fail to eist.

he mar"et failure that is #redicted in this case can !e studied with the hel# of

la!oratory e#eriments. hese e#eriments can !e run !y hand& using the

instructions #rovided& or they can !e run with the Veconla! #rogram 1*.

I. Prod$ct <$a%it!

he mar"ets considered in this cha#ter have the #ro#erty that sellers can choose

the 'uality of the #roduct that they sell. A high 'uality good is more costly to

#roduce& !ut it is worth more to !uyers. hese costs and !enefits raise the issue

of whether or not there is an o#timal 'uality. 4any& or even most& mar"ets have

the #ro#erty that !uyers are diverse in their willingness to #ay for 'uality

increases& so there will ty#ically !e a variety of different 'uality levels !eing sold

at different #rices. *ven in this case& sellers who offer high+cost& high+'uality

items may face a tem#tation to cut 'uality slightly& es#ecially if !uyers cannoto!serve 'uality #rior to #urchase. Luality might !e maintained in these mar"ets&

even with asymmetric information a!out 'uality& if sellers can ac'uire and

maintain re#utations& re#orted or signaled !y warranties and return #olicies.

7efore considering the effects of such #olicies& it is useful to eamine how

mar"ets might fail if 'uality cannot !e o!served in advance !y !uyers.

?or sim#licity& consider a case where all !uyers demand at most one unit each

and have the same #references for 'uality& which is re#resented !y a numerical

grade& g . he maimum willingness to #ay for a unit of the commodity will !e an

increasing function of the grade& which will !e denoted !y V= g >. 6imilarly& let the

cost #er unit !e an increasing function& C= g >& of the grade. he net value for eachgrade g is the difference: V= g > C= g >. he o#timal grade maimi<es this

difference. he "ey thing to notice is that the o#timal grade may not !e the

maimum feasi!le grade. ?or eam#le& it is often #rohi!itively e#ensive to

remove all im#urities or reduce the ris" of #roduct failure to <ero. he im#ortant



!ehavioral issue in this ty#e of mar"et is the etent to which com#etition in the

mar"et #lace will force 'uality to near+o#timal levels.

II. - C%assroom "(periment

Holt and 6herman =,0> use % 'uality grades in a research e#eriment& !ut

some of the main conclusions can !e illustrated with a sim#ler classroome#eriment with grades =Holt and 6herman& ,>. *ach !uyer only demands

one unit of the commodity& and !uyers have identical valuations. he value of the

commodity de#ends on the 'uality grade: /.00 for grade ,& .0 for grade 2&

and ,.%0 for grade . hus& with four !uyers and a grade of ,& for eam#le& the

mar"et demand would !e vertical at a 'uantity of / units for any #rice !elow /&

as shown !y the demand curve in the lower #art of ?igure ,-.,. he demands for

higher grades are similar& with cutoff #rices of .0 and ,.%0 for grades 2 and

.

0ig$re ').'. Demand and S$pp%! -rra!s &! Grade

4ar"et su##ly is determined !y the sellers costs given to sellers. *ach seller has

a ca#acity to #roduce two units& with the cost of the second unit !eing , higher

than the cost of the first unit. ?or a grade of ,& the costs for the first and second

units #roduced are ,./0 and 2./0 res#ectively& so each individual seller)s su##ly



curve would have two ste#s& !efore !ecoming #erfectly inelastic at two units for

#rices a!ove 2./0. $ith three identical sellers& the mar"et su##ly will also have

two ste#s with three units on each ste#. he mar"et su##ly for grade , is la!eled

6, in the lower #art of ?igure ,-.,& and it crosses the (, curve at a #rice of 2./0.

he su##ly and demand curves for the other grades are shown a!ove those for

grade ,. he total sur#lus =for consumers and #roducers> corres#onds to the area

!etween the su##ly and demand curves for a given grade. t is a##arent from

?igure ,-., that the sum of consumer and #roducer sur#lus is maimi<ed at a

grade of 2& which is& in this sense& the o#timal 'uality.

he results of a classroom e#eriment are shown in a!le ,-.,. his was run !y

hand with the instructions from the A##endi& with / !uyers and sellers. n the

first three rounds& sellers chose #rice and grade and wrote them on record sheets.

$hen all had finished& these #rices and grades were written on the !lac"!oard&

and !uyers were selected one !y one in random order to ma"e #urchases. wo of

the three 'uality choices in the first round were at the maimum level. *ach of

these high+'uality sellers sold a single unit& !ut the seller who offered a grade of 2

sold 2 units and earned more. he !uyers #referred the grade 2 good since it #rovided more sur#lus relative to the #rice charged. All three sellers had settled

on the o#timal grade of 2 !y the third round& and the common #rice of 5.%0 was

a##roimately e'ual to the com#etitive level determined !y the intersection of

su##ly and demand at this grade.

a!le ,-., 3esults from a Classroom *#eriment

6ource: Holt and 6herman=,>

Se##er 1 Se##er Se##er 3

8eriod , =fullinformation>

,,.50grade sold

,

%.00grade 2 sold

2

,2.00grade sold

,

8eriod 2 =full

information>

5.-5

grade2 sold

2

5.50

grade 2 sold

,

,.0

grade , sold

,

8eriod =full

information>

5.%5

grade2 sold,

5.%0

grade 2 sold

2

5.%0

grade 2 sold

,

8eriod /

=only #rice information>

2./0

grade, sold

,

5.%0

grade2 sold

,

2./0

grade , sold

2

8eriod 5 2./0 ,.%5 5.50

=only #rice information> grade , grade , grade,

sold , sold , sold 2



he first full information rounds were followed !y two rounds in which sellers

selected #rice and grade as !efore& !ut only the #rice was written on the !oard for

!uyers to see while sho##ing. wo of the sellers immediately cut grade to , in

round /& although they also cut the #rice& so !uyers would !e ti##ed off if theyinter#ret a low #rice as a signal of a low grade. n the final round& seller offers a

low grade of , at a #rice of 5.50& which had !een the going #rice for a grade of

2. he !uyer who #urchased from this seller must have antici#ated a grade of 2&

and this !uyer lost money in the round.

0ig$re ').. Data from a econ%a& C%assroom "(periment

?igure ,-.2 shows the results of an e#eriment with the same structure& !ut which

was done with the Veconla! software& which made it #ossi!le to run more rounds.

here were - !uyers and 5 sellers& which determined the su##ly and demand

functions for the three grades as shown on the left side of ?igure ,-.2. Again&

most of the sales are at an o#timal grade of 2 in the full+information rounds =,+>&

although sellers managed to "ee# #rices a!ove com#etitive levels in these rounds&

es#ecially for the grade 2 items. $hen only #rice information was allowed to !e #osted in round & the grade levels immediately fell to ,& although most of the

sales were at #rice levels that would result in losses for !uyers.




here have !een a num!er of e#erimental studies of mar"ets with asymmetric

'uality information. 1ynch et al. =,%> document lemons outcomes in dou!le

auction mar"ets& and they show that #erformance can !e im#roved with certain

ty#es of warranties& re'uirements for truthful advertising& etc. he lemons mar"et

e#eriment summari<ed in this cha#ter is !ased on the setu# in Holt and 6herman

=,0>& who used la!oratory e#eriments to evaluate factors that affect the degree

to which 'uality deteriorates when it is not o!served !y !uyers. (eFong& ?orsythe&

and 1undholm =,5> allowed sellers to ma"e #rice and 'uality re#resentations&

!ut the 'uality re#resentation did not have to !e accurate& and the !uyer had

im#erfect information a!out 'uality even after using the #roduct. 4iller and 8lott

=,5> re#ort e#eriments in which sellers can ma"e costly decisions that ZsignalZ

high 'uality& which might #revent 'uality deterioration. his literature is

surveyed in (avis and Holt =,& cha#ter -> and Holt =,5>.

<$estions'. Use the unit costs and redem#tion values for the mar"et design in ?igure

,-., to calculate the total sur#lus for each grade level.

. n a com#etitive e'uili!rium for a given grade& the #rice is determined !y

the intersection of su##ly and demand& for that grade. Com#are the

com#etitive e'uili!rium #rofits for each grade for the mar"et structure in

?igure ,-.,.

/. ?or the mar"et structure shown in ?igure ,-.,& a small increase in the cost

for each seller s first unit& say !y 0.25& would mean that the com#etitive

e'uili!rium #rofits for each seller are lower for grade 2 than for the other

grades. $hat effect do you thin" this cost increase would have on the

tendency for grade choices !y sellers to converge to the o#timal grade =2>Nf there are conflicting mar"et forces& which one do you thin" would

dominateN



Cha#ter ,. Asset 4ar"ets and 8rice 7u!!les

4any electronic mar"ets for assets are run as call mar"ets in which traders

su!mit limit orders to !uy or sell& and these orders are used to generate a single&

mar"et+clearing #rice when the mar"et is called. hus all asset shares trade at thesame #riceB the shares are #urchased !y those who su!mitted !uy orders with a

maimum willingness to #ay limit at or a!ove this #rice& and the shares are sold

!y those who su!mitted sell orders with a minimum willingness to acce#t limit at

or !elow this #rice. he e#eriment discussed in this cha#ter involves trading of

shares that #ay a randomly determined dividend. raders are endowed with asset

shares and cash that can either !e used to #urchase shares or earn interest in a safe

account. he interest rate determines the value of the asset on the !asis of

fundamentals: dividends and associated #ro!a!ilities& and the redem#tion value of

each share after a #re+announced final round of trade. his fundamental value

serves as a !enchmar" from which #rice !u!!les can !e measured. his setu# is

easy to im#lement using the Veconla! 1imit Irder Asset 4ar"et #rogram. 1arge !u!!les and crashes are ty#ical for ine#erienced traders& and the resulting

discussion can !e used to teach lessons a!out #resent value& !ac"ward induction&

etc.

I. B$&&%es and Crashes

(es#ite the wides#read !elief that #rices in e'uity mar"ets rise steadily over the

long term& these mar"ets ehi!it strong swings in #rice that do not seem to !e

9ustified !y changes in the underlying economic fundamentals. Deynes

e#lanation for these swings was that many =or even most> investors are lessconcerned with the fundamentals that determine long+term future #rofita!ility of a

com#any than with what the stoc" might sell for in several wee"s or months.

6uch investors will try to identify stoc"s that they thin" other investors will floc"

to& and this herding may create its own self+confirming u#ward #ressure on #rice.

he #sychology !ehind this #rocess is descri!ed !y Charles 4ac"ay s =,/,>

account of the (utch tuli#mania in the ,-th century:

;o!les& citi<ens& farmers& mechanics& seamen& footmen&

maidservants& even chimney+swee#s and old clotheswomen&

da!!led in tuli#s. Houses and lands were offered for sale at

ruinously low #rices& or assigned in #ayment of !argains made atthe tuli#+mart. ?oreigners !ecame smitten with the same fren<y&

and money #oured into Holland from all directions.



If course& tuli#s are not scarce li"e diamonds. hey can !e #roduced& and it was

only a matter of time until the correction& which in 4ac"ay s words ha##ened:

At last& however& the more #rudent !egan to see that this folly

could not last for ever. 3ich #eo#le no longer !ought the flowers

to "ee# them in their gardens& !ut to sell them again at cent #er

cent #rofit. t was seen that some!ody must lose fearfully in the

end. As this conviction s#read& #rices fell& and never rose again.

Confidence was destroyed& and a universal #anic sei<ed u#on the

dealers.

*ven if traders reali<e that #rices are out of line with #roduction costs and #rofit

o##ortunities in the long run& a "ind of overconfidence may lead traders to !elieve

that they will !e a!le to sell at a high level& or that they will !e a!le to sell 'uic"ly

enough in a crash to #reserve most of the ca#ital gains that they have

accumulated. he #ro!lem is that there are no !uyers at anything li"e #revious

#rice levels in the event of a crash& and often all it ta"es is slight decline& #erha#sfrom an eogenous shoc"& to s#oo" all !uyers and stimulate the su!se'uent free

fall in #rice.

8rice !u!!les and crashes can !e recreated in com#uter simulations !y

introducing a mi of trend+!ased and fundamentals+!ased traders. hen #rice

surges can !e stimulated !y #ositive eogenous shoc"s& and such simulations can

even #roduce negative !u!!les that follow negative shoc"s =6teiglit< and 6ha#iro&

,>. n a negative !u!!le& the trend traders are selling !ecause they e#ect

#rior #rice decreases to continue and this sell #ressure draws #rices down if the

fundamentals+!ased traders do not have the resources to correct the situation.

Com#uter simulations are suggestive& es#ecially if they mimic the #rice and

trading volume #atterns that are o!served in stoc" mar"et !ooms and crashes.he o!vious 'uestion& however& is how long human traders would stic" to

mechanical trading rules as conditions change. Ine #ro!lem with running a

la!oratory e#eriment with human su!9ects& however& is to decide how owners of

unsold shares are com#ensated when the e#eriment ends.

6mith& 6uchane"& and $illiams =,> addressed the end#oint #ro!lem !y #re+

s#ecifying a redem#tion value for the asset after 20 #eriods. raders were

endowed with asset shares and cash& and they could !uy and sell assets at the start

of each #eriod =via dou!le auction trading>. Assets owned at the end of a #eriod

#aid a dividend that was ty#ically randomly determined from a "nown

distri!ution. Cash did not earn any interest& and all cash held at the end wasconverted to earnings at a #re+announced rate.

n most of their sessions& the final redem#tion value was set to <ero. o a ris"+

neutral #erson& each share in this mar"et then #rovides value that e'uals the

e#ected value of the dividend times the num!er of #eriods remaining. ?or



$000

$050

$100

$150

$200

$250

$300

$350

$400

9 10 11 122 14 150 1 3 4 5 6 7 8 13

eam#le& consider an asset that #ays 0.50 or ,.50& each with #ro!a!ility ,2& so

the e#ected dividend is ,.00. his asset would only !e worth a num!er of

dollars that e'uals the num!er of #eriods remaining. hus with 20 rounds& the

e#ected value of the asset in round t that is 20 t & i.e. the fundamental value of

the asset declines linearly over time.

0ig$re '2.'. -verage ransactions Prices in a '* Ro$nd Do$&%e -$ction 6ith

a Dec%ining 0$ndamenta% a%$e

$ith this declining+value setu#& #rice !u!!les were o!served in many =!ut not

all> sessions. ?igure ,., shows the results of a ty#ical #rice !u!!le for a sessionwhere the final redem#tion value was 0.00 and the e#ected dividend was 0.,%.

6ince there were ,5 rounds& the fundamental value line starts at 2./0 and

declines !y 0.,% #er #eriod& reaching 0.,% in the final round. he figure shows

the average transactions #rices for each round& as determined !y dou!le auction

trading. he #rices were !elow the fundamental value line in early rounds. hen

#rices increase steadily and rise a!ove the declining value line. 8eo#le who !uy

in one round and see the #rice rise& would often try to !uy again& and others

wanting to 9oin in on the gains would !egin to !uy as well. he #rice tra9ectory

flattened out and then fell as the final round a##roached& it would !ecome o!vious

that no!ody would #ay much more than the e#ected dividend in the final round.

6uch !u!!les were o!served with undergraduates& graduate students& !usiness

students& and even a grou# of commodity traders. (es#ite the lac" of realism of

having a fied end#oint& the fact was that s#eculative !u!!les could form& even if

#eo#le "new that there was a fied end #oint. hese #rice #atterns suggest that



there might !e more fren<y in mar"ets with no final #eriod. he o!served

!u!!les are sur#rising given the o!vious nature of the <erovalue in the final

round. 6mith& 6uchane"& and $illiams did run some sessions with a non+<ero

final redem#tion value& which was set to !e e'ual to the sum of the reali<ed

dividend #ayments for all rounds. hus the final redem#tion value would not !e

"nown in advance& !ut rather& would !e determined !y the random dividend

reali<ations& and would generally !e decreasing from #eriod to #eriod ='uestion

,>.

6ince most financial assets do not have #redicta!le& declining fundamental

values& it is instructive to set u# e#eriments with constant or increasing values.

7all and Holt =,> achieved this in a classroom e#eriment !y using a fied

#ro!a!ility of !rea"age to induce a #reference for the #resent =when the asset is

not !ro"en> over the future. n their setu#& there was a ,% chance that each share

would !e destroyed& and they threw a si+sided die se#arately for each asset share

at the end of each trading #eriod =after trades were made via dou!le auction and

after dividends of , #er share were #aid>. Any shares that remained after at the

end of a "nown final round could !e redeemed for at a #re+announced rate of % #er share.

Ine =incorrect> way to value this asset would !e to ta"e the final redem#tion

value of % and add the total dividends of , #er round& with some ad9ustment for

the fact that an asset only has a 5% chance of surviving for another #eriod& e.g. %

O =5%> O =5%>=5%> O .... he #ro!lem with this a##roach is that the % final

#ayment is unli"ely to !e received. *ven an asset that has survived u# to the

!eginning of the final #eriod has a 5% chance of #roducing the % redem#tion

value. 6o at the start of the final round& the value of the asset is the dividend of ,

#lus the 5% chance of getting %& which for a ris"+neutral #erson would !e , O

=5%>% J , O 5 J %. hus an asset that is worth % at the end of the final round is

also worth % at the !eginning of the round. $or"ing !ac"wards& the net ste# isto thin" a!out the value of the asset at the !eginning of the netto+last round.

his asset will #ay , in that round and have a 5% chance of surviving to !e

worth % at the !eginning of the last round& which for a ris"neutral #erson

!ecomes: , O =5%>% J %. hus the asset is valued at % in the last two rounds.

n this manner& it can !e shown that the fundamental value of the asset is % in all

rounds& assuming ris" neutrality.

his setu#& with a constant fundamental value of %& #roduced #rice !u!!les in

some& !ut not all& trading se'uences. n one se'uence& #rices started at a little

!elow % in round , and rose to near in round 5& !efore #rices fell near the

final round =>. An o!vious 'uestion is whether !u!!les would !ecome more

severe with more rounds& since the final round would seem more distant in early

rounds& although the asset destruction #rocess would cho"e off trade as the

num!er of shares fell. Another #ro!lem is that the dou!le auction trading used in

this e#eriment is relatively time+consuming& even if trading is limited to



minute #eriods. 6ection #resents an alternative a##roach !ased on limit order

trading with a single mar"et+clearing #rice when the mar"et is called. 7ut first& it

will !e useful to discuss how assets with time streams of future dividends should

!e valued in the #resent.

II. - Digression on Present a%$e

6u##ose that money can !e invested in a safe account that earns interest at a rate

of r #er #eriod. he sim#lest case to consider is an asset that only #ays a dividend

of ( dollars once& at the end of the first #eriod& and then the asset loses all value.

Valuing the asset means deciding what to #ay now for a dividend of ( that arrives

one #eriod later. his asset is worth less than ( now& since you could ta"e (

dollars& invest at an interest rate r & and earn =,Or > (& !y the end of the #eriod&

which is greater than (. hus it is !etter to have ( dollars now than to have an

asset that #ays ( dollars one #eriod from now. his is the essence of the

#reference for #resent over future #ayments& which causes one to discount such

future #ayments. o determine how much to discount& lets consider an amountthat is less than (& and in #articular consider (=,Or > dollars now and invest at

rate r & the amount o!tained at the end of the #eriod is the investment of (=,Or >

times ,Or & which e'uals (. o summari<e& the present va#ue of getting ( dollars

one #eriod from now is (=,Or >. 6imilarly& the #resent value of any amount 6 to

!e received one #eriod from now is 6 =,Or >. Conversely& an amount : invested

today yields : =,Or > one #eriod from now. hese o!servations can !e

summari<ed:

#resent value of future #ayment ?: V J ?=,Or>

future value of #resent investment V: 6 : =, r >.

6imilarly& to find the future value of an amount V that is invested for one #eriod

and then reinvested at the same interest rate& we multi#ly the initial investment

amount !y =,Or > and then !y =,Or > again. hus the future value of : dollars

invested for two #eriods is : =,Or >2. Conversely& the #resent value of getting ?

dollars two #eriods from now is ?=,Or >2.

#resent value of future #ayment ? =2 #eriods later>: V J ?=,Or>2

futur

e value =2 #eriods later> of #resent investment V: 6 : =, r > .2

n general& the #resent value of an amount ? received t #eriods into the future is ?

=,Or>t. $ith this formula& one can value a series of dividend #ayments for any

finite value of t. ?or eam#le& the #resent value of an asset that #ays a dividend

( for two #eriods and then is redeemed for ' at the end of #eriod 2 would !e:



(=,Or > O (=,Or >2 O '=,Or >2& where the final term is the #resent value of the

redem#tion value.

he final issue to !e addressed is how to value a series of dividend #ayments that

has no terminal #oint. he main result is:

( ( ( ( (

=,.,> V , r =, r >2 ... t ... r .

=, r > =, r >

As indicated !y the right+hand e'uality& this #resent value turns out to !e (r .

Ine way to verify this is to use a mathematical formula for finding the sum of an

infinite series:

=,.2> , 5 52 5 ..., if 5 J,.

, 5

o a##ly this formula& we need to regrou# the terms in e'uation =,.,> so that they !egin with a ,:

( K , , L=,.> V , M, r >

=, r >2 ...O r =,

;ow we can a##ly the formula in =,.2> to evaluate the sum of terms in the s'uare

!rac"ets in =,.> !y letting 5 J ,=,Or>& which is less than , as re'uired. 6ince ,

5 J , ,=,Or> J r =,Or>& it follows that ,=, 5> is =,Or >r & which e'uals the sum of

the terms in s'uare !rac"ets on the right side of =,.>. 7y ma"ing thissu!stitution into =,.> and letting the =,Or > terms cancel from the numerator and

denominator& we o!tain the resulting #resent value for the infinite series of

dividends of ( dollars #er #eriod is (r .

t will !e instructive to develo# an alternative derivation of the result that the

#resent value of an infinite sum of dividends& ( #er #eriod& is (r . ?irst& since the

future is infinite and the dividends do not change& it follows that the future always

loo"s the same& even as time #asses. hus the #resent value of the future stream

of dividends will !e the same in each #eriod. 1et this #resent value !e : . $e

can thin" of : as !eing the #resent value of getting a dividend at the end of the

round& #lus the #resent value of getting the asset =always worth : > !ac" at the end

of the round. hus : J (=,Or > O : =,Or >& where the first term is the effect of theend+of+#eriod dividend& and the second term re#resents what the asset could !e

sold for at the end of the round after getting the dividend. his e'uation can !e

solved to o!tain: : J (r .



III. he Limit 5rder Market "(periment

n this section& we consider a sim#le asset #ricing model with a safe asset that

#ays an interest rate of r #er #eriod =e.g. an insured savings account> and a ris"y

asset that either #ays a high dividend& A & or a low amount& -& each with

#ro!a!ility ,2. 1et the e#ected value of the dividend !e denoted !y ( J

= A O ->2. As shown in the #revious section& the value of the asset would !e (r if

the num!er of #eriods were infinite. n the e#eriment to !e discussed& A J

,.00& - J 0./0& so ( J 0.-0. $ith an interest rate of ,0 #er cent& r J 0.,& and

hence the #resent value of an infinite+#eriod asset would !e 0.-00., J -.00.

his is the #resent value of future dividends& and it does not change as time

#asses& since the future is always the same. he tric" in setting u# a finite

hori)on e#eriment with an asset value that is constant over time is to have the

final#eriod redem#tion value !e e'ual to (r & which would !e the #resent value of

the dividends if they were to continue forever. n this manner& the infinite future

is incor#orated into the redem#tion value #er share in the final #eriod& and the

resulting asset value will !e flat over time& even as the final round is a##roaching.

hus the redem#tion value was set to !e -.00.As mentioned in the introduction& trades were arranged !y letting #eo#le su!mit

limit orders to !uy andor sell& and then using these orders to determine a single

mar"et+clearing #rice at which all trades are eecuted when the mar"et is closed

or called. his setu# is ty#ically called a Call 4ar"et& since the #rocess ends and

trades are determined at a #re+s#ecified time& e.g. at the end of the !usiness day&

or an hour !efore the financial mar"ets o#en in the morning. he e#eriment

used the Veconla! 1I4 #rogram& and the call occurred when all traders had

su!mitted orders& or when the e#erimenter #ressed the 6to# !utton on the Admin

3esults #age.

A sell order involves s#ecifying the maimum num!er of shares to !e sold& and a

minimum #rice that will !e acce#ted. A !uy order also sti#ulates a maimumnum!er of shares& !ut the difference is that the #rice is the ma5imum amount that

the #erson is willing to #ay. 7uy orders are ran"ed from high to low in a #seudo

demand array& and the sell orders are ran"ed from low to high in a #seudo su##ly

array. f more than one trader su!mits the same limit order #rice& then a

randomly determined #riority num!er is used to decide who gets listed first& i.e.

who is highest in the !id 'ueue or who is lowest in the offer 'ueue. hen these

su##ly and demand arrays are crossed to determine the mar"et+clearing #rice and

'uantity. ?or eam#le& if one !uyer !ids 5 for 2 shares and if another !ids / for

2 shares& then the demand array would have ste#s at 5 and /. 6u##ose that the

only sell order involved shares at a limit #rice of . hen shares would trade

at a #rice of /& since there are / shares demanded at any #rice !elow / and only

2 shares demanded at any higher #rice =#ro!lem >.

he first e#eriment to !e discussed involved ,2 traders& each with an

endowment of 50 in cash and % shares. he other #arameters were as discussed



a!ove: an interest rate of ,0 #er cent& dividends of either 0./0 or ,.00 #er share&

and a final+#eriod redem#tion #ayment of - #er share at the end #eriod& 20.

raders in this and all su!se'uent mar"ets to !e discussed were #aid an amount

that was ,,00 of total earnings.

0ig$re '2.. - + Ro$nd Ca%% Market 6ith a Constant 0$ndamenta% a%$e of ).++

he trading activity is shown in ?igure ,.2& where the hori<ontal line at -.00

indicates the fundamental =#resent> e#ected value of the asset. he first round

results are to the left of the first vertical !lac" line on the left side of the figure.

he clearing #rice was ,0 in that round& as indicated !y the dar" hori<ontal line

segment at ,0 on the left side of the figure. he dar" line segments for

su!se'uent #eriods show the mar"et clearing #rices for those #eriods& and the

width of each segment is #ro#ortional to the num!er of shares traded. he !ids

and as"s for traded shares are shown in !y the light dots that are a!ove the

clearing #rice for !ids and !elow for as"s. here were more transactions in round

2& !ut the #rice stayed the same& !efore falling in round /. he second #rice dro#

in round - was followed !y a shar# rise and continued high #rices near ,2&

which #ersisted until a steady decline !egan in round , and continued until

#rices fell to near the -.00 level in the final round.

he second mar"et& shown in ?igure ,.& also involved ,2 traders and 20

rounds& with the same #arameters which yield a #resent value of - #er share. n

this case& the trading started at ,2& which had !een the high in the #revious case.8rices rise steadily to a level of a!out 2 that is four times the level of the

#resent value as calculated from the mar"et fundamentals. ;ote that #eo#le who

!ought at #rices a!ove this value may have done so #rofita!ly if they were a!le to

sell at higher #rices. Ince the #rice !egan to dro# dramatically in round ,& the



volume of trade fell shar#ly& since few !uyers were now willing to #ay #rices that

were anything close to the 2 level that had #revailed #rior to round ,-.

0ig$re '2./. - + Ro$nd Ca%% Market 6ith a Constant 0$ndamenta% a%$e of ).++

he shar# dro#s that sometimes occured 9ust #rior to the final rounds of the 20

round mar"ets was the motivation for conducting some with a /0 round hori<on&

!ut with all other #arameters unchanged. Ine of these mar"ets& again involving

,2 traders& is shown in ?igure ,./. Here #rices !egin at eactly - in #eriod ,&

and then !egin a steady rise in su!se'uent #eriods. 8rice increases !ecome more

dramatic after #eriod 25. 8rice reaches a high of 25- in round ,& which is over

0 times the fundamental valueG he crash& when it comes& starts slowly and then

accelerates& with a dro# from 200 to ,00 in round /& and continued dro#s& on

somewhat higher trading volume after that. ndividual earnings were highly

varia!le& with one #erson earning over -0 =after dividing final cash !y ,00>& and

with several #eo#le ending u# with less than 5. he e#eriment lasted a!out an

hour.



0ig$re '2.3. - 3+ Ro$nd Ca%% Market 6ith a Constant 0$ndamenta% a%$e of ).++

I. 5ther Research on the Ca%% Market Instit$tion

here is a large and growing literature in which e#eriments are used to study

!ehavior in financial mar"ets. Ine im#ortant to#ic is the etent to which traders

with inside information a!out the value of an asset are a!le to e#loit this

information and earn more than uninformed traders. n some cases& mar"et #ricesreveal inside information and the trading #rices converge to levels that would !e

e#ected if this information were #u!lic& i.e. to a rational e#ectations #rediction

=8lott and 6under& ,2>. n #articular& the uninformed traders in the $atts =,2>

e#eriment did not "now whether or not others were informed& and in this case&

insiders tended to earn higher #rofits& although #rice convergence #atterns were

generally su##ortive of rational e#ectations #redictions.

here is a second& related strand of the literature on !ehavior in call mar"ets

where !uyers and sellers su!mit limit #rices that determine the common& mar"et+

clearing #rice. hese are called call mar"ets !ecause the final mar"etclearing

#rice is calculated when the mar"et is called& usually at a #re+announced time.

Call mar"et trading is commonly used for electronic trading of stoc"s on

echanges where there is not sufficient trading volume to su##ort the continuous

trading of a dou!le auction #rocess li"e that used for the ;ew Mor" 6toc"



*change. And a call mar"et mechanism is used to determine the o#ening #rices

on the ;ew Mor" 6toc" *change =Cason and ?riedman& ,->.

A classic study of com#uteri<ed call mar"ets is that of 6mith& et al. =,2>& who

#rovided !uyers with redem#tion values for each unit and sellers with costs for

each unit. =his "ind of call mar"et can !e run with the Veconla! software using

the C4 #rogram on the 4ar"ets 4enu.> 6mith com#ared the outcomes of call

mar"ets with those of a dou!le auction& in a stationary environment where costs

and values stay the same from #eriod to #eriod. 8rice convergence to com#etitive

levels was generally faster and more relia!le in the dou!le auction& !ut one

variant of the call mar"et trading rules yielded com#ara!le efficiency. ?riedman

=,> re#orts #rice formation results for call mar"ets that are almost as relia!le

as those of com#ara!le dou!le auctions.

An eam#le of a call mar"et run with this software is shown in ?igure ,.5.

Values and costs were determined randomly at the start of the first round and

remained the same for all 5 rounds. hese values and costs determine the su##ly

and demand functions& shown on the left side of the figure. he clearing #rice

converges to the com#etitive #rediction& and the !id and as" arrays flatten out ata##roimately this level.

raders with multi#le units in a call mar"et may attem#t to eercise mar"et

#ower. ?or eam#le& a seller with a unit that has a cost close to the com#etitive

#rice may wish to hold it !ac" in an effort to shift the revealed su##ly curve u#

and there!y raise the #rice received on the seller s other =low+cost> units.

6imilarly& !uyers may see" to hold !ac" !ids for marginal units in an effort to

lower #rice. he 4cCa!e& 3assenti& and 6mith =,> results revealed that

traders on each side of the mar"et sometimes tried to eercise mar"et #ower in

call mar"ets& as discussed in Cha#ter ,%. (es#ite these incentives& the mar"ets

that were run tended to !e 'uite efficient& at levels com#ara!le to those of dou!le

auctions.Dagel and Vogt =,> re#ort results of two+sided call mar"ets where !uyer

values and seller costs were determined randomly at the start of each trading

#eriod. he resulting #rices are com#ared with #redictions determined !y a ;ash

e'uili!rium in which each #erson s !id or as" is a function of their reali<ed value

or cost =see the net cha#ter on #rivate value auctions for this ty#e of calculation>.

Again& the Dagel and Vogt result is that the call mar"et is slightly more efficient

than the dou!le auction. his result must !e considered in light of the fact that

call mar"ets are easier to administer and can !e run after hours&



0ig$re '2.*. - 0ive?Ro$nd C%assroom Ca%% Market

with traders who connect from remote locations and who do not need to monitor

trading activity as carefully& since all included trades will !e at the same #rice.

Cason and ?riedman =,-> also com#are call mar"et #rices to the ;ash

#redictions. hey find some #ersistent differences& and they use a model of

learning to e#lain these differences.

<$estions

,. Consider the setu# discussed in section & with no interest #ayments.

6u##ose that the e#eriment lasts for ,5 #eriods& with dividends that are

either 0.,2 or 0.2/& each with #ro!a!ility ,2& and with a final

redem#tion value that e'uals the sum of the dividends reali<ed in the ,5

#eriods. here is no interest #aid on cash !alances held in each round.

hus the actual final redem#tion value will de#end on the random

dividend reali<ations. Calculate the e#ected value of the asset at the start

of the first #eriod& !efore any dividends have !een determined. In

average& how fast will the e#ected value of the asset decline in each

roundN $hat is the highest amount the final redem#tion value could !e&

and what is the lowest amountN



2. Consider an asset that #ays a dividend of 2 #er #eriod and has a ,,0

chance of !eing destroyed =after the dividend #ayment> in each round.

he asset can !e redeemed for 20 at the end of round 5. $hat is the

fundamental value of the asset in each roundN

. ra#h the su##ly and demand arrays used in the eam#le in section

=!uyers !id 5 for 2 shares and / for 2 shares& and only sell order is for

shares at a minimum >.



8art V. Auctions

Auctions can !e very useful in the sale of #erisha!le commodities li"e fish and

flowers. he #u!lic nature of most auctions is also a desira!le feature when e'ual

treatment and a!ove+!oard negotiations are im#ortant& as in the #u!lic #rocurement of mil"& highway construction& etc. hese days& internet auctions

can !e #articularly useful for a seller of a rare or s#ecialty item who needs to

connect with geogra#hically dis#ersed !uyers. And auctions are !eing

increasingly used to sell !andwidth licenses& as an alternative to administrative or

!eauty contest allocations& which can generate wasteful lo!!ying efforts. =6uch

lo!!ying is considered in the cha#ter on rent see"ing in 8art V.>

4ost #artici#ants find auction games to !e eciting& given the winlose nature of

the com#etition. he two main classes of models are those where !idders "now

their own #rivate values& and those where the underlying common value of the

#ri<e is not "nown. 8rivate values may differ from #erson to #erson& even when

the characteristics of the #ri<e are #erfectly "nown. ?or eam#le& two #ros#ective

house !uyers may have different family si<es or num!ers of vehicles& and

therefore& the same s'uare footage may !e much less useful to one than to

another& de#ending on how it is configured into common areas& !edrooms& and

#ar"ing. An eam#le of a common value auction is the !idding for an oil lease&

where each !idder ma"es an inde#endent geological study of the li"ely recovery

rates for the tract of land !eing leased. $inning in such an auction can !e

stressful if it turns out that the !idder overestimated the #ri<e value& a situation

"nown as the winner s curse.

8rivate value auctions are introduced in Cha#ter ,& where the focus is on

com#arisons of !idding strategies with ;ash e'uili!rium #redictions for differentnum!ers of !idders. here is also a discussion of several interesting variations:

,> an all+#ay #rovision that forces all !idders to #ay their own !ids& whether or not

they win& 2> a 6anta Clause #rovision that returns a fraction of auction revenues to

the !idders =distri!uted e'ually>& and > a second+#rice rule that only re'uires the

high !idder to #ay the second highest !id #rice.

Common value auctions are considered in Cha#ter 2,& where the focus is

on the effects of the num!ers of !idders on the winner s curse. he !uyer s curse&

discussed in Cha#ter 20& is a similar #henomenon that arises with a !idder see"ing

to #urchase a firm of un"nown #rofita!ility from the current owner. n each case&

the curse effect arises !ecause !idders may not reali<e that having a successful !id

is an event that conveys useful information a!out others valuations or estimates of value.

he internet has o#ened u# many eciting #ossi!ilities for new auction

designs& and la!oratory e#eriments can !e used to test!ed #ossi!le #rocedures

#rior to the final selection of the auction rules. he eorgia rrigation Auction



descri!ed in Cha#ter 22 was largely designed and run !y e#erimental economists

=Cummings& Holt& and 1aury& 200/>. t #ermitted the eorgia *nvironmental

8rotection (e#artment to use a!out five million dollars to com#ensate some

farmers for not using their irrigation #ermits during a severe drought year. he

Veconla! game ca#tures some =!ut not all> features of the auction that was run in

4arch 200,.

he irrigation reduction auction is an eam#le of a multi+unit auction&

which raises a num!er of interesting #ossi!ilities. Ine #ossi!ility is to let

winning !idders !e #aid their #er+acre !id amounts& which means that some

farmers will !e receiving com#ensation rates that are higher than those received

!y others. his is called a discriminatory auction. An alternative would have

!een to #ay all at the #er+acre !id rate of the marginal !idder& i.e. the lowest !id

that was not acce#ted. his is called a uniform #rice auction. An alternative to

letting !idders su!mit their own !ids in multi+unit auction is to have the current

#ro#osed #rice !e determined !y the auctioneer& via an automatic #rocess or

cloc". ?or eam#le& the state of Virginia recently auctioned off a!out ,00

emissions #ermits =each for one ton of nitrous oide for a #articular year>. he #rice was set low and raised in a series of rounds. At each round& !idders could

say how many tons they were willing to #urchase at the current #rice. he #rice

was raised until the 'uantity demanded fell to a level that e'ualed the num!er of

availa!le licenses. 6ome of these and other variations are also discussed in

Cha#ter 22.

Cha#ter ,. 8rivate Value Auctions

nternet newsgrou#s and online trading sites offer a fascinating glim#se into the

various ways that collecta!les can !e sold at auction. 6ome #eo#le 9ust #ost a #rice& and others announce a time #eriod in which !ids will !e entertained& with

the sale going to the #erson with the highest !id at the time of the close. hese

!ids can !e collected !y email and held as sealed !ids until the close& or the

highest standing !id at any given moment can !e announced in an ascending !id

auction. rade is motivated !y differences in individual values& e.g. as some

#eo#le wish to com#lete a collection and others wish to get rid of du#licates. his

cha#ter #ertains to the case where individual valuations differ randomly& with

each #erson "nowing only their own #rivate value. he standard model of a

#rivate+value auction is one where !idders values are inde#endent draws from a

distri!ution that is uniform in the sense that each money amount in a s#ecified

interval is e'ually li"ely. ?or eam#le& one could throw a ,0+sided die twice& withthe first throw determining the tens digit and the second throw determining the

ones digit. he cha#ter !egins with the sim#lest case& where the !idder receives a

value and must !id against a simulated o##onent& whose !ids are in turn

determined !y a draw from a uniform distri!ution. ;et& we consider the case



where the other !idder is another #artici#ant. n each case& the slo#e of the

!idvalue relationshi# is com#ared with the ;ash #rediction. hese auctions can

!e done !y hand with dice =see the A##endi>& or with the Veconla! game 8V.

he we!game allows o#tions that include an all+#ay rule& a second #rice auction&

and a revenue+sharing #rovision that returns a fraction of auction revenue to the

!idders.

I. Introd$ction

he ra#id develo#ment of e+commerce has o#ened u# o##ortunities for creating

new mar"ets that coordinate !uyers and sellers at diverse locations. he vast

increase in the num!ers of #otential traders online also ma"es it #ossi!le for

relatively thic" mar"ets to develo#& even for highly s#eciali<ed commodities and

collecta!les. 4any of these mar"ets are structured as auctions& since an auction

#ermits !ids and as"s to !e collected 2/ hours a day over an etended #eriod.

ains from trade in such mar"ets arise !ecause different individuals havedifferent values for a #articular commodity. here are& of course& many ways to

run an auction& and different sets of trading rules may have different #erformance

#ro#erties. 6ellers& naturally& would !e concerned with selecting the ty#e of

auction that will enhance sales revenue. ?rom an economist s #ers#ective& there is

interest in finding auctions that #romote the efficient allocation of items to those

who value them the most. f an auction fails to find the highest+value !uyer& this

may !e corrected !y trading in an after mar"et& !ut such trading itself entails

transactions costs& which may !e five #er cent of the value of the item& and even

more for low+value items where shi##ing and handling costs are significant.

*conomists have ty#ically relied on theoretical analysis to evaluate

efficiency #ro#erties of alternative sets of auction rules. he seminal wor" onauction theory can !e found in a ,%2 Kourna# of 6inance #a#er !y $illiam

Vic"rey& who later received a ;o!el 8ri<e in economics. 8rior to Vic"rey& an

analysis of auctions would li"ely !e !ased on a 7ertrand+ty#e model in which

#rices are driven u# to !e essentially e'ual to the resale value of the item. t was

Vic"rey s insight that different #eo#le are li"ely to have different values for the

same item& and he devised a mathematical model of com#etition in this contet.

he model is one where there is a #ro!a!ility distri!ution of values in the

#o#ulation& and the !idders are drawn at random from this #o#ulation. ?or

eam#le& su##ose that a !uyer s value for a car with a large #assenger area and

low gas mileage is inversely related to the #erson s daily commuting time. here

is a distri!ution of commuting distances in the #o#ulation& so there will !e a

distri!ution of individual valuations for the car. *ach #erson "nows their own

needs& and hence their own valuation& !ut they do not "now for sure what other



!idders values are. 7efore discussing the Vic"rey model& it is useful to !egin with

an overview of different ty#es of auctions.

II. -$ctions: >p, Do6n, and the Litt%e Magica% "%fE

6u##ose that you are a collector of cards from Magic$ 8he Gathering .

hese cards are associated with a #o#ular game in which the contestants #lay therole of wi<ards who duel with s#ells from cards in a dec". he cards are sold in

random assortments& so it is natural for collectors to find themselves with some

redundant cards. n addition& rare cards are more valua!le& and some out+of+#rint

cards sell for hundreds of dollars. 1et s consider the thought #rocess that may

occur to you as you contem#late a selling strategy. 6u##ose that you log into the

newsgrou# site and offer a !undle of cards in echange for a !undle that may !e

#ro#osed !y someone else. Mou do receive some res#onses& !ut the !undles

offered in echange contain cards that you do not desire& so you decide to #ost a

#rice for each of your cards. 6everal #eo#le see" to !uy at your #osted #rice& !ut

su##ose that someone ma"es a #rice offer that is a little a!ove your initial #rice inantici#ation of ecess demand. his causes you to sus#ect that others are willing

to #ay more as well& so you #ost another note inviting !ids on the cards& with a

one+wee" deadline. he first !id you receive on a #articularly nice card is /0&

and then a second !id comes in at /5. Mou wonder whether the first !idder

would !e willing to go u# a !it& say to 55. f you go ahead and #ost the highest

current !id on the card each time a new high !id is received& then you are

essentially conducting an 4ng#ish auction with ascending !ids. In the other

hand& if you do not announce the !ids and sell to the high !idder at closing time&

then you will have conducted a firstprice sea#ed+id auction. (avid

1uc"ing3eiley =,> re#orts that the most commonly used auction method for

Magic cards is the *nglish auction& although some first+#rice auctions areo!served.

1uc"ing+3eiley also found a case where these cards were sold in a

descending+!id (utch auction. Here the #rice !egins high and is lowered until

someone agrees to the current #ro#osed #rice. his ty#e of descending+!id

auction is used in Holland to sell flowers. *ach auction room contains several

cloc"s& mar"ed with #rices instead of hours. Carts of flowers are rolled into the

auction room on trac"s in ra#id succession. $hen a #articular cart is on dec"& the

'uality grade and grower information are flashed on an electronic screen. hen

the hand of cloc" falls over the #rice scale& from higher to lower #rices& until the

first !idder #resses a !utton to indicate acce#tance. his #rocess #roceeds

'uic"ly& which results in a high sales volume that is largely com#lete !y late

morning. he auction house is located net to the Amsterdam air#ort& so that

flowers can !e shi##ed !y air to distant locations li"e ;ew Mor" and o"yo.



f you "now your own #rivate value for the commodity !eing auctioned&

then the (utch auction is li"e the sealed+!id auction. his argument is !ased on

the fact that you learn nothing of relevance as the cloc" hand falls in a (utch

auction =Vic"rey& ,%2>. Conse'uently& you might as well 9ust choose your

sto##ing #oint in advance& 9ust as you would select a !id to su!mit in a sealed+!id

auction. hese two auction methods also share the #ro#erty that the winning

!idders #ay a #rice that e'uals their own !id. n each case& a higher !id will raise

the chances of winning& !ut the higher !id lowers the value of winning. n !oth

auctions& it is never o#timal to !id an amount that e'uals the full amount that you

are willing to #ay& since in this case you would !e indifferent !etween winning

and not winning. o summari<e& the descending+!id (utch auction is strategically

e'uivalent to the first+#rice& sealed+!id auction in the case of "nown #rivate #ri<e

values.

his e'uivalence raises the issue of whether there is a ty#e of sealed+!id

auction that is e'uivalent to the ascending+!id *nglish auction. $ith a "nown

#ri<e value& the !est strategy in an *nglish auction is to stay in the !idding until

the #rice 9ust reaches your own value. ?or eam#le& su##ose that one #erson svalue is 50& another s is /0& and a third #erson s value is 0. At a #rice of 20&

all three are interested. $hen the auctioneer raises the #rice to ,& the third

!idder dro#s out& !ut the first two continue to nod as the auctioneer raises the !id

amount. At /,& however& the second !idder declines to nod. he first !idder&

who is willing to #ay more& will agree to /, !ut should feel no #ressure to

e#ress an interest at a higher #rice since no!ody else will s#ea" u#. After the

usual going once& going twice warning& the #ri<e will !e sold to the first !idder at

a #rice of /,. ;otice that the #erson with the highest value #urchases the item&

!ut only #ays an amount that is a##roimately e'ual to the second highest value.

he o!servation that the !idding in an *nglish auction sto#s at the

secondhighest value led Vic"rey to devise a secondprice sea#ed+id auction. Asin any sealed !id auction& the seller collects sealed !ids and sells the item to the

#erson with the highest !id. he winning !idder& however& only has to #ay the

secondhighest !id. Vic"rey noted that the o#timal strategy in this auction is to !id

an amount that 9ust e'uals one s own value. o see why this is o#timal& su##ose

that your value is ,0. f you !id ,0 in a second+#rice auction& then you will

only win when all other !ids are lower than your own. f you decide instead to

raise your !id to ,2& you increase your chances of winning& !ut the increase is

on# in those cases where the second highest !id is a!ove ,0& causing you to lose

money on the win. ?or eam#le& if you !id ,2 and the net !id is ,,& you #ay

,, for an item that is only worth ,0 to you. hus it is never o#timal to !id

a!ove value in this ty#e of auction. ;et consider a reduction to a !id& say to .

f the second highest !id were !elow & then you would win anyway and #ay the

second !id with or without the !id reduction. 7ut if the second !id were a!ove

& say at & then your !id reduction would cause you to lose the auction in a



case where you would have won #rofita!ly. n summary& the !est !id is your own

value in a second+#rice auction. f every!ody does& in fact& !id at value& then the

high+value #erson will win& and will #ay an amount that e'uals the second highest

value. 7ut this is eactly what ha##ens in an *nglish auction where the !idding

sto#s at the second+highest value. hus the second+#rice auction is& in theory&

e'uivalent to the *nglish auction.

1uc"ing+3eiley =2000> #oints out that stam# collectors have long used

Vic"rey+li"e auctions as a way of including !idders who cannot attend an auction.

?or eam#le& if two distant !idders mail in !ids of 0 and /0& then the !idding

would start at ,. f no!ody entered a higher !id& then the #erson with the higher

!id would #urchase at ,. f some!ody else agreed to that #rice& the auctioneer

would raise the #rice to 2. he auctioneer would continue to go one u# on any

!idder #resent until the higher mail+in !id of /0 is reached. his miture of an

*nglish auction and a sealed+!id second #rice auction was achieved !y allowing

#roy !idding& since it is the auctioneer who is entering !ids !ased on the limit

#rices su!mitted !y mail. t is a natural etension to entertain only mailin !ids

and to simulate the *nglish auction !y awarding the #ri<e to the high !idder at thesecond !id. 1uc"ing+3eiley =2000> found records of a #ure second#rice stam#

auction held in ;ortham#ton& 4assachusetts in ,. He also notes that the most

#o#ular online auction& e7ay& allows #roy !idding& which is e#lained: nstead of

having everyone sit at their com#uters for days on end waiting for an auction to

end& we do it a little differently. *veryone has a little magical elf =a.".a. #roy> to

!id for them and all you need to do is tell your elf the most that you want to

s#end& and he ll sit there and out!id the others for you& until his limit is reached.

III. Bidding -gainst a >niform Distri&$tion

his section descri!es an e#eriment conducted !y Holt and 6herman =2000>in which !idders received #rivate values& and others !ids were simulated !y the

throw of ten+sided dice. his e#eriment lets one !egin to study the tradeoffs

involved in o#timal !idding without having to do a full game+theoretic analysis of

how others !ids are actually determined in a mar"et with real =non+simulated>

!idders. At the start of each round& the e#erimenter would go to each #erson s

des" and throw a ten+sided die three times to determine a random num!er

!etween 0.00 and .. his would !e the #erson s #rivate value for the #ri<e

!eing auctioned. 6ince each #enny amount in this interval is e'ually li"ely& the

#o#ulation distri!ution of values in this setu# is uniform. After finding out their

value& each #erson would select a !id "nowing that the other #erson s !id would

!e randomly determined !e three throws of the ten+sided die. f the randomly

determined other #erson s !id turns out to !e lower than the !idder s own !id&

then the !idder would earn the difference !etween his #rivate value and his !id. f

the other s !id were higher& then the !idder would earn nothing.



6u##ose that the first three throws of the die determine a value that will !e

denoted !y v& where v is now some "nown dollar amount !etween 0.00 and

.. he only way to win money is to !id !elow this value& !ut how much

lowerN he strategic dilemma in an auction of this ty#e is that a higher !id will

increase the chances of winning& !ut the value of winning with a higher !id is

diminished !ecause of the higher #rice that must !e #aid. I#timal !idding

involves finding the right #oint in this tradeoff& given one s willingness to tolerate

ris"& which can !e considera!le since the low !idder in the auction earns nothing.

his strategic tradeoff can !e understood !etter !y considering the !idder s

e#ected #ayoff under a sim#lifying assum#tion of ris" neutrality. his e#ected

#ayoff consists of two #arts: the #ro!a!ility of winning and the #ayoff conditional

on winning. A #erson with a value of v who wins with a !id of + will have to #ay

that !id amount& and hence will earn v +. hus the e#ected #ayoff is the #roduct

of the winner s earnings and the #ro!a!ility of winning:

=,.,> *#ected 8ayoff J =v +>8r=winning with +>.

he #ro!a!ility of winning with a !id of + is 9ust the #ro!a!ility that this !id is

a!ove the simulated other !id& i.e. a!ove the result of the throws of the ,0+sided

die. he other !id is e'ually li"ely to !e any #enny amount: 0.00& 0.0,& ..

?or sim#licity& we will ignore ties and assume that the other #erson will win in the

event of a tie. hen a !id of 0 would win with #ro!a!ility 0& a !id of ,0.00

would win with #ro!a!ility ,. his suggests that the #ro!a!ility of winning is

+,0& which is 0 for a !id of 0 and , for a !id of ,0. ?or a !id of 5& the

#ro!a!ility of winning is eactly ,2 according to this formula& which is correct

since there are 500 ways that the other !id is !elow 5 =0.00& 0.0,& /.>& and

there are 500 ways that the !id is greater than or e'ual to 5 =5.00& 5.0,&

.>. Using this formula =+,0> for the #ro!a!ility of winning& the e#ression

for the !idder s e#ected #ayoff in =,.,> can !e e#ressed:

=,.2> *#ected 8ayoff =v + +>= ,0> v+,0 +2 ,0.

his e#ected #ayoff ehi!its the strategic dilemma discussed earlier. he #ayoff

conditional on winning& v +& is decreasing in the !id amount& !ut the #ro!a!ility

of winning& +,0& is increasing in +. he o#timal !id involves finding the right

!alance !etween these two good things& high #ayoff and high #ro!a!ility of

winning.

he !idder "nows the value& v& at the time of !idding& so the function on the rightside of =,.2> can !e gra#hed to find the highest #oint& as in ?igure ,., for the

case of v J . he !id& +& is on the hori<ontal ais& and the function starts with a

height of 0 when + J 0 since a 0 !id has no chance of winning. hus at the other



bQ $4

$000$025$050$075$100$125$150$175$200

:0pecte! Payof

f

end& a !id that e'uals the value v will also yield a 0 e#ected #ayoff& since the

#ayoff for !idding the full value of the #ri<e is 0 regardless of whether one wins

or loses. n !etween these two #oints& the e#ected #ayoff function shows a hill+

sha#ed gra#h& which rises and then falls as one moves to higher !ids =from left to

right>.

0 , 2 / 5 % - ,0

7id

0ig$re '4.'. "(pected Pa!offs 6ith a Private a%$e of 2.++

Ine way to find the !est !id is to use e'uation =,.2> to set u# a s#readsheet to

calculate the e#ected #ayoff for each #ossi!le !id and find the !est one ='uestion

,>. ?or eam#le& if v J /& then the e#ected #ayoffs are: 0 for a !id of 0& 0.0

for a !id of ,& 0./0 for a !id of 2& 0.0 for a !id of & and 0.00 for a !id of

/. ?illing in the #ayoffs for all #ossi!le !ids in #enny increments would confirmthat the !est !id is 2 when one s value is /. 6imilarly& it is straightforward to

show that the o#timal !id is 2.50 when one s value is 5. hese calculations

suggest that the !est strategy =for a ris"+neutral #erson> is to !id one half of one s

value.

he gra#hical intuition !ehind !idding half of value is shown in ?igure

,.,. $hen the value is & the e#ected #ayoff function starts at the origin of the

gra#h& rises& and falls !ac" to 0 when one s !id is e'ual to the value of . he

e#ected #ayoff function is 'uadratic& and it forms a hill that is symmetric around

the highest #oint. he symmetry is consistent with the fact that the maimum is

located at /& halfway !etween 0 and the value of .

At the #oint where the function is flat& the slo#e of a dashed tangent line is

<ero& and the tangency #oint is directly a!ove a !id of / on the hori<ontal ais.

his #oint could !e found gra#hically for any s#ecific #rivate value.

Alternatively& we can use calculus to derive a formula that a##lies to all #ossi!le



values of v. =he rest of this #aragra#h can !e s"i##ed !y those who are already

familiar with calculus& and those who need even more review should see the

a##endi to Cha#ter ,.> A reader who is not familiar with calculus should go

ahead and read the discussion that followsB the only thing you will need !esides a

little intuition is a cou#le of rules for calculating derivatives =finding the slo#es of

tangent lines>. ?irst consider a linear function of +& say /+. his is a straight line

with a slo#e of /& so the derivative is /. his rule generali<es to any linear

function with a constant slo#e of k : the derivative of k+ with res#ect to + is 9ust ".

he second rule that will !e used is that the derivative of a 'uadratic function li"e

+2 is linear: 2+. he intuition is that the slo#es of tangent lines to a gra#h of the

function +2 !ecome stee#er as + increases& which the slo#e& 2+& is an increasing

function of + =the 2 is due to the num!er 2 in the e#onent of +2>. his formula is

easily modified to allow for multi#licative constants& e.g. the derivative of +2 is

=2+>& or the derivative of +2 is 2+ .

he e#ected #ayoff in e'uation =,.2> consists of two terms. he first

one& can !e written as =v,0>+& which is a linear function of + with a slo#e of v,0.

herefore the derivative of the e#ected #ayoff will have a v,0 term in it& as can !e seen on the right side of =,.>:

v 2+

=,.> (erivative of *#ected 8ayoff .

,0 ,0

he second term in the e#ected #ayoff e#ression =,.2> is +2,0& and the

derivative of this term is 2+,0 !ecause the derivative of +2 is 2+. o

summari<e& the derivative on the right side of =,.> is the sum of two terms& each

of which is the derivative of the corres#onding term in the e#ected #ayoff in

=,.2>.

he o#timal !id is the value of ! for which the slo#e of a tangent line is 0& so the

net ste# is to e'uate the derivative in =,.> to 0:

v 2+

=,./> 0.

,0 ,0

his e'uation is linear in +& and can !e solved to o!tain the o#timal !idding

strategy:

v

=,.5> +R =o#timal !id for ris" neutral #erson>.

2



he calculus method is general in the sense that it yields the o#timal !id for all

#ossi!le values of v& whereas the gra#hical and numerical methods had to !e done

se#arately for each value of v !eing considered.

o summari<e& the #redicted !id is a linear function of value& with a slo#e of 0.5.

he actual !id data in the Holt and 6herman e#eriment formed a scatter #lot with

an a##roimately linear sha#e& !ut most !ids were a!ove the half+value

#rediction. A linear regression yielded the estimate:

=,.%> + 0.,/ 0.%%- =3 J 0.,>&v

where the interce#t of ,/ cents& with a standard error of 0.%,& was not

significantly different from 0. he slo#e& with a standard error of 0.0,-& was

significantly different from ,2.

7y !idding a!ove one half of value& !idders are o!taining a higher chance of

winning& !ut a lower #ayoff if they win. A willingness to ta"e less money in order to reduce the ris" of losing and getting a <ero #ayoff may !e due to ris" aversion&

as discussed in the net section. here could& of course& !e other e#lanations for

the over+!idding& !ut the setu# with a simulated other !idder does #ermit us to

rule out some #ossi!ilities. 6ince the other !idder was 9ust a roll of the dice& the

over+!idding cannot !e due to issues of e'uity& fairness& or rivalistic desires to win

or reduce the other s earnings.

he e#eriment descri!ed in this section with simulated other !ids is essentially

an individual decision #ro!lem. An analogous game can !e set u# !y #roviding

each of two !idders with randomly determined #rivate values drawn from a

distri!ution that is uniform from 0 to ,0. As !efore& a high !id results inearnings of the difference !etween the #erson s #rivate value and the #erson s

own !id. A low !id results in earnings of 0. his is "nown as a firstprice

auction since the high !idder has to #ay the highest =first> #rice.

Under ris" neutrality& the ;ash e'uili!rium for this game is to !id one half of one

s #rivate value. he #roof of this claim is essentially an a##lication of the

analysis given a!ove& where the distri!ution of the other s !ids is uniform on a

range from 0 to ,0. ;ow su##ose that one #erson is !idding half of value. 6ince

the values are uniformly distri!uted& the !id of one+half of value will !e uniformly

distri!uted from 0 to a level of 5& which is one half of the maimum value. f

this #erson s !ids are uniformly distri!uted from 0 to 5& then a !id of 0 will not

win& a !id of 5 will win with a #ro!a!ility that is essentially ,& and a !id of 2.50will win with #ro!a!ility ,2. 6o the #ro!a!ility of winning with a !id of + is +5.

hus the e#ected+#ayoff function is given in e'uation =,.2> if the ,0 in each

denominator is re#laced !y a 5. *'uations =,.> and =,./> are changed



similarly. hen multi#lying !oth sides of the revised =,./> !y 5 yields the v2

!idding rule in =,.5>.

I. Bidding against a >niform Distri&$tion 6ith Risk -version

@optiona%A

he analysis in this net section shows that a sim#le model of ris" aversion can e#lain the general #attern of over!idding that is im#lied !y the

regression e'uation =,.%>. he analysis uses sim#le calculus& i.e. the derivative

of a #ower function li"e " 5 #& where " is a constant and the varia!le 5 is raised to

the #ower p. he derivative with res#ect to 5 of this #ower function is a new

function where the #ower has !een reduced !y , and the original #ower enters

multi#licatively. hus the derivative of k5 # is kp5 #,& which is called the #ower+

function rule of differentiation. A second rule that will !e used is that the

derivative of a #roduct of two functions is the first function times the derivative of

the second& #lus the derivative of the first function times the second. his is

analogous to calculating the change in room si<e =the #roduct of length and width>as the length times the change in width #lus the change in length times the width.

his #roduct rule is accurate for small changes. Using the #owerfunction and the

#roduct rules& those not familiar with calculus should !e a!le to follow the

arguments given !elow& !ut if you have trou!le& s"i# to the discussion that is 9ust

!elow the generali<ed !idding rule in =,.,,>.

he most convenient way to model ris" aversion in an auction is to

assume that utility is a nonlinear function& i.e. that the utility of a money amount

v+ is a #ower function =v+>,+r for 0 I r Q ,. $hen r J 0& this function is

linear& which corres#onds to the case of ris" neutrality. f r J ,2& then the #ower

,r is also [& so the utility function is the s'uare root function. A higher value

of r corres#onds to more ris" aversion. ;ow the e#ected #ayoff function in=,.2> must !e re#laced with an e#ected utility function& which is the utility of

the #ayoff times the #ro!a!ility of winning:

=,.-> *#ected Utility =v +>,rR +

TU.

V,0 W

As !efore& the o#timal !id is found !y e'uating the derivative of this function to

<ero. he e#ected utility on the right side of =,.-> is a #roduct of two functions

of !& so we use the #ower+function rule to o!tain the derivative =first functiontimes the derivative of the second #lus the derivative of the first times the second

function>. he derivative of !,0 is ,,0& so the first function times the derivative

of the second is the first term in =,.>. ;et& note that the #ower function rule



im#lies that the derivative of =v+>,+r is =,r >=v+>+r & which a##ears in the

second term in =,.>.

=,.> =v +>,r R , TU =,r v>= +>r R + TU.

V,0 W V,0W

his derivative can !e rewritten !y #utting #arts common to each term in the

#arentheses& as shown on the left side of =,.>:

R =v +>r T R =v +>r T=,.> =v +> U =,r +> U 0.

V ,0 W V ,0 W

4ulti#lying !oth sides !y ,0=v+>+r & one o!tains:

=,.,0> v + =,r +> 0.

his e'uation is linear in !& and can !e solved to o!tain the o#timal !idding

strategy:

v

=,.,,> +R =o#timal !id with ris" aversion>.

2r

his !idding rule reduces to the o#timal !idding rule for ris" neutrality =!idding

half of value> when r J 0. ncreases in r will raise the !ids. $hen r J ,2& the

!ids will !e two+thirds of value& which is consistent with the results of the

regression e'uation =,.%>.

. Bidding Behavior in a 6o?Person, 0irst?Price -$ction

3ecall that the #redicted !idding strategy is linear& with a slo#e of ,2

when !idders are ris" neutral. 7ids eceeded the v2 line in an e#eriment with

simulated other !ids& and the same #attern emerges when the other !idder is a

su!9ect in the e#eriment. ?igure ,.2 shows the !idvalue com!inations for ,0

rounds of a classroom e#eriment done with the Veconla! software. here were,0 !idding teams& each com#osed of one or two students on a networ"ed 8C. he

teams were randomly matched in a series of rounds. he !idding #attern is

a##roimately linear& ece#t for a leveling off with high values& which is a #attern

that has also !een noted !y (orsey and 3a<<olini =2002>. Inly a small fraction of



0

0

0Probability

the !ids are at or !elow the ;ash #rediction of v2& which is gra#hed as the solid

line. n fact& the ma9ority of !ids are a!ove the dashed line with slo#e of .%%-

o!tained from a linear regression for the case discussed in the #revious sectionB

the !id+to+value ratio averaged over all !ids in all rounds is a!out 0.-. his

#attern of over!idding relative to the ;ash #rediction is ty#ical& and the most

commonly mentioned e#lanation is ris" aversion.

0 , 2 / 5 % - ,0

8rivate Value

0ig$re '4.. 5&served Bidding Strategies for en S$&ects in a C%assroom "(periment

An analysis of ris" aversion for this game #arallels the #revious section s analysis

of !idding against a simulated !id that is uniform on the interval from 0 to ,0

dollars. 6u##ose that the e'uili!rium !idding strategy with ris" aversion is linear:+ J Xv& where 0 Q X Q ,. hen the lowest !id will !e 0& corres#onding to a value

of 0& and the highest !id will !e ,0X& corres#onding to a value of ,0. he

distri!ution of !ids is re#resented in ?igure ,.. ?or any #articular value of X&

the !ids will !e uniformly distri!uted from 0 to ,0X& as indicated !y the dashed

line with a constant height re#resenting a constant #ro!a!ility for each #ossi!le

!id. he figure is drawn for the case of X J 0.%& so !ids are uniform from 0 to

%. A !id of 0 will never win& a !id of ,0X J % will win with #ro!a!ility ,& and an

intermediate !id will win with #ro!a!ility of +%. n the general case& a !id of +

will win with #ro!a!ility +,0X.



0 , 2 / 5 % - ,0

7id

0ig$re '4./. - >niform Distri&$tion Bids on +, 1H

n a two+#erson auction when the other s !id is uniform from 0 to ,0 X& the

#ro!a!ility of winning in the e#ected #ayoff function will !e +,0X instead of the ratio !,0 that was used in section . he rest of the analysis of o#timal

!idding in that section is unchanged& with the occurrences of ,0 re#laced !y ,0X&

which cancels out of the denominator of e'uation =,.> 9ust as the ,0 did. he

resulting e'uili!rium !id is given in e'uation =,.,,> as !efore. his !idding

strategy is again linear& with a slo#e that is greater than one half when there is ris"

aversion =r P 0>. 3ecall that a ris" aversion coefficient of r J ,2 will yield a !id

line with slo#e 2& so the !ids in ?igure ,.2 are roughly consistent with a ris"

aversion coefficient of at least ,2.

I. "(tensions and 0$rther Reading6ome of the earliest e#erimental tests of the Vic"rey model are re#orted !y

Co##inger& 6mith& and itus =,0>. n #articular& they o!served over!idding

relative to ;ash #redictions in a #rivate+value& first+#rice auction& assuming ris"

neutrality. he effects of ris" aversion on individual !ehavior was e#lored in a

series of #a#ers !y Co& 3o!erson& and 6mith =,2>& and Co& 6mith& and

$al"er =,5& ,>. 6ee Dagel =,5> for a survey of auction e#eriments.

oeree& Holt& and 8alfrey =2002> #rovide an analysis of ris" aversion and noisy

!ehavior in first+#rice auctions.

U# to now& we have only considered two+#erson auctions. Auctions with more

!idders are more com#etitive& which will cause !ids to !e closer to values. he

formula in =,.5> can !e generali<ed to the case of F ris"+neutral !idders drawing

from the same uniform #rivate value distri!ution:

=,.,2> +R = F

,>v =o#timal !id with ris" neutrality>&

F

with a further u#ward ad9ustment for the case of ris" aversion. ;otice that =,.,2>

s#ecifies !idding half of value when F J 2& and that !ids rise as a #ro#ortion of

value for larger grou# si<es. As the num!er of !idders !ecomes very large& the

ratio& = F ,> F & !ecomes closer and closer to ,& and !ids converge to values. his

increase in com#etition causes e#ected #ayoffs to converge to <ero as !idvalue

differences shrin".



$i!

he Veconla! software #ermits a num!er of other interesting variations. Ine

o#tion is to re'uire all !idders to #ay their own !ids& with the #ri<e still going to

the highest !id. ?or eam#le& if a !idder with a value of / !ids , and if a

second !idder with a value of !ids 5& then the first earns , and the second

earns 5 J /. n this strategic setting& !idders with low values =and low

chances of winning> will reduce their !ids towards 0& to avoid having to #ay in the

event of a loss. hose with a relatively high value will ris" a higher !id& so that

the !idding strategy will have a curved sha#e& only rising significantly a!ove 0 at

very high values. his curved #attern can !e seen in the !ids in ?igure ,./&

which were o!served in a research e#eriment conducted at the University of

Virginia =with all #ayments made in cash>. his all+#ay auction is sometimes

used to model lo!!ying com#etition or a #atent race& where the contender with the

highest effort will win& !ut even the losers incur the costs associated with their

efforts.

A second o#tion #rovides for returning a fraction of the auction revenue to all

!idders& divided e'ually among the !idders. his is analogous to using the

revenue from the sale of #ri<e to reduce taes for all !idders. 6uch revenuere!ates tend to raise the !ids.

0ig$re '4.3. 5&served Bids for '+ S$&ects in an -%%?Pa! -$ction "(periment



<$estions

,. his 'uestion lets you set u# a sim#le s#readsheet to calculate the o#timal

!id. 7egin !y #utting the tet V J in cell A, and any numerical value& e.g.

& in cell 7,. hen #ut the #ossi!le !ids in the A column in fifty+cent

increments: 0 in A& 0.5 in A/& , in A5& etc. =Mou can use #enny amountsif you #refer.> he e#ected #ayoff formula in column 7& !eginning in

cell 7& should contain a formula with terms involving 7, =the #ri<e

value> and A =the !id>. Use the e#ected #ayoff formula in e'uation

=,.2> to com#lete this formula& which is then co#ied down to the lower

cells in column 7. Verify the e#ected #ayoff num!ers for a value of /

that were re#orted in section . 7y loo"ing at the e#ected #ayoffs in

column 7& one can determine the !id with the highest e#ected #ayoff. f

the value of is in cell 7,& then the o#timal !id should !e /. hen change

the value in cell 7, to 5 and show that the o#timal !id is 2.50.

2. 3ecall that e'uations =,.,>+=,.5> #ertain to the case of a !idder who is !idding against a simulated other !idder. $rite out the revised versions of

e'uations =,.2>& =,.>& =,./>& and =,.5> for the two+#erson auction&

assuming ris" neutrality.

. $rite out the revised versions of e'uations =,.2>& =,.>& =,./>& and

=,.5> for the two+#erson auction under ris" aversion with a #owerfunction

utility and #arameter r .

Cha#ter 20. he a"eover ame

his cha#ter #ertains to a situation in which a !uyer cannot directly o!serve the

underlying value of some o!9ect. A !uyer s !id will only !e acce#ted if it is

higher than the value to the current owner. he danger is that a #urchase is more

li"ely to !e made #recisely when the owner s value is low& which may lead to a

loss for the !uyer. he tendency to #urchase at a loss in such situations is called

the !uyer s curse. he Veconla! a"eover ame sets u# this situation& or

alternatively& the instructions in the a##endi can !e used with ,0+sided dice.

I. Wall Street @the 0i%mA

n the ,- Hollywood film =a## Street & 4ichael (ouglas #lays the role of acor#orate raider who ac'uires *1(A3 *nter#rises& with the intention of

increasing its #rofita!ility !y firing the union em#loyees. After the ac'uisition&

the new owners !ecome aware of some #reviously hidden !usiness #ro!lems =a

defect in an aircraft model under develo#ment> that drastically reduce the firm s



#rofit #otential. As the stoc" is falling& (ouglas turns and gives the order (um#

it. his event is re#resentative of a wave of aggressive ac'uisitions that swe#t

through $all 6treet in the mid+,0 s. 4any of these ta"eovers were motivated

!y the !elief that com#anies could !e transformed !y infusions of new ca#ital and

!etter management techni'ues. he mood later turned less o#timistic as ac'uired

com#anies did not achieve #rofit goals.

$ith the advantage of hindsight& some economists have attri!uted these

failed mergers to a selection !ias: it is the #ess profita+#e com#anies that are more

li"ely to !e sold !y owners with inside information a!out #ro!lem areas that

raiders may not detect. n a !idding #rocess with a num!er of com#etitors& the

!idder with over+o#timistic e#ectations is li"ely to end u# ma"ing the highest

tender offer& and the result may !e an ac'uisition #rice that is not 9ustified !y

su!se'uent #rofit #otential. he tendency for the winning !idder to lose money

on a #urchase is "nown as the winner s curse. *ven with only a single #otential

!uyer& a !id is more li"ely to !e acce#ted if it is too high& and the resulting

#otential for losses is sometimes called the !uyer s curse.

n a sense& winning in a !idding war can !e an informative event. A !idthat is acce#ted& at a minimum& indicates that the !id eceeds the current owner s

valuation. hus there would !e no incentive for trade if the value of the com#any

were the same for the owner and the !idder. 7ut even when the !idder has access

to su#erior ca#ital and management services& the !idder may end u# #aying too

much if the intrinsic valuation cannot !e determined in advance.

II. he B$!er9s C$rse "(periment

6ome elements that affect a firm s intrinsic #rofita!ility are revealed !y

accounting data and can easily !e o!served !y !oth current and #ros#ective

owners. Ither as#ects of a firm s o#erations are #rivate& and internal #ro!lemsare not li"ely to !e revealed to outsiders. he model #resented in this section is

highly styli<ed in the sense that all #rofita!ility information is #rivate and "nown

only !y the current owner. he #ros#ective !uyer is unsure a!out the eact

#rofita!ility and has only #ro!a!ilistic information on which to ma"e a !idding

decision. he #ros#ective !uyer& however& has a #roductivity advantage that will

!e e#lained !elow.

he first scenario to !e considered is one where the value of the firm to the

current owner is e'ually li"ely to !e any amount !etween 0 and ,00. his current

owner "nows the eact value& : & and the #ros#ective !uyer only "nows the range

of #ossi!le e'ually li"ely values. n an e#eriment& one could throw a tensided

die twice at the owner s des" to determine the value& with the throw !eing

uno!served !y the !uyer. o #rovide a motivation for trade& the !uyer has !etter

management s"ills. n #articular& the value to the !uyer is ,.5 times the value to

the current owner. ?or eam#le& if a !idder offers %0 for a com#any that is only



bid

0

0005

001

0015

Probability

worth 50 to the current owner& then the sale will go though and the !idder will

earn ,.5R50 minus the !id of %0& for a total of ,5.

a!le 20., 3ound , 3esults from a Classroom *#eriment=University of Virginia& *con /2& 6#ring 2002>

8ro#oser ,

8ro#oser 2

8ro#oser

8ro#oser

/

8ro#oser 5

8ro#oser

%

7uyer 7id %0 / 50 % 50 0

Iwner Value 2, 2 , % / 5-

Iwner 3es#onse acce#t acce#t acce#t acce#t acce#t re9ect

7uyer Value 2 5 /% ,0 %5 %

7uyer *arnings ?2 ?'3 ?3 ?1 '* +

a!le 20., shows the first round results from a classroom e#eriment. he

#ro#oser names =and their comments> have !een removed. ;otice that #ro#osers

in this round were #articularly unfortunate. *ce#t for #ro#oser % =who "new too

much as we will see !elow>& the #ro#oser !ids averaged /. All of these !ids

were acce#ted& and the owner values =2,& 2& ,& %& and / for 8ro#osers , to 5

res#ectively> averaged 25. he first four #ro#osers lost money& as shown !y the

!ottom row of the ta!le. he fifth #ro#oser earned ,.5 times / on an acce#ted

!id of 50& for earnings of ,5 cents. his tendency for !uyers to ma"e losses is not

sur#rising e5 post & since a !id of a!out 50 will only !e acce#ted if the seller valueis lower than 50. he seller values averaged only 25& and ,.5 times this amount is

-.5& which is still !elow 50.

0.00 0.,0 0.20 0.0 0./0 0.50 0.%0 0.-0 0.0 0.0 ,.00

Iwner)s Value



0ig$re +.'. "(pected 5$tcomes for a Bid of 1+

he tendency for !uyers to lose money is illustrated in ?igure 20.,& where

#ayoffs are in #ennies. Iwner values are uniformly distri!uted from 0 to & with

a #ro!a!ility of ,,00 of each value& as indicated !y the dashed line with a height

of 0.0,. A !id of %0 will only !e acce#ted if the owner s value is lower& and sinceall lower !ids are e'ually li"ely& the e#ected owner value is 0 for an acce#ted

!id of %0. his average owner value of 0 translates into an average value of ,.5

times as high& as indicated !y the vertical line at /5. 6o an acce#ted !id of %0 will

only yield e#ected earnings of /5. his analysis is easily generali<ed. A !id of +

will !e acce#ted if the owner value is !elow +. 6ince owner values for acce#ted

!ids are e'ually li"ely to !e anywhere !etween 0 and +& the average owner value

for an acce#ted !id is +2. his translates into a value of ,.5=+2>& or =/>+&

which is less than the acce#ted !id& +. hus any #ositive !id of ! will generate

losses on average in this setu#& and the o#timal !id is <ero.

he feed!ac" received !y !idders is somewhat varia!le& so it is difficult to

learn this through e#erience. n five rounds of !idding in the classroome#eriment& only three of the , acce#ted !ids resulted in #ositive earnings& and

losses resulted in lower !ids in the su!se'uent round in all cases. 7ut acce#ted

!ids with #ositive earnings were followed !y !id increases& and re9ections were

followed !y !id increases in a!out half of the cases =ecluding the #ro#oser who

!id 0 and was re9ected every time>. he net effect of these reactions caused

average !ids to stay at a!out 0 cents for rounds 2+5. he only #erson who did

not finish round 5 with a cumulative loss was the #erson who !id 0 in all rounds.

his #erson& a second+year #hysics and economics ma9or& had figured out the

o#timal !idding strategy on his own. he results of this e#eriment are ty#ical of

what is o!served in research e#eriments summari<ed net.

III. 0$rther Reading

he first !uyer s curse e#eriment was re#orted in 7a<erman and

6amuelson =,>& who used a discrete set of e'ually li"ely owner values =0& 20&

/0& %0& 0& and ,00>. 7all& 7a<erman& and Caroll =,0> used a grid from 0 to ,00

with 4.7.A. su!9ects& and the !ids did not decline to <ero. he average !id

stayed a!ove 0& and the modal !id was around 50.

n all of the setu#s descri!ed thusfar& the o#timal !id is <ero. his is no

longer the case when the lowest #ossi!le owner value is #ositive. $hen seller

values range from 50 to ,00 and !uyer values are ,.5 times the seller value& for eam#le& the o#timal !id is at the u##er end of the range& i.e. at ,00 =see 'uestions

2+/>. Holt and 6herman =,/> ran e#eriments with this setu# and found !ids to

!e well !elow the o#timal level. 6imilarly& this high+owner+value setu# was used

in #eriods %+,0 of the classroom e#eriment descri!ed a!ove& and the average



!ids were -5& -%& -& & and in these rounds& well !elow the o#timal level of

,00. Holt and 6herman attri!uted the failure to raise !ids to another ty#e of error&

i.e. the failure to reali<e that an increase in the !id will #ic" u# relatively

highvalue o!9ects at the margin. ?or eam#le& a !id of -0 will #ic" u# o!9ects

with seller values from 50 to -0& with an average of %0. 7ut raising the !id from

-0 to -, will #ic" u# a #urchase if the seller value is at the u##er end of this

range& at -0. ?ailure to recogni<e this factor may lead to !ids that are too low&

which they termed the loser s curse.

<$estions

,. How would the analysis of o#timal !idding in section change if the

value to the !idder were a constant @ times the value to the owner& where

0 Q @ Q 2 and the owner s value is e'ually li"ely to !e any amount

!etween 0 and ,00N =hus an acce#ted !id yields a #ayoff for the !idder

that is @ times the owner s value& minus the amount of an acce#ted !id.>

n #articular& does the o#timal !id de#end on @ N2. 6u##ose that the owner values are e'ually li"ely to !e any amount from 50

to & so that a !id of ,00 will always !e acce#ted. Assume that owners

will not sell when they are indifferent& so a !id of 50 will !e re9ected and

will #roduce <ero earnings. he value to the !idder is ,.5 times the value

to the owner. 6how that the lowest !id of 50 is not o#timal. =Hint:

Com#are the e#ected earnings for a !id of 50 with the e#ected earnings

for a !id of ,00.>

. ?or the setu# in 'uestion 2& a !id of 0 will !e acce#ted a!out four+fifths

of the time. f 0 is acce#ted& the e#ected value to the owner is !etween

50 and 0& with an average of -0. Use this information to calculate the

e#ected #ayoff for a !id of 0& and com#are your answer with thee#ected #ayoff for a !id of ,00 o!tained earlier.

/. he analysis for your answers to 'uestions 2 and =with owner values

!etween 50 and ,00> suggests that the !est !id in this setu# is ,00. 1et s

model the #ro!a!ility of a !id + !eing acce#ted as =+ 50>50& which is 0

for a !id of 50 and , for a !id of ,00. ?rom the !idder s #oint of view& the

e#ected value to the owner for an acce#ted !id of + is 50 #lus half of the

distance from 50 to +. he !idder value is ,.5 times the owner value& !ut

an acce#ted !id must !e #aid. Use this information in a s#readsheet to

calculate the e#ected #ayoff for all !ids from 50 to ,00& and there!y to

find the o#timal !id. he five columns of the s#readsheet should !e

la!eled: =,> 7id +& =2> Acce#tance 8ro!a!ility for a 7id +& => *#ected

Value to Iwner =conditional on + !eing acce#ted>& =/> *#ected Value to

7idder =conditional on ! !eing acce#ted>& =5> *#ected 8ayoff for 7idder

of 4a"ing a 7id + =which is the #roduct of columns 2 and />.



Alternatively& you may use this information to write the !idder s e#ected

#ayoff as a 'uadratic function of +& and then use calculus !y setting the

derivative of this function to <ero and solving for +.

;ote: A classroom e#eriment for the setu# in 'uestions 2+/ resulted in average

!ids of a!out 0. his information will not hel# you find the o#timal !id.



Cha#ter 2,. he $inner s Curse

n many !idding situations& the value of the #ri<e would !e a##roimately the

same for all !idders& although none of them can assess eactly what this common

value will turn out to !e. n such cases& each !idder may o!tain an estimate of the #ri<e value& and those with higher estimates are more li"ely to ma"e higher !ids.

As a result& the winning !idder may overestimate the value of the #ri<e and end

u# #aying more than it is worth. his winner s curse is analogous to the !uyer s

curse discussed in the #revious cha#ter. An auction where each !idder has #artial

information a!out an un"nown #ri<e value can !e run using ,0+sided dice to

determine common value elements& as e#lained in the A##endi. Alternatively&

it can !e run with the Veconla! game Common Value Auction game& which has

default settings that #ermit an evaluation of increases in the num!er of !idders.

his increase in num!ers tends to #roduce a more severe winner s curse.

I. I on the -$ction B$t I ish I 7adn9tE

Ince the author as"ed several flooring com#anies to ma"e !ids on re#lacing

some "itchen floor tile. *ach !idder would estimate the amounts of materials

needed and the time re'uired to remove the old tile and install the floor around the

various odd+sha#ed doorways and #antry area. Ine of the !ids was somewhat

lower than the others. nstead of e#ressing ha##iness over getting the 9o!& he

ehi!ited considera!le aniety a!out whether he had miscalculated the cost&

although in the end he did not withdraw the !id. his story illustrates the fact that

winning an auction can !e an informative event& or e'uivalently& the fact that youwin #roduces new information a!out the un"nown value of the #ri<e. A rational

!idder should antici#ate this information in ma"ing the !id originally. his is a

su!tle strategic consideration that is almost surely learned !y =unha##y>

e#erience.

he #ossi!ility of #aying too much for an o!9ect of un"nown value is #articularly

dangerous for one+time auctions in which !idders are not a!le to learn from

e#erience. 6u##ose that two #artners in an insurance !usiness wor" out of

se#arate offices and sell se#arate ty#es of insurance& e.g. !usiness insurance from

one office and life from the other. *ach #artner can o!serve the other s

!ottomline earnings& !ut cannot determine whether the other one is really

wor"ing. Ine of the #artners is a single mother who wor"s long hours& with goodresults des#ite a low earnings+#er+hour ratio. he other one& who en9oys

somewhat of a luc"y mar"et niche& is a!le to o!tain good earnings levels while

s#ending a large fraction of each day sociali<ing on the internet. After several

years of ha##y #artnershi#& each decides to try to !uy the whole !usiness from the



stoc"holders& who are current and former #artners. $hen they ma"e their !ids&

the one with the #rofita!le niche mar"et is li"ely to thin" the other office is as

#rofita!le as the #erson s own office& and hence to overestimate the value of the

other office. As a result& this #artner could end u# ac'uiring the !usiness for a

#rice that eceeds its value.

he tendency for winners value estimates to !e !iased has long !een "nown in

the oil industry& where drilling com#anies must ma"e rather im#erfect estimates

of the li"ely amounts of oil that can !e recovered on a given tract of land !eing

leased. ?or eam#le& 3o!ert $ilson& a #rofessor in the 6tanford 7usiness 6chool&

was once consulting with an oil com#any that was considering !idding at less

than half of the estimated lease value. $hen he in'uired a!out the #ossi!ility of a

higher !id& he was told that firms that !id this aggressively on such a lease are no

longer in this !usiness =source: #ersonal communication at a conference on

auction theory in the early ,0 s>.

he intuition underlying the un#rofita!ility of aggressive !idding is that

the firm with the highest !id in an auction is li"ely to have overestimated the lease

value. he resulting #ossi!ility of winning at a loss is more etreme when thereare many !idders& since the highest estimate out of a large num!er of estimates is

more li"ely to !iased u#ward& even though each individual estimate is e5 ante

un!iased. ?or eam#le& su##ose that a single value estimate is un!iased. hen

the higher of two un!iased estimates will !e !iased u#ward. Analogously& the

highest of three un!iased estimates will !e even more !iased& and the highest of

,00 estimates may show an etreme !ias toward the largest #ossi!le estimation

error in the u#ward direction. Dnowing this& a !idder in an auction with many

!idders should treat their own estimates as !eing inflated& which will li"ely turn

out to !e true if they win& and the resulting !ids may end u# !eing low relative to

the estimates. his num+ers effect can !e #articularly sinister& since the normal

strategic reaction to increased num!ers of !idders is to !id higher& as in a #rivatevalue auction.

$ilson =,%> s#ecified a model with this common+value structure and

showed that the ;ash e'uili!rium with fully rational !idders involved some

downward ad9ustment of !ids in antici#ation of a winner s+curse effect. *ach

!idder should reali<e that their !id is only relevant for #ayoffs if it is the highest

!id& which means that they have the highest value estimate. 6o #rior to ma"ing a

!id& the !idder should consider what they would want to !id "nowing that all

other estimates are lower than theirs in the event that they win the auction. he a

priori correction of this over+estimation #roduces !ids that will earn #ositive

#ayoffs on average.



II. - Simp%e Shoe?Bo( Mode%

Consider a sim#le situation where each of two !idders can essentially o!serve

half of the value of the #ri<e& as would !e the case for two !idders who can drill

test holes on different halves of a tract of land !eing leased for an oil well.

Another eam#le would !e the case for the two insurance #artners discussed in

the #revious section. n #articular& let the o!served value com#onents or signals

for !idders , and 2 !e denoted !y v, and v2 res#ectively. he #ri<e value is the

average of the two signals:

v v

=2,.,> 8ri<e Value J .

7oth !idders "now their own estimates& !ut they only "now that the other #erson s

estimate is the reali<ation of a random varia!le. n the e#eriment to !e

discussed& each #erson s value estimate is drawn from a distri!ution that is

uniform on the interval from 0 to ,0. 7idder ,& for eam#le& "nows v, and that

v2 is e'ually li"ely to !e any amount !etween 0 and ,0.

A classroom auction was run with these #arameters& and one !idder had a

relatively high value signal of .% in the first round. his #erson su!mitted a

!id of 5.0 =#resuma!ly the three+cent increase a!ove 5 was intended to

outguess anyone who might !id an even 5>. he other #erson s signal was

0.%0& so the #ri<e was only worth the average of .% and 0.%0& which is

/.%/. he !idder won this #ri<e& !ut #aid a #rice of 5.0& which resulted in a

loss. hree of the five winning !idders in that round ended u# losing money& and

a!out one out of five winners lost money in each of the remaining rounds.

he #ri<e value function in =2,.,> can !e generali<ed for the case of a larger

num!er of !idders with inde#endent signals& !y ta"ing an average& i.e. dividingthe sum of all signals !y the num!er of !idders. n a se#arate classroom

e#eriment& conducted with ,2 !idders& the sole winning !idder ended u# losing

money in three out of five rounds. As a result& aggregate earnings were <ero or

negative for most !idders. A ty#ical case was that of !idder & who su!mitted the

high !id of %.,0 on a signal of .%/ in the fifth and final round. he average of

all ,2 signals was 5./5& so this #erson lost %5 cents for the round. hese results

illustrate why the winner s+curse effect is often more severe with large num!ers of

!idders.

Ince an economist at the University of Virginia was as"ed to consult for a ma9or

telecommunications com#any that was #lanning to !id in the first ma9or U.6.

auction for radio wave !andwidth to !e used for #ersonal communicationsservices. his com#any was a ma9or #layer& !ut incredi!ly& it decided not to !id at

the last minute. he com#any re#resentative mentioned the danger of

over#ayment for the licenses at auction. his story does illustrate a #oint: that



#layers who have an o#tion of earning <ero with non+#artici#ation will never !id

in a manner that yields negative e#ected earnings. he technical im#lication is

that e#ected earnings in a ;ash e'uili!rium for a game with an eit o#tion

cannot !e negative. his raises the issue of how !idders rationally ad9ust their

!ehavior to avoid losses in a ;ash e'uili!rium& which is the to#ic of the net

section.

III. he 8ash "#$i%i&ri$m

As was the case in the cha#ter on #rivate+value auctions& the e'uili!rium !ids will

end u# !eing linear functions of the signals& of the form:

=2,.2> +i Xvi & where 0QXJ, and i ,&2.

6u##ose that !idder 2 is using a s#ecial case of this linear !id function !y !idding

eactly one half of the signal v2. n the net several #ages& we will now use this

assumed !ehavior to find the e#ected value of !idder , s earnings for any given !id& and then we will show that this e#ected #ayoff is maimi<ed when !idder ,

also !ids one half of the signal v,& i.e. X J 0.5. 6imilarly& when !idder , is !idding

one half of their signal& the !est res#onse for the other !idder is to !id one half of

their signal too. hus the ;ash e'uili!rium for two ris"+neutral !idders is to !id

half of one s signal.

6ince the arguments that follow are more mathematical than most #arts of the

!oo"& it is useful to !rea" them down into a series of ste#s. $e will assume for

sim#licity that !idders are ris" neutral& so we will need to find !idder , s e#ected

#ayoff function !efore it can !e maimi<ed. =t turns out to !e the case that ris"

aversion has no effect on the ;ash e'uili!rium !idding strategy in this case& as

noted later in the cha#ter.> n an auction where the #ayoff is 0 in the event of a

loss& the e#ected #ayoff is the #ro!a!ility of winning times the e#ected #ayoff

conditional on winning. 6o the first ste# is finding the #ro!a!ility of winning

given that the other !idder is !idding one half of their signal value. he second

ste# is to find the e#ected #ayoff& conditiona# on ;inning ;ith a particu#ar +id .

he third ste# is to multi#ly the #ro!a!ility of winning times the e#ected #ayoff

conditional on winning& to o!tain an e#ected #ayoff function& which will !e

maimi<ed using sim#le calculus in the final ste#. he result will show that a

!idder s !est res#onse is to !id half of the signal when the other one is !idding in

the same manner& so that this strategy is a ;ash e'uili!rium.

7efore going through these ste#s& is may !e useful to review how we willgo a!out maimi<ing a function. hin" of the gra#h of a function as a hill& which

is increasing on the left and decreasing on the right& as for ?igure ,., in the

cha#ter on #rivate+value auctions. At the to# of the hill& a tangent line will !e

hori<ontal =thin" of a graduation ca# !alanced on the to# of someone s head>. 6o



to maimi<e the function& we need to find the #oint where the slo#e of a tangent

line is 0. he slo#e of a tangent line can !e found !y drawing the gra#h carefully&

drawing the tangent line& and measuring its slo#e& !ut this method is not general

since s#ecific num!ers are needed to draw the lines. n general& the slo#e of the

tangent line to a function can !e found !y ta"ing the derivative of the function.

7oth the #ro!a!ility of winning and the conditional e#ected #ayoff will turn out

to !e linear functions of the #erson s !id +& so the e#ected+#ayoff #roduct to !e

maimi<ed will !e 'uadratic& with terms involving + and +2. A linear term& li"e

+2& is a straight line through the origin with a slo#e of ,2& so the derivative is the

slo#e& ,2. he 'uadratic term& +2& also starts at the origin& and it increases to ,

when + J ,& to / when + J 2& to when + J & and to ,% when + J /. ;otice that

this ty#e of function is increasing more ra#idly as ! increases& i.e. the slo#e is

increasing in +. Here all you need to "now is that the derivative of a 'uadratic

e#ression li"e +2 is 2+& which is the slo#e of a straight line that is tangent to the

curved function at any #oint. his slo#e is& naturally& increasing in +. Armed with

this information& we are ready to find the e#ected #ayoff function and maimi<e

it.

Step 1. 6inding Bidder 19s Pro+a+i#it of =inning for a Given Bid

6u##ose that the other #erson =!idder 2> is "nown to !e !idding half of their

signal. 6ince the signal is e'ually li"ely to !e any value from 0 to ,0& the

second !idder s signal is e'ually li"ely to !e any of the ,000 #enny amounts

!etween 0 to ,0& as shown !y the dashed line with height of 0.00, in ?igure

2,.,. $hen +2 J v22& then !idder , will win if +, P v22& or e'uivalently& when

the other s value is sufficiently low: v2 Q 2+,. he #ro!a!ility of winning with a

!id of +, is the #ro!a!ility that v2 Q 2+,. his #ro!a!ility can !e assessed with the

hel# of ?igure 2,.,. 6u##ose that !idder , ma"es a !id of 2& for eam#le& as

shown !y the short vertical !ar. $e have 9ust shown that this !id will win if theother s value is less than 2+,& i.e. less than / in this eam#le. ;otice that

fourtenths of the area under the dashed line is to the left of /. hus the

#ro!a!ility that the other s value is less than / is 0./& calculated as /,0 J 2+,,0.

A !id of 0 will never win& and a !id of 5 will always win& and in general& we

have the result:

=2,.> 8ro!a!ility of $inning =with a !id of > J +, .



b

0

00005

0001

00015

Probability

0 , 2 / 5 % - ,0

Ither)s 6ignal

0ig$re '.'. - >niform Distri&$tion on the Interva% +, '+H

Step . 6inding the 45pected Paoff Conditiona# on =inning

6u##ose that !idder , !ids +, and wins. his ha##ens when v2 Q 2+,. ?or

eam#le& when the !id is 2 as in ?igure 2,.,& winning would indicate that v2 Q

/& i.e. to the left of the higher vertical !ar in the figure. 6ince v2 is uniformlydistri!uted& it is e'ually li"ely to !e any #enny amount less than /& so the

e#ected value of v2 would !e 2 once we find out that the !id of 2 won. his

generali<es easilyB the e#ected value of v2 conditional on winning with a !id of

+, is 9ust +, ='uestion ,>. 7idder , "nows the signal v, and e#ects the other signal

to !e e'ual to the !id +, if it wins& so the e#ected value of the #ri<e is the average

of v, and +,:

=2,./> Conditional *#ected 8ri<e Value =winning with a 7id of > J +, +,

v, .

2

Step 3. 6inding the 45pected Paoff 6unction

he e#ected #ayoff for a !id of +, is the #roduct of the #ro!a!ility of winning in

=2,.> and the difference !etween the conditional e#ected #ri<e value in =2,./>

and the !id:

=2,.5> *#ected 8ayoff J 2+, R +, v, +, UT J +v, , +,2 .

,0 V 2 W ,0 ,0

Step &. Ma5imi)ing the 45pected Paoff 6unction

n order to maimi<e this e#ected #ayoff& we will set its derivative e'ual to 0.

he e#ression on the far right side of =2,.5> contains two terms& one that is



'uadratic in +, and one that is linear. 3ecall that the derivative of =+,>2 is 2+, and

the derivative of a linear function is its slo#e coefficient& so:

v, 2+,

=2,.%> *#ected 8ayoff (erivative J .

,0 ,0

6etting this derivative e'ual to 0 and multi#lying !y ,0 yields: v, 2+, J 0& or

e'uivalently& +, J v,2. o summari<e& if !idder 2 is !idding half of value as

assumed originally in =2,.2>& then !idder , s !est res#onse is to !id half of value

as well& so the ;ash e'uili!rium !idding strategy is for each #erson to !ehave in

this manner.

vi i ,&2. =e'uili!rium !id>.

=2,.-> +i J

2

;ormally in a first+#rice auction& one should !id !elow value& and it can !e seen

that the !id in =2,.-> is less than the conditional e#ected value in =2,./>. ?inally&

recall that this analysis !egan with an assum#tion that !idder 2 was using the

strategy in =2,.2> with X J ,2. his may seem li"e an ar!itrary assum#tion& !ut it

can !e shown that the only linear !idding strategy for this auction must have a

slo#e of ,2 ='uestion >.

I. he inner9s C$rse

$hen !oth !idders are !idding half of the value estimate& as in =2,.->& then theone who wins will !e the one with the higher estimate& i.e. the one who

overestimates the value of the #ri<e. Another way to see this is to thin" a!out

what a na ve calculation of the #ri<e value would entail. Ine might reason that

since the other s value estimate is e'ually li"ely to !e any amount !etween 0 and

,0& the e#ected value of the other s estimate is 5. hen "nowing one s own

estimate& say v,& the e#ected #ri<e value is =v, O 5>2. his e#ected #ri<e value&

however& is not conditioned on winning the auction. ;otice that this

unconditional e#ected value is greater than the conditional e#ected value in

=2,./> whenever the #erson s !id is less than 5& which is always the case when

!ids are half of the value estimate. *ce#t for this !oundary case where the !id is

eactly 5& the na ve value calculation will result in an overestimate of value&which can lead to an ecessively high !id and negative earnings.

Holt and 6herman =2000> conducted a num!er of common+value auctions with a

#ri<e value that was the average of the signals. 6u!9ects !egan with a cash

!alance of ,5 to cover any losses. ?igure 2,.2 shows !ids and signals for the



b ;<2

$0

$1

$2

$3

$4

$5

$6

$7

$8

$i!

final 5 rounds of a single session. Ine of the !idders can !e distinguished from

the others !y the large s'uare mar"s. ;otice that this #erson is !idding in an

a##roimately linear manner& !ut that this #erson s !ids are !idding a!ove the v2

line& which re#resents the ;ash e'uili!rium. he dashed line shows the

regression line that was fit to all !ids over all sessions& so we see that this session

was a little high !ut not aty#ical in the nature of the !idvalue relationshi#.

0 , 2 / 5 % - ,0

Iwn 6ignal

0ig$re '.. Bids for "ight S$&ects in the 0ina% 0ive Ro$nds of a Common?a%$e -$ction

Je!: So%id Line: 8ash "#$i%i&ri$m, Dashed Line: Regression for -%% Sessions

@7o%t and Sherman, +++A

he e#eriment session shown in ?igure 2,.2 was a research e#eriment& with

students recruited from a variety of economics classes at the University of

Virginia. ?igure 2,. shows analogous data for a classroom e#eriment with one #erson #aid #artial earnings at random. hese were students in an e#erimental

economics class& who had already read a!out #rivate+value auctions and were

relatively well+versed in game theory. ;otice that the !ids are lower& !ut still



$0

$1

$2

$3

$4

$5

$6

$7

$8

$i!

generally a!ove the ;ash #rediction. 6ome #eo#le were !idding a!out half of

their signals& however.

0 , 2 / 5 % - ,0

Iwn 6ignal

0ig$re './. Bids for en S$&ects in a C%assroom Common?a%$e -$ction

@fina% * ro$ndsA


he winner s curse was first discussed in $ilson =,%>& and a##lications to oil

lease drilling were descri!ed in Ca#en& Cla##& and Cam#!ell =,-,>. Dagel and1evine =,%> #rovided e#erimental evidence that it could !e re#roduced in the

la!oratory& even after su!9ects o!tain some e#erience. he literature on

common+value auctions is surveyed in Dagel =,5>.



<$estions

,. 6"etch a version of ?igure 2,., for the case where !idder , s !id is ,.

6how the #ro!a!ility distri!ution for the other #erson s value& v2&

conditional on the event that the first #erson s !id is the high !id. hen

calculate the e#ected value of v2& conditional on !idder , winning with a

!id of ,. hen e#lain in words why the e#ected value of v2 conditional

on the winning !id +, is e'ual to +,.

2. =his 'uestion is somewhat mechanical& !ut it lets you test your

"nowledge of the !id derivations in the reading.> 6u##ose that the #ri<e

value in e'uation =2,.,> is altered to !e the sum of the signals instead of

the average. he signals are still uniformly distri!uted on the range from

0 to ,0. a> $hat is the #ro!a!ility of winning with a !id of +, if the

other #erson !ids an amount that e'uals their signal& i.e. if +2 J v2. !> $hat

is the e#ected #ri<e value conditional on winning with a !id of +,N =Hint:

remem!er that the #ri<e value is determined !y the sum of the signals& not

the average.> c> $hat is the e#ected #ayoff function for !idder ,N =Hint:

#lease do not forget to su!tract the !id from the e#ected #ri<e value& sinceif you win you have to #ay your !id. Mou should get a function that is

'uadratic in +,.> d> =Calculus re'uired.> 6how that the !idder , s e#ected

#ayoff is maimi<ed with a !id that is e'ual to the !idder s own signal.

. he goal of this 'uestion is to show that the only linear strategy of the

form in e'uation =2,.2> is one with a slo#e of ,2 when the #ri<e value is

the average of the signals. a> $hat is the #ro!a!ility of winning with a !id

of +, if the other #erson !ids: if +2 J Xv2. =Hint: your answer should im#ly

=2,.> when X J ,2.> !> $hat is the e#ected #ri<e value conditional on

!idding with a !id of +,N c> $hat is the e#ected #ayoff function for

!idder ,N d> =Calculus re'uired.> 6how that the !idder , s e#ected #ayoff

is maimi<ed with a !id that e'uals one half of the !idder s own signal.

Cha#ter 22. 4ulti+Unit Auctions

his cha#ter #ertains to situations where a single auction is used to arrangemulti#le transactions at the same time. he first #art of the cha#ter descri!es an

auction that was used to determine which tracts of land in drought+stric"en

6outhwest eorgia would not !e irrigated during the 200, growing season. he

auction design was !ased on a series of la!oratory e#eriments in which



#artici#ants were #ut in the role of !eing farmers with one or more tracts of land&

each consisting of a s#ecified num!er of acres. n each round of the auction& the

!ids are ran"ed from low to high& with low !ids !eing #rovisionally acce#ted =u#

to a total e#enditure set !y the e#erimenter>. 7idders do not "now which round

will !e the final round that actually determines who must forego irrigation for that

growing season and !e com#ensated. he game can !e used to motivate a

discussion of the advantages of auctions& and of the many #ossi!le alternative

ways that such an auction can !e run. he actual auction was conducted with a

we!+!ased networ" of com#uters at eight different locations& and a similar

structure is used in the Veconla! rrigation 3eduction Auction #rogram.

Alternatively& a discussion of whether or not to use a uniform #rice can !e

motivated !y the hand+run instructions in the a##endi. hese instructions

re'uire that #artici#ants first !e endowed with some small items& li"e !all #oint

#ens& which can !e sold to the e#erimenter. he second #art of the cha#ter

contains a discussion of the widely discussed ?ederal Communications

Commission =?CC> auctions for communications !andwidth. hese auctions are

run simultaneously for distinct !andwidth licenses& with !id+driven #riceincreases. An alternative to the simultaneous ascending !id auctions of this ty#e

would !e to have the #ro#osed !id #rice !e increased automatically !y a cloc"&

and to let !idders indicate whether they are still willing to !uy after each u#ward

clic" of the cloc". 6uch a cloc"+auction was used to sell ;itrous Iide #ollution

allowances in Virginia in 200/. n all of the multi+unit auctions discussed in this

cha#ter& la!oratory e#eriments have !een used etensively to hel# design and

test the #rocedures that were used.

I. Dr! J

n early 2000& 9ust after the #u!licity associated with the new century and theM2D com#uter !ugs& a severe drought #lagued much of the 6outheastern United

6tates. 6ome of the hardest hit localities were in 6outh eorgia& and one of the

Atlanta news#a#ers ran a regular (ry 2D u#date on conditions and conservation

measures. If #articular concern were the record low levels for the ?lint 3iver&

which threatened wildlife and fish in the river and the oyster fishery in ?lorida.

As a result& #ressure was mounted to release water from a la"e north of Atlanta

into a #arallel river to #rotect the oyster fishery& and the threat to Atlanta drin"ing

water was a factor in the final decision to ta"e drastic action.

n A#ril 2000& the eorgia legislature #assed the ?lint 3iver (rought 8rotection

Act& mandating the use of an auction+li"e #rocess to restrict agricultural irrigation

in certain areas if the (irector of the eorgia *nvironmental 8rotection

(e#artment called a drought emergency. he uns#ecified nature of the mandated

auction made an ideal situation for la!oratory testing of alternative auction

mechanisms that might !e recommended. his cha#ter summari<es the



e#eriments and su!se'uent auction results& which are re#orted in Cummings&

Holt& and 1aury =200>.

he ?lint 3iver& which !egins from a drainage #i#e near the Atlanta Hartsfield

Air#ort& grows to a si<e that su##orts some !arge traffic !y the time it reaches the

?lorida state line and later em#ties into the ulf. A!out -0 #ercent of the water

usage in this river !asin is agricultural irrigation. ?armers have #ermits& which

were o!tained without charge& for #articular circular irrigation systems that

ty#ically cover areas from 50 to 00 acres. $ater is not metered& and therefore& it

is li!erally dum#ed into the fields during dry #eriods& creating the green circles

visi!le from the air& which ma"es restrictions on irrigation easy to monitor. he

idea !ehind the law was to use the economic incentives of a !idding #rocess to

select relatively low+use+value land to retire from irrigation. ?armers would !e

com#ensated to reduce any negative #olitical im#acts. he state legislature set

aside ,0 million dollars& ta"en from its share of the multi+state o!acco industry

settlement& to #ay farmers not to use one or more #ermits for irrigation.

herefore& what was !eing #urchased !y the state was the right to irrigate& not the

use of the land itself. n reality& it is unli"ely that land would !e #lanted in adrought year in the a!sence of irrigation #ermit. 7esides !eing non+coercive and

sensitive to economic use value& the auction format had the advantage of !eing

fair and easy to im#lement relative to administrative #rocesses. 6#eed was also

an im#ortant factor& given the limited time !etween the 4arch , deadline for the

declaration of a drought emergency and the o#timal time for #lanting cro#s

several wee"s later. ?inally& the diverse geogra#hic locations of the farmers

re'uired that an auction collect !ids from diverse locations& which suggested the

use of we!+!ased communications !etween officials at a num!er of !idding sites.

Any auction would involve a single !uyer& the state *nvironmental 8rotection

(e#artment =*8(>& and many sellers& the farmers with #ermits. =echnically

s#ea"ing& the *8( officials were careful not to use #urchase or sale terminology&which would have ownershi# im#lications& !ut this terminology will !e used here

for sim#licity.> 8ermits varied in si<e in terms of the num!ers of acres covered&

and in terms of whether the water would come from the surface =river or cree"s>

or from wells. his is a multi+unit auction& since the state could #urchase many

#ermits& i.e. com#ensate the #ermit holders for not irrigating the covered areas for

the s#ecified growing season.

he goals of the #eo#le running the auction were that the auction not !e viewed

as !eing ar!itrary or unfair& and that the auction ta"e out as much irrigation as

#ossi!le =measured in acres covered !y re#urchased #ermits> given the !udget

availa!le to !e s#ent. If course& economists would also !e concerned with

economic efficiency& i.e. that the auction would ta"e less #roductive land out of

irrigation.

A num!er of different ty#es of auctions could !e considered& and all involved

!ids !eing made on a #er+acre !asis so that !ids for different si<ed trac"s could !e



com#ared and ran"ed& with the low !ids !eing acce#ted. Ine method& a

discriminative auction& would have #eo#le su!mitting sealed !ids& with the

winning low !idders each receiving the amounts that they !id. ?or eam#le& if

the !ids were ,00& 200& 00& and /00 #er acre for / #ermits& and if the two

lowest were acce#ted& then the low !idder would receive ,00 #er acre and the

second low !idder would receive 200 #er acre. his auction is discriminative

since different #eo#le receive different amounts for a##roimately the same

amount of irrigation reduction #er acre. n contrast& a uniform+#rice auction

would esta!lish a cutoff #rice and #ay all !idders at or !elow this level an amount

that e'uals the cutoff #rice. f the cutoff #rice were 200 in the a!ove eam#le&

then the two low !idders would each receive 200& des#ite the fact that one !idder

was willing to acce#t a com#ensation of only ,00 #er acre. his uniform+#rice

auction would have !een a multi+unit& low+!idder+wins version of the second#rice

auction discussed in Cha#ter ,& whereas the discriminative auction is analogous

to the first+#rice auction discussed in that cha#ter. Fust as !idding !ehavior will

differ !etween first and second+#rice auctions& !idding !ehavior will differ

!etween discriminative and uniform #rice multi+unit auctions& so it is not o!viouswhich one will #rovide the greatest irrigation reduction for a given e#enditure.

his is where the e#eriments are used. n addition& it was widely !elieved that

collusion !etween farmers might !ecome #ro!lematic& given that many of them

are friends and relatives& and that many shared a distrust of Atlanta!ased officials.

Hence it was desira!le to evaluate alternative auction #rocedures under conditions

where the !idders could discuss !idding strategy and results freely.

he initial e#eriments were run in 4ay of 2000& almost a year !efore the first

actual auction& so we will discuss the e#eriments first. *arly discussions with

state officials indicated that discriminative auctions were #referred& to avoid the

a##arent waste of #aying someone more than they !id& which would ha##en in a

uniform+#rice auction in which all !idders would !e #aid the same amount #er acre. After some initial e#eriments& it !ecame clear that a multi+round auction

would remove a lot of the uncertainty that !idders would face in such a new

situation. n a multi+round auction& !ids would !e collected& ran"ed& and

#rovisional winners would !e #osted& !ut the results would not im#lemented if the

officials running the auction decided to acce#t revised !ids in a su!se'uent round.

7ids that were unchanged !etween rounds would !e carried over& !ut !idders

would have the o#tion of lowering or raising their !ids& !ased on the #rovisional

results. his #rocess was #erceived as allowing farmers to find out a##roimately

what the going #rice would !e and then to com#ete at the margin to !e included in

the irrigation reduction.

6ome of the early e#eriments were run a!out a year !efore the actual auction

and were watched !y state *8( officials. A conscious decision was made to

#rovide an amount of contet and realism that is somewhat unusual for la!oratory

e#eriments. 8artici#ants were recruited in grou#s ranging in si<e from to over



/2& and were told that they would have the role of farmers. 6ome of the

#artici#ants were students from Atlanta& and others were farmers and locals from

the 6outh eorgia area. Ine final test involved over 50 local #artici#ants !idding

simultaneously at different locations near Al!any& to test the software and

communication with officials in Atlanta. he final auction& conducted in 4arch

200,& involved a!out 200 farmers in different locations.

6ince most farmers had multi#le tracts of differing si<es and

#roductivities& each #artici#ant in the e#eriment was given three tracts of land. A

tract was descri!ed !y a s#ecific num!er of acres and a use value& which is the

amount of money that would !e earned if that tract would !e irrigated and farmed

for the current growing season. 8artici#ants were told that the land would not !e

farmed if the irrigation #ermits were sold.

?or eam#le& consider a #erson with three #ermits& with acreage and use values

#er acre as shown in the first two columns of a!le 22.,. he first #ermit covers

,00 acres& and has a use value of ,00 #er acre& so the !idder would re'uire an

amount a!ove ,00 #er acre as com#ensation for not irrigating and using the land.

n this eam#le& the !id was ,20 #er acre on #ermit ,& and this !id was acce#ted&so the earnings are ,20 =#er acre> times ,00 =acres> so the total shown in the right+

hand column is ,2&000. f this !id had !een re9ected& the farmer would have

farmed the land and would have earned the use value of ,00 #er acre times the

num!er of acres& for a total that would have !een ,0&000. o !e sure that you

understand these calculations& #lease calculate the earnings for #ermit 2& with an

acce#ted !id& and for #ermit & with a re9ected !id. hese calculations can !e

entered in the far+right column of the ta!le.

After some e#erimentation with alternative sets of #rocedures and some

consultation with *8( officials& we ended u# focusing on two alternative setu#s.

a!le 22., 6am#le *arnings Calculations for the rrigation 3eduction Auction

8ota# !cres >se :a#ue 0per

acre2

Bid

0per acre2

!uction

/utcome

4arnings

8ermit , ,00 ,00 ,20 acce#ted ,2&000

8ermit 2 50 200 250 acce#ted SSSSSS

8ermit ,00 00 /00 re9ected SSSSSS

Ine #ro#osed auction design involved a single+round& sealed+!id discriminativeauction& and another involved a multi+round version of the same auction. n either

case& all !ids would !e collected on !id sheets and ran"ed from low to high. hen

starting with the low !ids& the total e#enditures would !e calculated for adding

#ermits with higher !ids. A cutoff for inclusion was determined when the total



e#enditures reached the amount allotted !y the auctioneer. ?or eam#le& if the

amount to !e s#ent had !een announced to !e 50&000& and if the lowest 20 !ids

yielded a total e#enditure of /&500& and a 2,st !id would ta"e the e#enditure

a!ove this limit& then only 20 !ids would !e acce#ted. his cutoff would

determine earnings in the single+round sessions& with earnings on #ermits with

re9ected !ids !eing determined !y their use values. n the sessions with a

multiround setu#& the cutoff would !e calculated& and the #ermits with !ids !elow

the cutoff would !e announced as !eing #rovisionally acce#ted. hen new !ids

would !e acce#ted& which would then !e ran"ed& with a new announcement of

which tracts had #rovisionally acce#ted !ids. his #rocess would continue until

the e#erimenter decided to sto# the auction& at which time the acce#ted !ids

would !e used to determine earnings.

A ty#ical session !egan with a trainer auction for colored writing #ens =see the

instructions in the a##endi to this cha#ter>. his #ractice auction was intended

to convey the main features of su!mitting different !ids for different tracts& which

would !e ran"ed& with some low !ids !eing acce#ted and others not. 8artici#ants

were allowed to collude on !ids in any manner& ece#t that they were not #ermitted to "ee# #eo#le away from the !id su!mission area. 6ome #eo#le

discussed #rice in small grou#s& and others made #u!lic suggestions.

?igure 22., shows the results of two sessions run with identical cost

structures& ece#t that the larger one had !een scaled u# to accommodate more

#artici#ants. he line la!eled land values shows the #ermit use vales #er acre&

with each ste# having a width e'ual to the num!er of acres for the #ermit with a

use value at that ste#. hese o##ortunity costs& arrayed in this manner from low

to high& constitute a su##ly function. n each of these sessions& we announced a

fied total !udget to !e used to #urchase #ermits. 6o a low total acreage can !e

#urchased at a high #rice #er acre& and a high total acreage can !e #urchased at a

low #rice #er acre. his fied !udget then generates a curve that is analogous to ademand function. f B is the total !udget availa!le to #urchase < total acres at a

#rice #er acre of P & then all money is s#ent if P< J B& or if P J B<& which

generates the negative relationshi# !etween P and < that is gra#hed on the left

side of ?igure 22.,. he connected lines on the right side of the figure show the

average #rice #er acre of the #rovisionally acce#ted !ids& !y round& for each of the

sessions. ;otice that these #rice averages converge to the com#etitive

#redictions& des#ite the fact that #artici#ants could freely discuss the !idding

#rocess& announce suggested !ids& etc. 8artici#ants were not told in advance how

many rounds there would !e.

cost per acre



0

50

100

150

200

acres 151051oer reision round

!0person auction

4!person auction

budget locus

land alues

0ig$re .' Res$%ts of 6o Sessions 6ith M$%tip%e Ro$nds

So$rce: C$mmings, 7o%t, and La$r! @++/A

Ine advantage of la!oratory e#eriments is that #rocedures can !e test!edded

and unantici#ated #ro!lems can !e discovered and fied !efore the real auction.

n one of the sessions with student su!9ects& a #artici#ant as"ed what would

ha##en if there ha##ened to !e a num!er of !idders tied at the cutoff !id that

ehausted the announced !udget& and if there was not enough money in the

!udget to cover all !ids at the tie level. his #ossi!ility was not covered clearly in

the instructions& and the e#erimenter in charge of the session announced that all

of those tied would !e included as #rovisional winners& or as final winners if thiscost per acre



0

10

20

30

40

50

60

7080

90

100

110

120

130

140

acres 1 5oer reision round

budget locus

land alues

random tie"brea#ing rule

inclusie tie"brea#ing rule

0ig$re . Res$%ts of 6o Sessions 6ith Different ie Breaking R$%es


were the final round. n this session& a tie arose at a #rice a!out 5 #ercent a!ove

the com#etitive level& and all tied !ids were #rovisionally included. n the

su!se'uent round& more !ids came in at the tie level from the #revious round& and

this accumulation of !ids at the focal tie #oint continued. he resulting #ayment

to su!9ects& which were needed to include all tied !ids& ended u# !eing twice the

!udgeted amount. his would have !een analogous to s#ending 20 million in the

actual auction instead of the ,0 million !udgeted !y the legislatureG

A second session was run later that day with different #eo#le. he #rocedureswere identical& ece#t that it was announced that the !ids to !e included would !e

randomly determined in the event of a tie. he #rices converged to the

com#etitive level& as indicated in ?igure 22.2.

here were a num!er of other #rocedural changes that were im#lemented as a

result of the e#eriments. ?or eam#le& we noticed that low !ids tended to

increase when we announced the cutoff !id& i.e. the highest #rovisionally acce#ted

!id& at the end of each round. his #attern of increasing low !ids is intuitive&

since the low !idders faced less ris" with !id increases if they "new a!out how

high they could have gone in the #revious round. his induced us to run some

sessions where we only announced the #ermit num!ers =!ut not the actual !ids>

for those #ermits that were #rovisional winners at the end of each round. hisreduction in information resulted in less of an u#ward cree# of low !ids in

successive rounds& and as !efore& the high !ids tended to fall as !idders scram!led

to !ecome included. hese modifications in tie !rea"ing+rules and the #ost+round



announcement #rocedures were incor#orated into the auction rules used !y the

*8( in the su!se'uent auction.

he final auction was conducted in A#ril 200,& with assistance from a num!er of

e#erimental economists and graduate students and staff from eorgia 6tate

University& where the e#eriments had !een done and where the results were

collected and dis#layed to to# *8( officials& who met in the *#erimental

*conomics 1a!oratory at eorgia 6tate. Almost 200 farmers turned u# at

locations at a.m. on the designated 6aturday morning& along with numerous

television re#orters and s#ectators. he #rocedures were 'uite close to those that

had !een im#lemented in the e#eriments& ece#t that there were no redem#tion

values& and all acreage amounts were those registered with the *8(. *ach !id

was a signed contract& with one co#y going to the !idder and another to the !id

officials on site. All !ids were entered in com#uters !y auction officials& as had

!een done in the e#eriments& and the resulting !ids from the eight locations were

ran"ed and #ro9ected in Atlanta& where officials then discussed whether to sto# the

auction& and if not& how much money to release to determine #rovisional winners

for that round. his non+fiity of the !udget differed from the #rocedures that had !een used in the e#eriments& !ut it did not contradict any of the #u!lished

auction rules. he changes in the #rovisionally released !udget caused the cutoff

!id to stay roughly the same in the first four rounds& ending u# at ,25 #er acre in

round /. he director of the *8( then decided to release more money and raise

the cutoff !id to 200 in round 5& when the auction was terminated. ?igure 22.

shows the distri!utions of !ids in the relevant range& from ,0+2,0& a range that

covered the cutoffs !etween !ids that were #rovisionally acce#ted and those that

were not. As can !e seen from the figure& the !ids in this range fell from one

round to the net& ena!ling more acres to !e ac'uired for cutoffs in this range. f

a fied !udget had !een used& as in the e#eriments& then the cutoff !id would

have fallen from round to round& and the economists involved as advisors wereconcerned that the increasing !udgets used from round to round may have

discouraged some farmers from ma"ing !id reductions at the margin. n addition&

the dramatic final+round increase in the !udget e#enditure and in the cutoff !id

might have serious conse'uences for !idding in a future auction that used the

same #rocedures.

he 200, rrigation Auction was considered to !e a success. n #articular& !ids

were received on a!out %0 #ercent of the acres that were eligi!le to !e retired

from irrigation. n total& a!out &000 acres were ta"en out of irrigation& at an

average #rice of a!out ,5 #er acre.

n 2002& the state officials& in consultation with e#erimental economists =1aury

and Cummings> decided to run the auction as a single+round discriminative

auction& with sealed !ids !eing acce#ted my mail. his method was less

e#ensive to administer& and attendance at the auction site was less im#ortant&



since farmers were already familiar with the com#ensation and low!ids+win

features of the auction that had !een im#lemented in the #revious year.

n addition& the mail+in #rocedures may have ena!led more #eo#le to #artici#ate&

since attendance at the auction site was not re'uired. A reserve #rice =maimum

!id> of ,50 was im#osed& so that !ids a!ove this amount would not !e

considered. n this second auction& /,&000 acres were removed from irrigation& at

an average cost of ,/ #er acre =4c(owell& 2002>.

\\

Iffers from ,0 + 2,0

0ig$re ./ Res$%ts from the 0ina% Irrigation Red$ction -$ction


II. 0CC Band6idth -$ctions

overnment agencies in a num!er of countries have recently used simultaneous

auctions to allocate #ortions of !roadcast fre'uency !andwidth in different

regional mar"ets. hese auctions ty#ically involve a large num!er of licenses&

which are defined !y geogra#hic region and ad9acent fre'uency intervals. he

rationale for conducting the auctions simultaneously is that there may !e

com#lementarities in valuation as !idder strive to o!tain contiguous licenses that

may allow them to en9oy economies of scale and to #rovide more valua!le

services to consumers. ?or eam#le& a #erson who signs u# for cell#hone service



with a #articular com#any would ty#ically !e willing to #ay more if the com#any

also #rovides service =free of roaming fees > in ad9acent geogra#hic areas. $ith a

few ece#tions& these simultaneous auctions are run as *nglish auctions& with

simultaneous ascending !ids for each license. 7ids are collected in a se'uence of

rounds. At each round& !idders who are eligi!le to !id for a #articular license !ut

are not currently the high !idder& can maintain their active status !y !idding a!ove

the current high !id !y a s#ecified increment. he activity rules can !e com#le&

!ut the main idea is sim#leB they are intended to force !idders to "ee# !idding in

order to !e eligi!le to !id in later rounds. he #ur#ose of this forced !id activity

is to "ee# the high !ids moving u#ward& so that valuation information is revealed

during the course of the auction& and not in sudden !id 9um#s in the final rounds.

he auctions sto# when no !ids for any license are changed in a given round.

his ty#e of simultaneous& ascending+!id auction was used !y the U.6. ?ederal

Communications Commission =?CC> to sell !andwidth for #ersonal

communications services& !eginning in the ,0 s. he amounts of money raised

were sur#risingly large& and similar auctions have !een im#lemented in *uro#e.

he auctions have #roved to !e fast& efficient& and lucrative as com#ared with theadministrative = !eauty+contest > allocations that were used #reviouslyB see

Cha#ter 2 on rent+see"ing for more discussion of the #otential inefficiencies of

these administrative #rocedures.

here are& however& some reasons to 'uestion the efficiency of simultaneous

auctions for individual licenses. A !idder s strategy in a single+unit& ascending+

!id auction with a "nown #rivate value is fairly sim#le& one must "ee# !idding

actively until the high !id eceeds the !idder s #rivate value. n this manner& the

!idding will eclude the low value !uyers and the #ri<e will !e awarded to the

!idder with the highest value. he strategic environment is more interesting with

common values elements& since information a!out the un"nown common value

might !e inferred from o!serving when other !idders dro# out of the !idding.he case of valuation com#lementarities is even more com#le& as can !e seen !y

considering a sim#le eam#le. 6u##ose that there are two contiguous licenses

!eing sold& A and 7& with three com#eting !idders& & & and . 7idder is a

local #rovider in region A and has a #rivate value of ,0 for A and 0 or 7.

Conversely& !idder is a local #rovider in the other region and has a value of 0

for A and ,0 for 7. he third !idder is a national #rovider& with values of 5 for A

alone& 5 for 7 alone& and 0 for the A7 com!ination. hese valuations are shown

in the left column of a!le 22.2& where the su!scri#ts indicate the license=s>

o!tained& A alone& 7 alone& or the A7 com!ination.

6u##ose that the !idding se'uence for the first rounds is shown in a!le 22.2.

After the first round of !idding& !idder is the high !idder for A& and !idder is

the high !idder for 7. 6u##ose that the activity rule re'uires !idder to raise

these !ids to 5 to stay active in each region& as shown in the outcome for round 2.



n res#onse& !idders and raise the !ids for their #referred licenses to %& to stay

active& as shown in the 3ound column.

a!le 22.2 7id 6e'uence for a 6imultaneous Ascending+7id Auction:

he *#osure 8ro!lem

:a#ues 'ound 1 'ound 'ound 3

7idder

VA J ,0& V7 J 0& VA7 J ,0

3 for -& 0 for 7 no change 1 for -

7idder

VA J 0& V7 J ,0& VA7 J ,0

0 for A& 3 for B no change 1 for B

7idder

VA J 5& V7 J 5& VA7 J 0

for A& for 7 * for -, * for B no change

As round / !egins& !idder faces a dilemma if the #rivate values of the first

two !idders are not "nown. t ma"es no sense for !idder to com#ete for a

single license& since the !idding has already to##ed the !idder s #rivate value of 5for each single license. 7ut to !id a!ove this value of 5 for !oth licenses #roduces

an e#osure #ro!lem& since !idder does not "now whether the other two !idders

will #ush the individual !ids for the two licenses u# to a level where the sum is

greater than 0& which is value of the #ac"age to !idder . f this e#osure ris"

causes !idder to dro# out& then the outcome is inefficient in the sense that the

total value =,0 for !idder & who gets license A& and ,0 for !idder who gets

license 7> is less than the value of the A7 #ac"age =0> to !idder . Ine

widely discussed solution to the e#osure #ro!lem is to allow !idding on

#ac"ages& so that there would !e three simultaneous auctions& for A alone& for 7

alone& and for the A7 #ac"age. $hen the !idding sto#s& the seller would then

select the final allocation that maimi<es the sales revenue. his is sometimescalled com!inatorial !idding& and with large num!ers of licenses& it is

com#licated !y the fact that it may !e difficult to calculate the

revenuemaimi<ing allocation of licenses& at least in finite time. *conomists and

o#erations researchers have wor"ed on algorithms to deal with the

revenuemaimi<ation #ro!lem in a timely manner& and as a result& the U.6.

?ederal Communications Commission considered and tested the use of

com!inatorial !idding for !andwidth licenses.

Com!inatorial !idding& of course& may introduce #ro!lems that are not merely

due to com#utation. Consider the eam#le shown in a!le 22.& where the

valuations are as !efore& with the ece#tion that the A7 #ac"age for !idder has

!een reduced from 0 to ,5. n this eam#le& the !idding in the first round is the

same as !efore& ece#t that !idder !ids % for the A7 #ac"age instead of

!idding for each license. 6ince and each su!mit !ids of / for their #referred

licenses& the revenue+maimi<ing allocation after the first round would !e to



aware the licenses se#arately& for a total revenue of . 7idder to#s this in the

second round !y !idding ,0 for the A7 #ac"age& and the other !idders res#ond

with !ids of for A and 5 for 7 in round & for a total of ,. o stay com#etitive&

!idder res#onds with a !id of ,/ for the A7 #ac"age& which would !e the

revenue+maimi<ing result if the !idding were to sto# after round /. n order for

!idders and to o!tain their #referred licenses& it is necessary for them to raises

the sum of their !ids& !ut each would #refer that the other !e the one to do so. ?or

eam#le& !idder would li"e to maintain a !id of and have !idder come u# to

to defeat the #ac"age !id of . 7ut !idder would #refer to see some of the

9oint increase come from !idder ,. his coordination #ro!lem is magnified if

there are greater num!ers of !idders involved and if they do not "now one another

s values. t is easy to imagine that a coordination failure might result& as some

!idders try to free ride and let others !ear the cost of raining the !id total. A

coordination failure in this eam#le would result in an allocation of the A7

#ac"age to !idder & for a total value of ,5& which is !elow the sum of #rivate

values =,0 O ,0> if the licenses are awarded se#arately to !idders and .

a!le 22. 7id 6e'uence for a 6imultaneous Ascending+7id Auction:

he Coordination 8ro!lem

:a#ues 'ound 1 'ound 'ound 3 'ound &

7idder

VA J ,0& V7 J 0& VA7 J ,0

3 for -& 0 for 7 no change 2 for - no change

7idder

VA J 0& V7 J ,0& VA7 J ,0

0 for A& 3 for B no change * for B no change

7idder

VA J 5& V7 J 5& VA7 J ,5

% for A7 '+ for -B no change '3 for -B

hese coordination and free+riding incentives raise interesting !ehavioral issues

that can !e investigated with la!oratory e#eriments.

e#eriments& cloc" auction

III. - C%ock -$ction: the ++3 irginia 8itro$s 5(ide -$ction

I. "(tensions?or a nice discussion of the considerations involved in designing multiunit

auctions& see Dlem#erer =2002>.



8art V. 7argaining and ?airness

7ilateral !argaining is #ervasive& even in a develo#ed economy& es#ecially for

large #urchases li"e automo!iles or s#ecialty items li"e housing. 7argaining is

also central in many legal and #olitical dis#utes& and it is im#licit in family

relations& care of the elderly& etc. he highly fluid give and ta"e of face+to+facenegotiations ma"es it difficult to s#ecify convincing structured models that #ermit

the calculation of ;ash e'uili!ria. *#erimental research has res#onded in two

ways. ?irst& it is #ossi!le to loo" at !ehavior in unstructured !argaining situations

to s#ot interesting #atterns of !ehavior& li"e the well "nown deadline effect& the

tendency to delay agreements until the last moment. he second a##roach is to

limit the timing and se'uence of decisions& in order to learn something a!out

fairness and e'uity considerations in a sim#lified setting. he ultimatum

!argaining game discussed in Cha#ter 2 is an eam#le of this latter a##roach. n

an ultimatum game& one #erson ma"es a #ro#osal a!out how to s#lit a fied

amount of money& and the other either acce#ts or re9ects& in which case !oth earn

<ero. 6eemingly irrational re9ections in such games have fascinated economists&

and more recently& anthro#ologists. A generali<ation of the ultimatum game is

one where the res#onder can either acce#t the original #ro#osal or ma"e a counter

#ro#osal& which is then either acce#ted or re9ected !y the original #ro#oser& and

there is a discussion of !ehavioral #atterns in such multi+stage games.

he scenarios discussed in Cha#ter 2/ also have alternating decision structures&

!ut the focus is on mani#ulations that highlight issues of fairness& trust& and

reci#rocity. he trust game !egins when one #erson is endowed with some cash&

say ,0& of which #art or all may !e #assed to the other #erson. he money

#assed is augmented& e.g. tri#led& and any #art of the resulting amount may !e

#assed !ac" to the original #erson. A high level of trust would !e indicated if thefirst #erson #assed the full ,0& e#ecting the second #erson to reci#rocate and

#ass !ac" at least a third of the resulting 0. If course& the #ro#oser might

#refer to #ass nothing if no #ass+!ac" is antici#ated. he second setu# that is

considered& the reci#rocity game& has more of a mar"et contet& !ut the

underlying !ehavioral factors are similar. Here& em#loyer announces a wage& and

seeing this& the wor"er chooses an effort level& which is costly for the wor"er !ut

which !enefits the em#loyer. he issue is whether fairness considerations& which

are clearly #resent in !ilateral negotiations& will have an effect on mar"et clearing

in more im#ersonal mar"et settings. he final #art of the cha#ter also #ertains to

a two+stage game in which the #rinci#al =em#loyer> selects a contract& which is

either acce#ted or re9ected !y the agent =wor"er>& who then chooses an effort. hemenu of contracts allows incentives =targets and #ro!a!ilistic #unishments> andor

non+!inding !onuses that are #rovided e5 post .



Cha#ter 2. Ultimatum 7argaining

his game #rovides one #erson with the a!ility to #ro#ose a final offer that must

!e acce#ted or re9ected& 9ust as a mono#olist may #ost a #rice on a ta"e+it+or+leave

it !asis. he difference !etween the usual mono#oly situation and ultimatum !argaining is that there is only one !uyer of a single item in ultimatum !argaining&

so a re9ection results in <ero earnings for !oth !uyer and seller. Although

re9ections of #ositive amounts of money& however small& may sur#rise some& they

will not sur#rise anyone who has #artici#ated in this ty#e of !argaining& e.g. with

the Ultimatum ame on the Veconla! site. he we!game also has a setu# o#tion

that allows the res#onder a range of intermediate res#onses that involve an

e'ually #ro#ortional reduction = s'uish > of the #ro#osed earnings for each #erson.

his s'uish would !e analogous to a #artial #urchase in the !ilateral mono#oly

situation& at least under constant cost and value conditions.

I. his is M! 0ina% 5ffer, ake it or Leave ItE

4any economic situations involve a final offer from one #erson to another& where

re9ection means <ero earnings on the transaction for !oth. ?or eam#le& su##ose

that a local mono#olist can #roduce a unit of a commodity for 5. he sole !uyer

needs the #roduct and is willing to #ay any amount u# to ,5 !ut not a #enny

more& since ,5 is what it would cost to !uy the #roduct elsewhere and #ay to

have it shi##ed into the local mar"et. hen the sur#lus to !e divided is ,0& since

this is the difference !etween the value and the cost. he !uyer "nows the seller

s cost& and hence "nows that a #rice of ,0 would s#lit the sur#lus. A higher #rice

corres#onds to offering a lower amount to the !uyer. he seller has a strategicadvantage if the seller can ma"e a ta"e+it+or+leave+it offer& which we assume to !e

the case. his setu# is called an ultimatum game since the #ro#oser =seller>

ma"es a single offer to the res#onder =!uyer>& who must either acce#t the

#ro#osed s#lit of the sur#lus =as determined !y the #rice> or re9ect& which results

in <ero earnings for !oth.

he ultimatum game was introduced !y uth et al. =,2> !ecause it highlights

an etreme conflict !etween the dictates of selfish& strategic !ehavior and notions

of fairness. f each #erson only cares a!out his or her own earnings& and if more

money is #referred to less& then the #ro#oser should !e a!le to get away with

offering a very small amount to the res#onder. he mono#olist in the #revious

#aragra#h s eam#le could offer a #rice of ,/.& "nowing that the !uyer wouldrather #ay this #rice than a #rice of ,5.00 =including shi##ing> from a seller in

another mar"et. he ,/. #rice is unfair in the sense that . of the availa!le

sur#lus goes to the seller& and only a #enny goes to the !uyer. *ven so& the !uyer



who only cares a!out getting the lowest #ossi!le #rice should acce#t any #rice

offer !elow ,5.00& and hence the seller should offer ,/..

t is easy to thin" of situations in which a seller might hesitate to e#loit a strong

strategic advantage. etting a reservation for dinner after graduation in a college

town is the "ind of thing one tries to do si months in advance. 3estaurants seem

to shy away from allocating the scarce ta!le s#ace on the !asis of #rice& which is

usually not raised on graduation day. 6ome moderate #rice increases may !e

hidden in the form of re'uiring the #urchase of a s#ecial graduation meal& !ut this

"ind of #rice #remium is nowhere near what would !e needed to remove ecess

demand& as evidenced !y the long lead time in acce#ting reservations. A #ossi!le

e#lanation is that an eor!itant #rice might !e widely discussed and re#orted&

with a !ac"lash that might harm future !usiness. he higher the #rice& the lower

the cost of re9ecting the deal. n the mono#oly eam#le discussed a!ove& a #rice

of ,/. would !e re9ected if the !uyer is willing to incur a one+cent cost in

order to #unish the seller for charging such a #rice.

6everal variants of the ultimatum game have !een widely used in la!oratory

e#eriments !ecause this game maimi<es the tension !etween fairnessconsiderations and the other etreme where #eo#le only care a!out their own

earnings. 4oreover& in the la! it is often #ossi!le to set u# a one+shot situation

with enough anonymity to eliminate any considerations of re#utation& reward& and

#unishment. he net section descri!es an ultimatum e#eriment where

la!oratory control was a #articularly difficult #ro!lem.

II. Bargaining in the B$sh

Fean *nsminger =200,> conducted an ultimatum e#eriment in a num!er

of small villages in *ast Africa. All #artici#ants were mem!ers of the Irma clan.

he Irma offer an interesting case where there is considera!le variation in theetent of integration into a mar"et economy& which may affect attitudes towards

fairness. he more nomadic families raise cattle and live largely off of the mil"

and other #roducts& with very little !eing !ought or sold in any mar"et. Although

some nomadic families have high wealth& which is "e#t in the herd& they ty#ically

have low incomes in terms of wage #ayments. Ither Irma& in contrast& have

chosen a more sedentary lifestyle for a variety of reasons& including the

encroachment of gra<ing lands. hose who live sedentary lives in villages

ty#ically #urchase food with money income o!tained as wages or cro# revenues.

hus e#osure to a mar"et economy is 'uite varia!le and is well measured as

direct money income. 6uch income is not highly correlated with wealth& since

some of the wealthiest are self+sufficient nomadic families with large herds.

4any !argaining situations in a mar"et contet end u# with individuals agreeing

to s#lit the difference& so a natural con9ecture is that !ehavior in an ultimatum

game will !e related to the degree of e#osure to a mar"et economy.



001

02

03

04

05

06

.elati;e Yr eAuency

@ccepte!.eZecte!

A survey was com#leted for each household& and at least one adult was recruited

from each household to #lay fun games for real money. he e#eriments were

conducted in grass houses& which ena!led *nsminger to isolate grou#s of #eo#le

during the course of the e#eriment. A grand master& who was "nown to all

villagers& read the instructions. his #erson would turn away to avoid seeing

decisions as they were made. he amount of money to !e divided !etween the

#ro#oser and res#onder in each #air was set to !e a##roimately e'ual to a ty#ical

day s wages =,00 Denyan shillings>. he game was only #layed once. *ach

#ro#oser would ma"e an offer !y moving some of the shillings to the other side of

the ta!le& !efore leaving the room while the res#onder was allowed to acce#t or

re9ect this offer. *nsminger re#orts that the #eo#le en9oyed the games& des#ite

some amusement at the insanity and foolishness of $estern ways. 6he tried to

move from one village to another !efore word of results arrived.

0 0., 0.2 0. 0./ 0.5 0.% 0.- 0. 0.

Iffer as a ?raction of 6ta"e 6i<e

0ig$re /.'. Distri&$tion of 5ffers in an >%timat$m Game 6ith the 5rma

@*1 Pairs, 5ne?Da! age Stake So$rce: "nsminger, ++'A

he data for 5% !argaining #airs are shown in ?igure 2.,. he average offer was

// #ercent of the sta"e& with a clear mode at 50 #ercent. he lowest offer was 0

#ercent& and even these une'ual s#lits were rarely re9ected =as indicated !y the

light #art at the !ottom of the !ar>. f #eo#le had foreseen that the /0 #ercent

would all !e acce#ted& then they might have lowered their offers.



t is clear that the modal offer is not o#timal in the sense of e#ected+#ayoff

maimi<ation against the actual e5 post re9ection #attern. 8eo#le who made these

generous offers almost always mentioned fairness as the 9ustification in follow+u#

interviews. *nsminger was sus#icious& however& and a##roached some relia!le

informants who revealed a different #icture of the tal" of the village. he

#ro#osers were a##arently o+sessed with the #ossi!ility that low offers would !e

re9ected& even though re9ections were thought to !e unli"ely. 6uch o!sessions

suggest that e#ected+#ayoff maimi<ation may not !e an a##ro#riate assum#tion

for large amounts of money& e.g.& a day s income. his con9ecture would !e

consistent with the #ayoff+scale effects on ris" aversion discussed #reviously in

Cha#ter /.

;evertheless& it cannot !e the case that individuals ma"ing relatively high offers

were doing so so#e# out of fairness considerations& since offers were much lower

in a second e#eriment in which the #ro#oser s s#lit of the same amount of money

was automatically im#lemented. his game& without any #ossi!ility of re9ection&

is called a dictator game. he average offer fell from // #ercent in the ultimatum

game to , #ercent when there was no #ossi!ility of re9ection. *ven though thereseems to !e some strategic reaction !y #ro#osers to their advantage in the dictator

game& the modal offer was still at the fair or fiftyfifty division& and less than a

tenth of the #ro#osers "e#t all of the money.

*nsminger used a multi#le regression to evaluate #ro#oser offers in !oth the

ultimatum and dictator games. n !oth cases& the #resence of wage income is

significantly related to offersB those with such mar"et interactions tend to ma"e

higher offers. Her con9ecture is that #eo#le e#osed to face+to+face mar"et

transactions may !e more used to the notion of s#litting the difference. Varia!les

such as age& gender& education& and wealth =in cattle e'uivalents> are not

significant. his intra+cultural finding is also evident in a cross+cultural study

involving ,5 small+scale societies on five continents =Henrich et al.& 200,>. hisinvolved one+shot ultimatum games with com#ara!le #rocedures that were

conducted !y anthro#ologists and economists. here was considera!le variation&

with the average offer ranging from 0.2% to 0.5 of the sta"e. he societies were

ran"ed in two dimensions: the etent of economic coo#eration and the etent of

mar"et integration. 7oth varia!les were highly significant in a regression&

e#laining a!out %, #ercent of the variation& whereas individual varia!les li"e

age& se& and relative wealth were not. he lowest offers were o!served in

societies where very little #roduction occurred outside of family units =e.g&& the

4achiguenga of 8eru>. he highest offers were o!served in a society where

#roduction involved 9oint effortB offers a!ove one half for the 1amelara of

ndonesia& who are whale hunters in large sea+going canoes.

A one+shot ultimatum game& #layed for money& is a strange new

e#erience for these #eo#le& and !ehavior seemed to !e influenced !y #arallels

with social institutions in some cases. he Ache of 8araguay& for eam#le& made



generous offers& a!ove 0.5 on average. he #ro#osed sharing in the ultimatum

game has some #arallels with a #ractice where!y Ache hunters with large "ills

will leave them at the edge of cam# for others to find and divide.

III. Bargaining in the La&

Ultimatum games have !een conducted in more standard& student su!9ect #ools inmany develo#ed countries& and with sta"es that are usually a!out ,0. he mean

offer& as a fraction of the sta"e& is ty#ically a!out 0./& and there seems to !e less

variation in the mean offer than was o!served in the small+scale societies. 3oth et

al. =,,> re#ort ultimatum game e#eriments that were run in four universities&

each in a different country. he modal offer was 0.5 in the U.6. and 6lovenia& as

was the case for the Irma and for other studies in the U.6. =e.g.& ?orsythe et al.&

,>. he modal offer was somewhat lower& 0./& in srael and Fa#an. 3e9ection

rates in srael and Fa#an were no higher than in the other two countries& des#ite

the lower mean offers& which led the authors to con9ecture that the differences in

!ehavior across countries were due to different cultural norms. hese differencesin !ehavior re#orted !y 3oth et al. =,,> were& however& lower than the

differences o!served in the ,5 small+scale societies& which are less homogeneous

in the nature of their economic #roduction activities.

3ecall that the Irma made no offers !elow 0.. n contrast& some student

su!9ects in develo#ed countries ma"e offers of 0.2 or lower& and these low offers

are re9ected a!out half of the time. ?or eam#le& consider the data in ?igure 2.2&

which is for a one+shot ultimatum game that was done in class& !ut with full

money #ayments and a ,0 sta"e. he mean offer was a!out 0./ =as com#ared

with 0.// for the Irma>& !ut a!out a 'uarter of the offers were 0.2 or !elow& and

these were re9ected a third of the time.

Ultimatum !argaining !ehavior is relatively sensitive to various #rocedural details& and for that reason etreme care was ta"en in the Henrich et al.

cross+cultural study. Hoffman et al. =,/> re#ort that the median offer of 5 in an

ultimatum game was reduced to / !y #utting the game into a mar"et contet. he

mar"et terminology had the #ro#oser #lay the role of a seller who chose a ta"e+

itor+leave+it #rice for a single unit. nteractions with #osted #rices are more

anonymous than face+to+face negotiations& and mar"et #rice terminology in the

la!oratory may stimulate less generous offers for this reason. he median offer

fell again& to & when high scores on a trivia 'ui< were used to decide which

#erson in each #air would #lay the role of the seller in the mar"et. 8resuma!ly&

this role+assignment effect is due to the fact that #eo#le may !e more willing to

acce#t an aggressive =low> offer from a #erson who earned the right ma"e the

offer.



0

01

02

03

04

05

06

.elati;e Yr eAuency

@ccepte!.eZecte!

0 0., 0.2 0. 0./ 0.5 0.% 0.- 0. 0.


0ig$re /.. Distri&$tion of 5ffers in a 5ne?Shot >%timat$m Game

@'' Pairs, '+ Stake, Cash Pa!ments, >niversit! of irginia, Spring ++'A

t is somewhat unusual for a seller to offer only one unit for sale to a

!uyer& and a multi+unit setu# #rovides the !uyer with an o#tion for #artial

re9ection. ?or eam#le& su##ose that there are ,0 units for sale. *ach costs 0 to

#roduce& and each unit is worth , to the !uyer. he seller #osts a #rice for the ,0

units as a grou#& !ut the !uyer can decide to #urchase a smaller num!er& which

reduces each #arty s earnings #ro#ortionately. ?or eam#le& su##ose that theseller #osts a #rice of %& which would #rovide earnings of % for the seller and /

for the !uyer. 7y #urchasing only half of the units& the !uyer reduces these

earnings to for the seller and 2 for the !uyer. his #artial re9ection o#tion is

im#lemented in the Veconla! software as a s'uish o#tion =Andreoni& Castillo& and

8etrie& 200>. f this o#tion is #ermitted& the res#onder can choose a fraction that

indicates the etent of acce#tance: with 0 !eing full re9ection and , !eing full

acce#tance. his o#tion was used in the classroom e#eriment shown in ?igures

2. =first round> and 2./ =fifth and final round>. he etreme offer of 0.2 in the

first round was s'uished !y a half. Iffers in the fifth round were more clustered

a!out 0.5.



0

01

02

03

04

05

06

.elati;e Yr eAuency

@ccepte!.eZecte!

0

01

02

03

04

05

06

.elati;e YreAuency

@ccepte!.eZecte!

0 0., 0.2 0. 0./ 0.5 0.% 0.- 0. 0.


0ig$re /./. Distri&$tion of 5ffers in Ro$nd ' of a C%assroom >%timat$m Game

@1 Pairs, >niversit! of irginia, Spring ++A

0 0., 0.2 0. 0./ 0.5 0.% 0.- 0. 0.




0ig$re /.3. Distri&$tion of 5ffers in Ro$nd * of a C%assroom >%timat$m Game

@1 Pairs, >niversit! of irginia, Spring ++A

I. M$%ti?Stage Bargaining

?ace+to+face negotiations are ty#ically characteri<ed !y a series of offers andcounter+offers. he ultimatum game can !e transformed into a game with many

stages !y letting the #layers ta"e turns ma"ing #ro#osals a!out how to s#lit an

amount of money. f there is an agreement at any stage& then the agreed s#lit is

im#lemented. A #enalty for delay can !e inserted !y letting the si<e of the sta"e

shrin" from one stage to the net. ?or eam#le& the amount of money that could

!e divided in the first stage may !e 5& !ut a failure to reach an agreement may

reduce this sta"e to 2 in the second stage. f there is a #re+announced final stage

and the res#onder in that stage does not agree& then !oth earn <ero. hus the final

stage is li"e an ultimatum game& and can !e analy<ed as such.

;ow consider a game with only two stages& with a money sta"e& M& in the first

stage and a lower amount& X& in the final stage. ?or sim#licity& su##ose that the

initial sta"e is ,5& which is reduced to ,0 if no agreement is reached in the first

round. =arning$ the ana#sis in this paragraph is +ased on the 0ver

unreasona+#e2 assumption that peop#e are perfect# rationa# and care on# a+out

their o;n paoffs. 8ut yourself in the #osition of !eing the #ro#oser in the first

stage of this game& with the "nowledge that !oth of you would always #refer the

action with the highest #ayoff& even if that action only increases one s #ayoff !y a

#enny. f you offer the other #layer too little in the first stage& then this offer will

!e re9ected& since the other #layer !ecomes the #ro#oser in the final =ultimatum

game> stage. 6o you must figure out how much the other #erson would e#ect to

earn in the final stage when they ma"e a #ro#osal to s#lit the ,0 that remains. ntheory& the other #erson could ma"e a second+stage offer of a #enny& which you

would acce#t under the assum#tion that you #refer more money =a #enny> to less

=<ero>. Hence the other #erson would e#ect to earn . in the final stage&

assuming #erfect rationality and no concerns for fairness. f you offer them less

than . in the first stage& they will re9ect& and if you offer them more they will

acce#t. he least you could get away with offering in the first stage is& therefore&

9ust a little more than .& so you offer ,0& which is acce#ted.

n the #revious eam#le& the theoretical #rediction is that the first+stage

offer will e'ual the amount of the sta"e that would remain if !argaining were to

#roceed to the second stage. his result can !e generali<ed. f the si<e of the

money sta"e in the final stage is X& then the #erson ma"ing the offer in that stageshould offer the other a #enny& which will !e acce#ted. he #erson ma"ing the

#ro#osal in the final stage can o!tain essentially the whole sta"e& i.e. X 0.0,.

hus the #erson ma"ing an offer in the first stage can get away with offering a



slightly higher amount& i.e. X& which is acce#ted. n this two+stage game& the

initial #ro#oser earns the amount !y which the #ie shrin"s& M X& and the

other #erson earns the amount remaining in the second stage& X.

An e#eriment with this two+stage structure is re#orted in oeree and Holt

=200,>. he si<e of the sta"e in the initial stage was 5 in !oth cases. n one

treatment& the #ie was reduced to 2.00& so the first+stage offer should !e 2.00& as

shown in the middle column of a!le 2,.,. he average first+stage offer was

2.,-& 'uite close to this #rediction. n a second treatment& shown on the right

side of the ta!le& the #ie shrun" to 0.50& so the #rediction is for the first+stage

offer to !e very ine'uita!le =0.50>. he average offer was somewhat higher& at

,.%2& as shown in the right column of the ta!le. he reduction in the average

offer was much less than #redicted !y the theory& and re9ections were 'uite

common in this second treatment. ;otice that the cost of re9ecting a low offer is

low& and "nowing this& the initial #ro#osers were reluctant to e#loit their

advantage fully in this second treatment.

a!le 2,.,. A wo+6tage 7argaining ame 8layed Ince =6ource:oeree and Holt& 200,>

reatment , reatment 2

6i<e of 8ie in ?irst 6tage 5.00 5.00

6i<e of 8ie in 6econd 6tage 2.00 0.50

6elfish ;ash ?irst+6tage Iffer 2.00 0.50

Average ?irst+6tage Iffer .') '.1

he effects of #ayoff ine'uities are even more dramatic in a second twostage

e#eriment re#orted !y oeree and Holt =2000>. he initial #ro#osers in this

design made seven choices corres#onding to seven different !argaining situations&

with the understanding that only one of the situations would !e selected

afterwards& at random& !efore the #ro#osal for that situation was communicated to

the other #layer. he initial #ie si<e was 2./0 in all seven cases& !ut the second+

stage #ie si<e varied from 0.00 to 2./0& with five intermediate cases.

he two etreme treatments for this e#eriment are shown in a!le 2,.2.

n the middle column& the #ie shrin"s from 2./0 to 0.00. his gives the initial

#ro#oser a large strategic advantage& so the initial offer would !e a #enny in a



game !etween two selfish& rational #layers. his outcome would #roduce a shar#

asymmetry in the #ayoffs in favor of the #ro#oser. Adding insult to in9ury& the

instructions for this case indicated that the initial #ro#oser would receive a fied

#ayment of 2.%5 in addition to the earnings from the !argaining& whereas the

initial res#onder would only receive 0.25. Inly if the initial #ro#oser were to

offer the whole #ie of 2./0 in the first stage would final earnings !e e'uali<ed.

his is listed as the egalitarian first+stage offer of 2./0 in the middle column. he

average of the actual offers =,.5> shown in the !ottom row& is well a!ove the

;ash #rediction& and is closer to the egalitarian offer of 2./0.

he other etreme treatment is shown in the right+hand column of a!le

2,.2. Here the #ie does not shrin" at all in the second stage& so the theoretical

#rediction is that the first+stage offer will !e 2./0. he initial res#onder now has

the strategic advantage& since the #ie remains high in the final =ultimatum> stage

when this #erson has the turn to ma"e the final offer. o ma"e matters even more

asymmetric& this strategic advantage is com#lemented with a high fied #ayment

to the initial res#onder. he only way for the disadvantaged initial #ro#oser to

o!tain e'ual earnings would !e to offer nothing in the first stage& des#ite the factthat the theoretical #rediction =assuming selfish !ehavior> is 2./0. he average

of the o!served offers& 0.%& is much closer to the egalitarian offer.

a!le 2,.2. A wo+6tage 7argaining ame with Asymmetric ?ied 8ayments

=6ource: oeree and Holt& 2000>

6i<e of 8ie in ?irst 6tage 2./0 2./0

6i<e of 8ie in 6econd 6tage 0.00 2./0

8ro#oser ?ied 8ayment 2.%5 0.25

3es#onder ?ied 8ayment 0.25 2.%5

*galitarian ?irst+6tage Iffer .3+ +.++

6elfish ;ash ?irst+6tage Iffer 0.0, 2./0

Average ?irst+6tage Iffer '.*4 +.1/

o summari<e& the first+stage offers in this e#eriment should !e e'ual to the

remaining #ie si<e& !ut the asymmetric fied #ayments were structured so that theegalitarian offers would !e inversely related to the remaining #ie si<e. his

inverse relationshi# was generally #resent in the data for the seven treatments.

he authors show that the data #atterns are roughly consistent with an enriched

model in which #eo#le care a!out relative earnings as well as their own earnings.



?or eam#le& a #erson may !e willing to give u# some money to avoid having the

other #erson earn more& which is an aversion to disadvantageous ine'uity.

3oughly s#ea"ing& thin" of this as an envy effect. t is also #ossi!le that #eo#le

might wish to avoid ma"ing significantly more than the other #erson& which

would !e an aversion to advantageous ine'uity. hin" of this as a "ind of guilt

effect& which is li"ely to !e wea"er than the envy effect. hese two effects are

ca#tured !y a formal model of ine'uity aversion #ro#osed !y ?ehr and 6chmidt

=,>. oeree and Holt =2000> estimate guilt and envy #arameters for their data

and conclude that the envy effect is more #ronounced.


he first ultimatum e#eriment was re#orted !y uth et al. =,2>& and the

results were re#licated !y ?orsythe et al. =,>& who introduced the dictator

game. *conomists and others have !een fascinated !y !ehavior in these games

where there is a high tension !etween notions of fairness and strategic& narrowly

self+interested !ehavior. *nsminger =200,> re#orts that one of her Africansu!9ects 9ovially remar"ed: will !e s#ending years trying to figure out what this

all meant.

4any economists were initially s"e#tical of the high degree of seemly

non+strategic #lay and costly re9ections. Ine reaction was that irrational

re9ections would diminish when the sta"es of the game are increased. Hoffman et

al. =,%> increased the sta"es from ,0 to ,00& which did not have much effect

on initial #ro#osals. he effects of high sta"es have also !een studied !y 6lonim

and 3oth =,> and 1ist and Cherry =2000>.

3e9ections are not& of course& irrational if individuals have #references that

de#end on relative earnings. ?or eam#le& a res#onder may #refer that !oth earn

e'ual <ero amounts to a situation with ine'uita!le #ositive earnings. 7olton andIc"enfels =,> #ro#osed a model with #references !ased on relative earnings&

and ?ehr and 6chmidt =,> develo#ed a closely related model of ine'uity

aversion that was mentioned a!ove.

?inally& there have !een many e#eriments in other& less+structured

negotiations& where the order of #ro#osal and res#onse is not im#osed !y the

e#erimenter. ?or eam#le& Hoffman and 6#it<er =,2& ,5> used an o#en&

unstructured setting to evaluate the a!ility of !argainers to agree on efficient

outcomes& irres#ective of #ro#erty rights =the Coase theorem>. 6ee 3oth =,5>

for a survey of the literature on !argaining e#eriments.



Cha#ter 2/. rust& 3eci#rocity& and

8rinci#alAgent ames

Although increases in the si<e of the mar"et may #romote #roductives#eciali<ation and trade& the accom#anying increase in anonymity raises the need

to trust in trading relationshi#s. he trust game sets u# a styli<ed situation where

one #erson can decide how much of an initial sta"e to "ee# and how much to #ass

to the other #erson. All money #assed is augmented& and the res#onder then

decides how much of this augmented amount to "ee# and how much to #ass !ac"

to the initial decision ma"er. he trust game e#eriment #uts #artici#ants into this

situation so that they can e#erience the tension !etween #rivate motives and the

#otential gains from trust and coo#eration.

he gift+echange view of wage setting is that em#loyers set wages a!ove

mar"et+clearing levels in an effort to elicit high effort res#onses& even though

those res#onses are not e#licitly rewarded e5 post after wages have !een set.he reci#rocity game e#eriment is one where each em#loyer is matched with a

wor"er who first sees the wage that is offered and then decides on the level of

costly effort to su##ly. he issue is whether notions of trust and reci#rocity may

have noticea!le effects in a mar"et contet.

he final #art of the cha#ter #ertains to another em#loyerem#loyee relationshi#&

where the em#loyer !egins !y selecting the ty#e of contract& from a menu that

includes either e5 ante incentives or less+structured e5 post !onuses. All three of

the games considered in this cha#ter can !e run with the Veconla! software& !y

selecting the desired e#eriment under the 7argaining?airness menu. he trust

and reci#rocity games& which are often run as one+shot interactions& can also !e

easily done with #encil and #a#er.

I. he r$st Game

he trust game was eamined in an e#eriment !y 7erg& (ic"haut& and 4cCa!e

=,5>& with the o!9ective of studying trust and reci#rocity in a controlled setting.

A standard version !egins when one #erson in each #air is given ,0. he first

mover must decide how much =if any> of this money to #ass to the other #erson&

and how much to "ee#. 4oney that is #assed is tri#led !efore it is given to the

second mover& who decides how much =if any> to return to the first #erson. he

first mover earns the amount "e#t initially #lus any money that is returned. hesecond mover earns the amount that is "e#t in the second stage. he game would

ty#ically !e e#lained as an investment game& in order to avoid the suggestive

trust terminology. f this game is only #layed once& as is often the case in

e#eriments& then the su!game #erfect ;ash e'uili!rium for selfish #layers is for



the second #erson to "ee# all that is #assed& and hence for the first mover to #ass

nothing. he action of #assing money initially would signal that the first mover

trusts the second mover to return a reasona!le amount of money& #erha#s due to a

feeling of reci#rocity generated !y the initial action.

7erg& (ic"haut and 4cCa!e ran this e#eriment with 2 #airs of #artici#ants in a

single round interaction. If these& almost all of the senders =0 of 2> #assed a

#ositive amount of money& and a!out a third of the res#onders returned more than

was sent. he average amount #assed was 5.,% =out of ,0>& and after !eing

tri#led& the average amount returned was , #ercent. hese !ehavior #atterns are

inconsistent with the #rediction of a su!game+#erfect ;ash e'uili!rium for #urely

selfish #layers.

a!le 2/., shows the amounts #assed and returned for % #airs of individuals in a

single+round demonstration e#eriment& which was run at the University of

Virginia with full #ayment and with the standard #arameters =,0 given to

#ro#osers& with the amount #assed !eing tri#led>. hus& in the aggregate&

#ro#osers are given %0. 7ased on the 7erg& (ic"haut& and 4cCa!e results& one

would e#ect a!out half =0> to !e #assed& which when tri#led& would !ecome0& and a!out ,K of that =,%> to !e returned. ?or the si #airs shown in the

ta!le& the aggregate amount #assed was .00& which was tri#led to .00& !ut

only ,0.00 was returned. he res#onders in this e#eriment were clearly not

living u# to the e#ectations that #ro#osers had& unless #ro#osers were mainly

trying to increase res#onder earnings.

a!le 2/., A (emonstration rust ame *#eriment

I(1 I( I(3 I(& I(" I(*

Amount 8assed

0.00 ,0.00 2.00 ,0.00 ,.00 ,0.00Amount 3eturned

0.00 0.00 0.00 0.00 0.00 ,0.00

8ro#oser s *arnings

,0.00 0.00 .00 0.00 .00 ,0.00

3es#onder s *arnings

0.00 0.00 %.00 0.00 .00 20.00

he results of a classroom trust game are shown in ?igure 2/.,B the matchings

were random in the first treatment and fied in the second. his setting clearlyinduced a significant amount of money !eing #assed& on average& and return rate

that was 9ust enough& on average& to reim!urse the first movers.



0ig$re 3.' - C%assroom r$st Game

here has !een considera!le discussion of #ossi!le reasons for deviations of

!ehavior from the ;ash #redictions for selfish #layers. Co =,> re#licated the

7erg& (ic"haut& and 4cCa!e results& and com#ared the amounts #assed with

!ehavior in a second treatment where the res#onder did not have the o##ortunity

to reci#rocate. he amounts #assed were not significantly different for the two

treatments& which is evidence that the #ass !ehavior may not !e due to antici#ated

reci#rocity. Co suggested that the #ositive amounts ty#ically #assed and tri#led

were due to altruism& i.e. a "indness or other+regarding #reference for increasing

another #erson s earnings& even at some cost to oneself. Altruism is often

mentioned as an e#lanation for voluntary contri!utions in #u!lic goods games

discussed in a Cha#ter 2%. f altruism is driving these contri!utions& then it

cannot !e the whole story& since the res#onder !ehavior is not very coo#erative.

Ine difference is that what is #assed is tri#led and what is returned is not altered&

so the cost of altruism is lower for #ro#osers. Co argues that altruism is a factor

in res#onder !ehavior& !ut he notes that reci#rocity may also !e involved. 6uch

reci#rocity could arise !ecause #artici#ants may feel some social #ressure to

return a #art of the gains from money #assed& des#ite the anonymity of the

e#erimental #rocedures. A similar e#lanation would !e that concern for the

other s earnings goes u# if the other #erson has shown some initial "indness.

4any economic interactions in mar"et economies are re#eated& and this re#etition

may ena!le trading #artners to develo# trust and reci#rocal arrangements. ?or one thing& either #erson has the freedom to !rea" out of a !inary trading

relationshi# in the event that the other s #erformance is not satisfactory. *ven

when a !rea"u# is not #ossi!le& the re#etition may increase levels of coo#eration.

?or eam#le& Cochard& Van 8hu& and $illinger =2000> re#ort results of a trust



game that was run !oth as a single+shot interaction and as a re#eated interaction

for - #eriods& !ut with #ayoff #arameters scaled down to account for the higher

num!er of #eriods. here were some other minor #rocedural differences that will

not !e discussed here. he main result is that the amounts #assed and returned

were higher in the re#eated+interaction treatment. n the final #eriod& which was

"nown in advance& the amount #assed stayed high& !ut the amount returned was

very low.

II. - La&or Market Reciprocit! Game

his game im#lements a setu# where #artici#ants are matched in #airs& with one

setting a wage and the other choosing an effort level. he #erson in the em#loyer

role is free to choose any wage !etween s#ecified limits. his wage is #aid

irres#ective of the wor"er s su!se'uent effort decision& which must !e !etween 0

and some u##er limit. A higher level of wor"er effort is costly to the wor"er and

!eneficial to the em#loyer. his is sometimes called a reci#rocity game& since the

em#loyer may offer a high wage in the ho#e that the wor"er will reci#rocate withhigh effort. n the e#eriment to !e discussed& the wage was !etween 0 and ,0&

and the wor"er effort was re'uired to !e !etween 0 and ,0. *ach additional unit

of effort reduced the wor"er s earnings !y 0.25& and added .00 to the em#loyer

s earnings. hus economic efficiency would re'uire an effort of ,0 in this

contet. *m#loyers did not "now wor"ers costs& and wor"ers did not "now the

value of effort to the em#loyer.

he results of a classroom reci#rocity game with these #arameters are shown in

?igure 2/.2. he first ,0 #eriods were done with random matchings& which gives

wor"ers little incentive to #rovide acce#ta!le efforts. he average efforts& shown

!y the small dots& leveled off at a!out 2 in rounds +& followed !y a shar# fall at

the end. he final #eriod efforts of 0 indicate that wor"ers were aware that theyhad no incentive to !e coo#erative if the em#loyers had no o##ortunity to

reci#rocate. he second treatment was done with fied matchings& and the

increased incentive to coo#erate resulted in much higher effort levels =until the

final #eriod> and higher earnings. t is curious that the re#eated nature of the

matching #rocess increased trust and reci#rocity more in this game than in the

trust game in ?igure 2/.,.



0ig$re 3.. - C%assroom Reciprocit! Game

III. - Principa%?-gent Game

he #rinci#al+agent model is one in which one #erson& the #rinci#al& hires the

other& the agent& to #erform some tas". he game !egins when the #rinci#al

selects the nature of a contract #ayment& which can !e contingent on o!served and

verifia!le outcome measures. hen the agent may either re9ect the contract& and

!oth earn default #ayments& or acce#t and #erform the re'uired tas"& with

#ayment from the #rinci#al de#ending on outcomes as s#ecified in the contract.

As !efore& a high effort !y the agent is costly to the agent and !eneficial to the

#rinci#al. his model has !een widely a##lied in la!or economics and contract

law& and generali<ations are used in many other two+#erson& se'uential+decision

settings where #ayments are determined !y e#licit or im#licit contracts.

?igure 2/. shows data from a classroom #rinci#al+agent e#eriment run in a

Contract 1aw class at the University of Virginia 1aw 6chool. *ach of the ,2

em#loyers !egan !y choosing a contract that s#ecified a suggested effort and a

fied wage to !e #aid regardless of the wor"er)s effort. *m#loyers also chose

!etween an Zincentive contractZ with a #otential #unishment and a Z!onus

contractZ with a non+!inding #romise of an e5 post !onus if a suggested level of

effort was met. he wage had to cover the wor"er)s effort cost at the suggested

effort level& and u##er limits on #ossi!le #enalty and !onus #ayments were

enforced. A #enalty could only !e im#osed if low effort could !e verified !y athird #arty& which occurred with a #re+s#ecified #ro!a!ility of ,. $or"ers could

either re9ect the contract or acce#t and select an effort from , to ,0. A!out

threefourths of the contracts were !onus contracts. *ffort levels declined& and



a!out half of the efforts ended u# at the minimum level of ,. *fforts were lower

for #enalty contracts.

0ig$re 3./. - C%assroom Principa%?-gent Game: -verage ages and "fforts &! Ro$nd

he #articular setu# used in this classroom eercise is !ased on the e#eriments

re#orted in ?ehr& Dlein& and 6chmidt =200,>& although their ZhandrunZ #rocedures

are not eactly re#licated !y the Veconla! interactive #rogram. he default

#ayoffs are o!tained !y dividing the ?ehr& et al. #arameters !y ,0 =division !y

would reflect the to"en to (4 to U.6. dollar conversion rate more accurately>.

he results of their research e#eriments are roughly consistent with the

classroom data #atterns shown in ?igure 2/.. n #articular& !onus contracts were

more widely used than incentive contracts.

I. "(tensions

rust games are !eing widely used to o!tain social #reference measures that are

com#ara!le across cultures. ?or e#erimental evidence of reci#rocity in a related

contet& see ?ehr& Dirchsteiger& and 3iedl =,>.

8art V. 8u!lic Choice



8u!lic e#enditures are ty#ically influenced !y voting and other #olitical

#rocesses& and the outcomes of such #rocesses can !e very sensitive to the rules

that govern voting. Cha#ter 25 is an introduction to the results of voting

e#eriments& and some related field e#eriments from the #olitical science

literature.

4any #u!lic #rograms and #olicies are designed to remedy situations where the

actions ta"en !y some #eo#le affect the well !eing of others. he classic

eam#le is that of the #rovision of a #u!lic good li"e national defense& which can

!e consumed freely !y all without crowding and the #ossi!ility of eclusion.

Cha#ter 2% introduces a model of voluntary contri!utions to a #u!lic good& where

the #rivate cost to each #erson is less than the social !enefit. he setu# of the

game allows the inde#endent alteration of =internal> #rivate cost and =eternal>

#u!lic !enefit. ;on+selfish motives for giving are discussed in the contet of

la!oratory e#eriments.

A similar situation with eternal !enefits arises when it only ta"es one volunteer

to #rovide the good outcome #referred !y all. he volunteer s dilemma discussed

in Cha#ter 2- is whether to incur the #rivate cost to achieve this good outcome.his game is different from a standard #u!lic goods setu# in that the #rivate

!enefit to the volunteer eceeds the cost. ?or eam#le& su##ose that you would

rather 9um# in the icy river and rescue someone& !ut you would #refer not to 9um#

if one or more other !ystanders decide to volunteer. (ata from volunteer s

dilemma games are often used to evaluate ;ash #redictions of the effects of

changing the num!er of #otential volunteers.

*ternal effects on others well !eing may arise in a negative sense as well& as is

the case with #ollution or overuse of a common resource. he common #ool

resource game in Cha#ter 2 is a setting where each #erson s efforts to secure

!enefits from a shared resource tend to diminish the !enefits that others derive

from their efforts& as might ha##en with ecessive harvests from a fishery. heresource is li"e a #u!lic good in the sense that the #ro!lem arises from non+

eclusion& !ut the difference is the #resence of congestion effects& i.e. it is not

non+rivaled. he sim#lest common #ool resource game is one in which #eo#le

select efforts inde#endently& and the !enefits achieved are diminished as total

effort increases. he focus of the discussion is on the etent of overuse.

he final cha#ter introduces a #ro!lem of wasteful com#etition that may arise in

non+mar"et allocation #rocedures. n !eauty contest com#etitions for a !roadcast

license& for eam#le& the contenders may s#end considera!le amounts of real

resources in the #rocess of lo!!ying. hese e#enditures are not recovered !y

unsuccessful contenders. 6uch lo!!ying e#enses are eam#les of rent see"ing&

and the total cost of such activities may even eceed the value of the #ri<e =full

dissi#ation of rents>. Cha#ter 2 is !ased on ordon ulloc" s model of rent

see"ing& where the #ro!a!ility of o!taining the #ri<e is the #ro#ortion of total



effort that one eerts. ?actors that affect rent dissi#ation in theory are evaluated

in the contet of la!oratory e#eriments.



Cha#ter 25. Voting =in #rogress>

(ecentrali<ed e'uili!rium outcomes are highly efficient in many economic

mar"ets& and as a conse'uence& economists sometimes fall into the tra# of thin"ing that the outcomes of a #olitical #rocess necessarily have some "ind of

intrinsic value. n contrast& #olitical scientists are well aware that the outcome of

a vote may de#end critically on factors li"e the structure of the agenda& whether

straw votes are allowed& and the etent to which #eo#le vote strategically or

naively =#olitical scientists use a less #e9orative term& sincerely >. Voting

e#eriments allow one to evaluate alternative #olitical institutions& and to study

how #eo#le actually !ehave in controlled la!oratory conditions. hese studies are

com#lemented !y field e#eriments that see" to create contets with more

eternal validity& e.g. having voters view media ads with differing formats. hese

la!oratory and field e#eriments are selectively surveyed in this cha#ter. Voting

e#eriments can !e run with the Veconla! Voting #rogram& or with the

instructions #rovided in the A##endi.

"(tensions

8arts of this cha#ter are !ased on Anderson and Holt =,-> and #arts are !ased

on a survey !y 4oore and Holt =2005>.



Cha#ter 2%. Voluntary Contri!utions

his cha#ter is !ased on the standard voluntary contri!utions game& in which the

#rivate net !enefit from ma"ing a contri!ution is negative unless others

reci#rocate later or unless the #erson receives satisfaction from the !enefit #rovided to others. he setu# ma"es it #ossi!le to evaluate inde#endent

variations of the #rivate internal !enefit and the #u!lic eternal !enefit to others.

hese and other treatment mani#ulations in e#eriments are used to evaluate

alternative e#lanations for o!served #atterns of contri!utions to a #u!lic good.

he #u!lic goods game& 8& should !e conducted #rior to class discussion.

Alternatively& a hand+run version using #laying cards is easy to im#lement using

the instructions #rovided in the a##endi. n some contets& a target level of

contri!utions is re'uired for a #u!lic good to !e #rovided& and this setu# can !e

im#lemented with the #rovision+#oint #u!lic goods #rogram =88>.

I. "conomists 0ree Ride, Does -n!one "%seE

he selfish caricature of homo economicus im#lies that individuals will

free ride off of the #u!lic !enefits #rovided !y others activities. 6uch free riding

may result in the under+#rovision of #u!lic goods. A #ure #u!lic good& li"e

national defense& has several "ey characteristics:

,> t is oint# provided and none5c#uda+#e in the sense that the #roduction

re'uired to ma"e the good availa!le to one #erson will ensure that it is

availa!le to all others in the grou#. n #articular& access cannot !econtrolled.

2> t is nonriva#ed in the sense that one #erson s consum#tion of the good is

not affected !y another s& i.e. there is no congestion.

$hen a single individual #rovides a #u!lic good& li"e shoveling a

sidewal"& the #rivate #rovision cost may eceed the #rivate !enefit& even though

the social !enefit for all others com!ined may eceed the #rovision cost incurred

!y that #erson. he resulting misallocations have !een recogni<ed since Adam

6mith s =,--%> discussion of the role of government in the #rovision of street

lam#s.

n many cases& there is not a !right+line distinction !etween #rivate and #u!lic goods. ?or eam#le& #ar"s are generally considered to !e #u!lic goods&

although they may !ecome crowded and re'uire some method of eclusion. A

good that is rivaled !ut non+ecluda!le is sometimes called a common+#ool



resource& which is the to#ic of a su!se'uent cha#ter. $ith common+#ool

resources li"e ground water resources& fisheries& or #u!lic gra<ing grounds& the

#ro!lem is ty#ically one of how to manage the resource to #revent overuse& since

individuals may not ta"e into account the negative effects that their own usage has

on what is availa!le for others. n contrast& most #u!lic goods are not #rovided !y

nature& and the ma9or #ro!lems often #ertain to #rovision of the a##ro#riate

amounts of the good. his cha#ter is mainly focused on the voluntary #rovision

of such goods& although there is some discussion of voting and mechanisms that

may !e used to im#rove the situation.

oods are often #roduced !y those who receive the greatest !enefit& !ut

under+#rovision can remain a #ro!lem as long as there are some #u!lic !enefits to

others that are not !e fully valued !y the #rovider. *ducation& for eam#le& offers

clear economic advantages to the student& !ut the #u!lic at large also !enefits

from having a well+educated citi<enry. A #u!lic goods #ro!lem remains when not

all of the !enefits are en9oyed !y the #rovider& and this is one of the rationales for

the heavy #u!lic involvement in school systems. he mere #resence of #u!lic

!enefits does not 9ustify #u!lic #rovision of such goods& since #u!lic #rovisionmay involve inefficiencies and distortions due to the need to collect taes. he

#olitical #ro!lems associated with #u!lic goods are com#licated when the !enefits

are une'ually distri!uted& e.g.& #u!lic !roadcasting of cultural materials.

n close+"nit societies& #u!lic goods and common+#ool resource #ro!lems

may !e mitigated !y the #resence of social norms that dictate or reward

otherregarding !ehavior. ?or eam#le& Haw"es =,> re#orts that large game

hunters in #rimitive societies are e#ected to share a "ill with all households in a

village& and sometimes even with those of neigh!oring villages. 6uch wides#read

sharing seems desira!le given the difficulty of meat storage and the diminishing

marginal value of ecess consum#tion in a short #eriod of time. hese social

norms transform a good that would normally !e thought of as #rivate into a goodthat is 9ointly #rovided and non+ecluda!le. his transformation is desira!le since

the #rivate return from large+game hunting is lower than the #rivate return from

gathering and scavenging. ?or eam#le& the GDung of 7otswana and ;ami!ia are

a hunter+gatherer society where large #rey =e.g. warthogs> are widely shared&

whereas small animals and #lant food are ty#ically "e#t within the household. In

the !asis of #u!lished accounts& Haw"es =,> estimates that& at one #oint& male

large+game hunters ac'uired an average of 2&000 calories #er day& with only

a!out a tenth of that =2&500> going to the #erson s own household. n contrast& the

collection of #lant foods yielded an estimated range around 5&000 calories #er day&

even after accounting for the etra #rocessing re'uired for the #re#aration of such

food. ;otice that large+game hunting !y males was more #roductive for the

village as a whole& !ut had a return for one s household of a!out half the level that

could !e o!tained from gathering activities. ;evertheless& many of the men

continued to engage in hunting activities& #erha#s due to the tendency for all to



share their "ills in a ty#e of reci#rocal arrangement. Haw"es& however&

'uestioned the reci#rocity hy#othesis since some men were consistently much

!etter hunters than others& and yet all were involved in the sharing arrangements.

6he stresseed the im#ortance of more #rivate& fitness+related incentives& and she

con9ectured that successful large+game hunters have more allies and !etter

o##ortunities for mating.

*conomists are sometimes ridiculed for ignoring social norms and sim#ly

assuming that individuals will ty#ically free+ride on others generosity. An early

#u!lic goods e#eriment is re#orted !y two sociologists& 4arwell and Ames

=,,>. heir e#eriment involved grou#s of high school students who could

either invest an initial endowment in a #rivate echange or a #u!lic echange.

nvestment in the #u!lic echange #roduced a net loss to the individual& even

though the !enefits to others were su!stantially a!ove the #rivate cost to the

individual su!9ect. ;evertheless& the authors o!served significant amounts of

investment in the #u!lic echange& with the ma9or ece#tion !eing when the

e#eriment was done with a grou# of economics doctoral students. he resulting

#a#er title !egan: *conomists ?ree 3ide& (oes Anyone *lseNhe 4arwell and Ames #a#er initiated a large literature on the etent to

which su!9ects in e#eriments incur #rivate costs in activities that !enefit others.

A ty#ical e#eriment involves dividing su!9ects into grou#s& and giving each one

an endowment of to"ens that can !e invested in a #rivate echange or account&

with earnings #er to"en that eceed the earnings #er to"en o!tained from

investment in the #u!lic account. ?or eam#le& each to"en might #roduce ,0

cents for the investor when invested in the #rivate account and only 5 cents for the

investor and for each of the others when invested in the #u!lic account. n this

eam#le& the social o#timum would !e to invest all to"ens in the #u!lic account as

long as the num!er of individuals in the grou#& F & is greater than 2& since the

social !enefit 5 F would !e greater than the #rivate !enefit of ,0 for each to"enwhen F P 2. he ,0+cent #rivate return can !e thought of as the o##ortunity cost

of investment in the #u!lic account. he ratio of the #er ca#ita !enefit to the

o##ortunity cost is sometimes called the marginal #er ca#ita return or 48C3&

which would !e 5,0 J 0.5 in this eam#le. A higher 48C3 reduces the net cost

of ma"ing a contri!ution to the #u!lic account. ?or eam#le& if the #rivate

account returns ,0 cents and the #er+ca#ita return on the #u!lic account is raised

to cents& there is only a , cent #rivate loss associated with investment. 4any of

the e#eriments involved changes in the 48C3.

A second treatment varia!le of interest is the num!er of #eo#le involved&

since a higher grou# si<e increases the social !enefit of an investment in the

#u!lic echange when the 48C3 is held constant. ?or eam#le& the social

!enefit in the eam#le from the #revious #aragra#h is 5 F & which is increasing in

F . Alternatively& thin" a!out what you would do if you could give u# ten dollars

in order to return a dollar to every mem!er of the student !ody at your university&



including yourself. Here the 48C3 is only 0.,& !ut the #u!lic !enefit is

etremely large. he motives for contri!ution to a #u!lic good are am#lified if

others are e#ected to reci#rocate in some future #eriod. hese considerations

suggest that contri!utions might !e sensitive to factors such as grou# si<e& the

48C3& and whether or not the #u!lic goods e#eriment involves re#etition with

the same grou#. n multi+round e#eriments& individuals are given new

endowments of to"ens at the start of each round& and grou#ings can either !e

fied or randomly reconfigured in su!se'uent rounds.

he u#shot is that the etent of voluntary contri!utions to #u!lic goods

de#ends on a wide variety of #rocedural factors& although there is considera!le

de!ate a!out whether contri!utions are #rimarily due to "indness& to reci#rocal

reactions to others "indness& or to confusion. ?or eam#le& there is li"ely to !e

more confusion in a single+shot investment decision& and many #eo#le may

initially divide their endowment of to"ens e'ually !etween the two ty#es of

investment& #u!lic and #rivate. his is analogous to the ty#ical choice of dividing

one s retirement fund contri!utions e'ually !etween stoc"s and !onds at the start

of one s career. 4any of the 4arwell and Ames e#eriments involved a singledecision& e.g. administered !y a 'uestionnaire that was mailed to high school

students. 6u!se'uent e#eriments !y economists =that did not involve economics

doctoral students> showed that re#etition ty#ically #roduced a declining #attern of

contri!utions. 6ome #eo#le may sto# contri!uting if others are o!served to !e

free riding. Ine motive for ma"ing contri!utions may !e to #revent others from

!ehaving in this manner& in the ho#e that contri!utions will !e reci#rocated in

su!se'uent rounds. his reci#rocity motive is o!viously wea"er in the final

rounds. Although contri!utions tend to decline in the final #eriods& some #eo#le

do contri!ute even in the final #eriod. $e !egin with a summary of an

e#eriment with a one+shot setu#. 4ulti+round e#eriments will !e considered in

later sections.

II. Sing%e?Ro$nd "(periments

*nsminger =200,> re#orts the results of a #u!lic goods e#eriment involving

young men of the Irma& a society from Denya that is descri!ed in Cha#ter 2.

8artici#ants were divided into grou#s of /& and each #erson was given 50 shillings

that could !e "e#t or invested in a grou# #ro9ect. nvestments were made !y

#lacing to"ens in envelo#es& which were shuffled to #reserve anonymity. hen

the contents were em#tied and #u!licly counted !y a mem!er of the grou# !efore

!eing dou!led !y the e#erimenter and divided e'ually among the four

#artici#ants. ?or eam#le& if one #erson contri!uted two shillings& these would !e

dou!led and the resulting four shillings would !e distri!uted& one #er #erson.

hus a contri!ution of two shillings yields a #rivate return of only ,& so the

48C3 is only 0.5.



0

01

02

03

04

.elati;e Yr eAuency

he data for this single+round game are shown in ?igure 2%.,. he modal

contri!ution is four+tenths of the endowment& with a fair amount of variation. n

fact& a 'uarter of the 2/ #artici#ants contri!uted the entire 50 shillings. he

average contri!ution is a!out %0K& which is at the high end of the /0+%0K range

o!served in the first round of most #u!lic goods e#eriments done in the U.6.

=1edyard& ,5>. *nsminger con9ectures that contri!utions were enhanced !y

familiarity with the Haram!ee institution used to arrange for funding of #u!lic

#ro9ects li"e schools. his #ractice involves the s#ecification of income+!ased

suggested voluntary contri!utions for each household& with social #ressures for

com#liance. n fact& some of the #artici#ants commented on the similarity of the

Haram!ee and the e#erimental setu#& and a ma9or Haram!ee solicitation was in

#rogress at the time of the e#eriment. he a!sence of com#lete free riding !y

anyone is nota!le in ?igure 2%.,.

0 0., 0.2 0. 0./ 0.5 0.% 0.- 0. 0. ,

Contri!ution as a ?raction of the 50+6hilling *ndowment

0ig$re 1.'. Contri&$tions to a Gro$p ProectE for Qo$ng 5rma Ma%es, 6ith

Gro$p SiOe 3 and an MPCR +.*. So$rce: "nsminger @++'A

MPC' 4ffects

he net to#ic is the etent to which contri!utions are affected !y treatment

varia!les li"e 48C3 and grou# si<e. oeree& Holt& and 1aury =2002> re#ort an

e#eriment in which the #artici#ants had to ma"e decisions for ten differenttreatments in which an endowment of 25 to"ens was allocated to #u!lic and

#rivate uses. 6u!9ects were #aid % and were told that only one of these

treatments would !e selected e5 post to determine additional earnings. his is a



single+round e#eriment in the sense that there is no feed!ac" o!tained !etween

decisions& and only one will count. A to"en "e#t was worth 5 cents in all ten

cases. n one treatment& a to"en contri!uted would return 2 cents to each of the /

#artici#ants. his yields an 48C3 of 0./ since each nic"el foregone from the

#rivate return yields 2 cents from the #u!lic return& and 25 J 0./. n another

treatment& a to"en contri!uted would return / cents to each of the / #artici#ants&

so these treatments hold the grou# si<e constant and increase the 48C3 from 0./

to 0.. his dou!ling of the 48C3 essentially dou!led average contri!utions&

from /. to ,0.% to"ens. If the 2 #artici#ants& 25 increased their contri!utions&

decreased& and / showed no change. he null hy#othesis& that increases are

9ust as li"ely as decreases& can !e re9ected using any standard statistical test.

Although these are the only two treatments that allow a direct test of the 48C3

effect& this effect has !een o!served in many other e#eriments& most of which

have involved more than a single round =see the net section>.

Interna# and 45terna# 'eturn 4ffects

n each of the treatments 9ust discussed& the !enefit to oneself for contri!uting iseactly e'ual to the !enefit that every other #erson receives. n many #u!lic

goods settings& however& the #erson ma"ing a voluntary contri!ution may en9oy a

greater #ersonal !enefit than others. ?or eam#le& !enefactors ty#ically give

money for #ro9ects that they #articularly value for some reason. t is 'uite

common for gifts to medical research teams to !e related to illnesses that are

#resent in the donor s family. *ven in Adam 6mith s streetlight eam#le& a #erson

who erects a light over the street would #ass !y that s#ot more often& and hence

would receive a greater !enefit than any other randomly selected #erson in the

town.

his difference !etween donor !enefits and other #u!lic !enefits can !e eamined

!y introducing the distinction !etween the internal return to the #erson ma"ing thecontri!ution and the eternal return that is en9oyed !y each of the other #eo#le in

the grou#. 3ecall that one of the oeree& Holt& and 1aury =2002> treatments

involved a return of 2 cents for oneself and for each other #erson when a to"en

worth 5 cents was contri!uted. hus the internal return to oneself is 25 J 0./&

and the eternal return to others is also 25 J 0./. here was another treatment in

which the return to oneself was raised from 2 cents to / cents& whereas the return

to each of the three others was held constant at 2 cents. hus the internal return

was increased from 0./ to 0. =J/5>& with the eternal return constant at 0./.

his increase in the internal return essentially lowered the cost of contri!uting

regardless of the motive for contri!uting =generosity& confusion& etc.> his

dou!ling of the internal return essentially dou!led the average o!served

contri!ution from /. to"ens to ,0.- to"ens. Again& the effect was highly

significant& with 25 #eo#le increasing their contri!utions& decreasing& and /



showing no change as the internal return increased holding the eternal return

constant.

;et consider the effects of changing the eternal return& holding the internal

return constant at 2 cents #er to"en contri!uted =for an internal return of ' I J 0./>.

he return to each of the other #artici#ants was raised from 2 cents to % cents&

which raised the eternal return from ' 4 J 25 J 0./ to ' 4 J %5 J ,.2. his tri#ling

of the eternal return a##roimately dou!led the average o!served contri!ution&

from /. to"ens to ,0.5 to"ens. =Contri!utions increased for 2 su!9ects&

decreased for 2& and showed no change for the other ->. he ten treatments

#rovide a num!er of other o##ortunities to o!serve eternal+return effects. a!le

2%., shows the average num!er of to"ens contri!uted with grou# si<e fied at /&

where the eternal return varies from 0./ to 0. to ,.2. he to# row is for a low

internal return of 0./& and the !ottom row is for a high internal return of 0.. $ith

one ece#tion& increases in the eternal return =moving from left to right> result in

increases in the average contri!ution. A similar #attern =again with one

ece#tion> is o!served in a!le 2%.2& which shows com#ara!le averages for the

treatments in which the grou# si<e was 2 instead of /.

a!le 2%.,. Average ;um!er of o"ens Contri!uted with rou# 6i<e J /

=6ource: oeree& Holt& and 1aury& 2002>

*ternal 3eturn

1ow

*ternal

' 4 J 0./

4edium

*ternal

' 4 J 0.

High

*ternal

' 4 J ,.2

Very High

*ternal

' 4 J 2./

1ow nternal 3eturn: ' I J 0./ /. + ,0.5 +

High nternal 3eturn: ' I J 0. ,0.- ,0.% ,/. +

a!le 2%.2. Average ;um!er of o"ens Contri!uted with rou# 6i<e J 2

=6ource: oeree& Holt& and 1aury& 2002>

*ternal 3eturn

1ow

*ternal

' 4 J 0./

4edium

*ternal

' 4 J 0.

High

*ternal

' 4 J ,.2

Very High

*ternal

' 4 J 2./

1ow nternal 3eturn: ' I J 0./ + + -.- +

High nternal 3eturn: ' I J 0. %.- ,2./ ,,.- ,/.5

?inally& consider the vertical com#arisons !etween two num!ers in the same

column of the same ta!le& i.e. where the internal return is increased and the

eternal return is held constant. he average contri!ution increases in all three



cases. Iverall& the lowest contri!utions are for the case of low internal and

eternal returns =u##er+left corner of a!le 2%.,>& and the highest contri!utions are

for the case of a high internal return and a very high eternal return =!ottomright

corner of a!le 2%.2>.

GroupSi)e 4ffects

A com#arison of the two ta!les affords some #ers#ective on grou#+si<e effects.

here are four cases where grou# si<e is changed& holding the returns constant&

and the increase in grou# si<e =from a!le 2%.2 to a!le 2%.,> raises average

contri!utions in all cases !ut one.

4conomic !#truism

hese com#arisons =and some su##orting statistical analysis> lead the

authors to conclude that contri!utions res#ond #ositively to increases in internal

return& eternal return& and grou# si<e& with the internal+return effect !eing

strongest for the e#eriment !eing re#orted here.

(es#ite the individual differences and the #resence of some une#lainedvariations in individual decisions& the overall data #atterns in these single+round

e#eriments are roughly consistent with a model in which #eo#le tend to hel#

others !y contri!uting some of their to"ens. 6uch contri!utions are more

common when ,> the #rivate cost of contri!uting is reduced& 2> the !enefit to each

other #erson is increased& and > the num!er of others who !enefit is increased.

*ven a #erfectly selfish free+rider may decide to contri!ute if it is thought that this

might induce others to contri!ute in the future& !ut there is no o##ortunity for

reci#rocity in these one+round e#eriments. hese results suggest that many

individuals are not #erfectly selfish free+riders. 4oreover& any altruistic

tendencies are not eclusively a warm+glow feeling from the mere act of

contri!uting& !ut rather& altruism is in #art economic in the sense that is it de#endson the costs to oneself and the !enefits to others.

III. M$%ti?Ro$nd "(periments

*conomists !egan running multi+round #u!lic goods e#eriments to determine

whether free+riding !ehavior would emerge as su!9ects gained e#erience and

!egan to understand the incentives more clearly. he ty#ical setu# involves ,0

rounds with a fied grou# of individuals. ?igure 2%.2 shows the fractions of

endowment contri!uted for a classroom e#eriment conducted with the Veconla!

software. here is considera!le varia!ility across individuals. 8layer , =solid thin

line> has relatively low contri!utions& with a s#i"e u#ward in round 5. 8layer 2

contri!utes the full amount in all #eriods& and #layer contri!utes nothing in any

#eriod. Average contri!utions for all 5 individuals =the thic" solid line> start at

a!out 0.5 and decline slowly& with an u#ward surge in later #eriods caused largely



!y #layer 5 s change from no contri!utions to a full contri!ution of all 25 to"ens.

Average contri!utions fall in the final #eriods& !ut not to <ero. he overall #attern

shown here is somewhat ty#ical of other ,0+round #u!lic goods e#erimentsB

contri!utions tend to start in the 0./+0.% range and decline over time& with erratic

movements as individuals !ecome frustrated and try to signal =with high

contri!utions> or #unish =with low contri!utions>. Contri!utions generally fall in

the final #eriod& when an attem#t to signal others to contri!ute in the future would

!e useless& !ut some #eo#le continue to contri!ute even in the final #eriod.

3ound

0ig$re 1. Data from a 0ive?Person C%assroom P$&%ic Goods Game 6ith an Interna%Ret$rn of +.1 and an "(terna% Ret$rn of +.2 @>-, "con 32, 0a%% ++'A

As is the case with one+round #u!lic goods e#eriments& there is evidence that

contri!utions in multi+round e#eriments res#ond to treatment varia!les that alter

the !enefits and costs of contri!utions. n a classic study& saac and

$al"er=,!> used two different 48C3 levels& 0. and 0.-5& with the order

!eing alternated in every other session. 6i sessions were conducted with grou#s

of si<e /& and si sessions were conducted with grou#s of si<e ,0. rou#

com#osition remained fied for all ,0 rounds. he average fractions of the

endowment contri!uted are shown in !old+faced font in the lower+left #art of

a!le 2%.. he increase in 48C3 a##roimately dou!les contri!utions for !oth

grou# si<es. he increase in grou# si<e raises contri!utions for the low 48C3&

!ut not for the high 48C3.



hese #atterns are reinforced !y the saac& $al"er& and $illiams =,/>

results in the far+right column of a!le 2%.. his column shows average

contri!utions for three grou#s of si<e /0& with an 48C3 that goes from very low

=0.0> to low =0.> to high =0.-5>. ;otice that !oth treatment varia!les tend to

have no effect when contri!utions go a!ove a!out /0 #er cent of the endowment&

as can !e seen from the entries in the lower+right corner of the ta!le.

a!le 2%.. Average ?raction of *ndowment Contri!uted for en 3ounds

=6ource: Isaac and a%ker, '422&B saac& $al"er& and $illiams& ,/>

F J /

rou# 6i<e

F J ,0 F J /0

Very 1ow 48C3 = 0.0 > +

+

0.,

1ow 48C3 = 0. > +.'2 +.1 0.//

High 48C3 = 0.-5 >+.3/ +.33

0.

he effects of varying the internal and eternal returns in multi+round

#u!lic goods e#eriments are re#orted in oeree& Holt& and 1aury =2002>.

6u!9ects were #aired in grou#s of si<e two& with new matchings in each round.

;o!ody was #aired with the same #erson twice. $ith the internal return fied at

' J 0.& the average num!er of to"ens contri!uted =out of an endowment of 25>

increased from 5 to -. to ,0 to ,,.2 as the eternal return was increased from 0./

to 0. to ,.2 to 2./. Holding the eternal return fied at '* J ,.2& a reduction inthe internal return from 0. to 0./ caused average contri!utions to fall from ,0 to

/./. A =random+effects> regression !ased on individual data was used to evaluate

the significance of these and other effects. he estimated regression =with

standard errors shown in #arentheses> is:

Contri+ution J ,.- 0.-R 'ound O .-R ' O .2R '* O 0.,%R/ther t,

=.,> =0.,> =.%> =,.2> =0.02>

he final /ther t1 varia!le is the contri!ution made !y the #artner in the #revious

#eriod. ;otice that contri!utions tend to decline over time& and that the internal+

return effect is stronger than the eternal return effect. he #ositive effect of the #revious #artner s contri!ution suggests that attitudes toward others may !e

influenced !y their !ehavior. his is consistent with the results re#orted !y van

(i9"& 6onnemans& and van $inden =,->& who measured attitudes toward others



!efore and after a multi+round #u!lic goods e#eriment. Changes in attitudes

were a##arent and de#ended on the results of the #u!lic goods game in an

intuitive manner.

I. 0$rther Reading

here is considera!le interest in the study of factors that influence voluntarycontri!utions. An analysis of these factors !ased only on economic theory may !e

insufficient& since many of the relevant issues are !ehavioral and must !e

addressed with e#eriments. ?or eam#le& 1ist and 1uc"ing+3eiley =200,> re#ort

a controlled field e#eriment in which different mail solicitations for donations

mentioned different amounts of seed money gifts& i.e. #re+eisting gifts that were

announced in the solicitation mailing. he #resence of significant seed money

had a very large effect on contri!ution levels. A related issue is the etent to

which the #u!lic #rovision of a #u!lic good will crowd out #rivate voluntary

donations =Andreoni& ,>.

here have !een many #u!lic goods e#eriments that focus on the effectsof factors li"e culture& gender& and age on contri!utions in #u!lic goods games&

with somewhat mied results. ?or eam#le& there is no clear effect of gender =see

the survey in 1edyard& ,5>. oeree& Holt& and 1aury =2002> found no gender

differences on average& although men in their sam#le were more li"ely to ma"e

etremely low or high contri!utions than women.

A second strand of the literature consists of #a#ers that eamine changes

in #rocedures& e.g. whether grou#s remain fied or are randomly reconfigured

each round =e.g.& Croson& ,%>& and whether or not the e#erimenter or the

#artici#ants can o!serve individual contri!ution levels. 1etting #eo#le tal" a!out

contri!utions !etween rounds tends to raise the level of coo#eration& an effect that

#ersists to some etent even if communication is su!se'uently sto##ed =saac and$al"er& ,a>. Contri!utions are also increased in the #resence of o##ortunities

for individuals to levy #unishments and rewards on those who did or did not

contri!ute =Andreoni& Har!augh& and Vesterlund& 200,>.

A third strand of the literature #ertains to variations in the #ayoff structure&

e.g. having the #u!lic !enefits !e nonlinear functions of the total contri!utions !y

grou# mem!ers. n the linear #u!lic goods games descri!ed thusfar& it is o#timal

for a selfish #erson to contri!ute nothing when the internal return is less than one.

n this case& the ;ash e'uili!rium involves <ero contri!utions !y all& which is

analogous to defection in a #risoner s dilemma. Using nonlinear #ayoff functions

ma"es it #ossi!le to design e#eriments in which the ;ash e'uili!rium for selfish

#layers is to contri!ute some& !ut not all& to"ens to the #u!lic good. 6ee 1aury

and Holt =2000> for a survey of e#eriments with interior ;ash e'uili!ria.

Alternatively& the #ayoff structure can !e altered !y re'uiring that total

contri!utions eceed a s#ecified threshold !efore any !enefits are derived



=7agnoli and 4cDee& ,,& and Croson and 4ar"s& 2000>. 6uch #rovision

#oints introduce a coordination #ro!lem& since it is not o#timal to contri!ute

unless one e#ects that other contri!utions will !e sufficient to reach the re'uired

threshold. =ames with contri!ution thresholds can !e im#lemented with the

8rovision+8oint 8u!lic oods Veconla! game.>

(es#ite the large num!er of #u!lic goods e#eriments& there is still a lively

de!ate a!out the #rimary motives for contri!ution. Ine a##roach is to model

individuals utility functions as de#ending on !oth the individual s own #ayoff and

on #ayoffs received !y others. Altruism& for eam#le& can !e introduced !y

having utility !e an increasing function of the sum of others #ayoffs. Anderson&

oeree& and Holt =,>& for eam#le estimate that individuals in the saac and

$al"er =,!> e#eriments are willing to give u# at most a!out ten cents to give

others a dollar& and similar estimates were o!tained !y oeree& Holt& and 1aury

=2002>. In the other hand& some #eo#le may not li"e to see others earnings go

a!ove their own& which suggests that relative earnings may matter =7olton and

Ic"enfels& ,B ?ehr and 6chmidt& ,>. n multiround e#eriments&

individuals attitudes toward others may change in res#onse to others !ehavior&and #eo#le may !e willing to contri!ute as long as enough others do so. 4any of

these alternative theories are surveyed in Holt and 1aury =,>.

All of the discussion thusfar has #ertained to voluntary contri!utions& !ut there

are many mechanisms that have !een designed to induce im#roved levels of

#rovision. A critical #ro!lem is that #eo#le may not "now how others value a

#u!lic good& and #eo#le have an incentive to understate their values if ta shares

or re'uired efforts de#end on re#orted values. n theory& there are some

mechanisms that do #rovide #eo#le with an incentive to re#ort values truthfully&

and these have !een tested in the la!oratory. ?or eam#le& see Chen and 8lott

=,%> or the survey in 1edyard =,5>. n #ractice& most #u!lic goods decisions

are either directly or indirectly made on the !asis of #olitical considerations& e.g.lo!!ying and log rolling or vote trading. n #articular& #rovision decisions may !e

sensitive to the #references of the median voter with #references that s#lit others

more or less e'ually. 4oreover& the identity of the median voter may change as

#eo#le move to locations that offer the mi of #u!lic goods that suits their own

#ersonal needs. 6ee Hewett et al. =2002> for a classroom e#eriment in which the

ty#es and levels of #u!lic goods are determined !y voting and !y relocation or

voting with feet. Anderson and Holt =2002!> survey e#eriments that focus on

such #u!lic choice issues.

<$estions

,. $hat was the a##roimate 48C3 for the large game hunters in the GDung

eam#le descri!ed in section N



2. 6u##ose that *nsminger s e#eriment =discussed in section > had used

grou#s of si<e instead of si<e /& with all contri!utions !eing dou!led and

divided e'ually. $hat would the resulting 48C3 have !eenN

. How would it have !een #ossi!le for *nsminger to dou!le the grou# si<e

and hold the 48C3 constantN How could she have increased the 48C3

from 0.5 to 0.-5& "ee#ing the grou# si<e fied at /N

/. *nsminger wrote a small code num!er on the inside of each #erson s

envelo#e& so that she could record which #eo#le made each contri!ution.

he others could not see the codes& so contri!utions were anonymous.

$hat if she had wanted contri!utions to !e dou!le anonymous in the sense

that no!ody& not even the e#erimenter& could o!serve who made which

contri!utionN Can you thin" of a feasi!le way to im#lement this treatment

and still !e a!le to #ay #eo#le !ased on the outcome of the gameN

5. 6u##ose that *nsminger had calculated #ayoffs differently& !y #aying each

of the four #eo#le in a grou# an amount that was one+third of the dou!led

total contri!utions of the other three #artici#ants. f contri!utions were ,&

2& & and / shillings for #eo#le with (s ,& 2& & and /& calculate each #erson s return from the grou# #ro9ect.

%. $hat would the internal and eternal returns !e for the eam#le in

'uestion 5N

-. he internal+return effects in a!les 2%., and 2%.2 are indicated !y the

vertical com#arisons in the same ta!le. he num!ers effects are indicated

!y the com#arisons from one ta!le to the matched cell in the other. How

many num!ers+effect com#arisons are there& and how many are in the

#redicted direction =more contri!utions with higher grou# si<e>N

. he eternal+return com#arisons in a!les 2%., and 2%.2 are found !y

loo"ing at averages in the same row& with higher average contri!utions

antici#ated as one moves to the right. How many eternal+returncom#arisons are there in the two ta!les com!ined& and how many are in

the #redicted directionN =Hint: (o not restrict consideration to averages in

ad9acent columns.>

. $hich two entries in a!le 2%., illustrate the 48C3 effectN

,0. Calculate the 48C3 for each of the two treatments descri!ed in the

nstructions A##endi to Cha#ter 2%.



Cha#ter 2-. he Volunteer s (ilemma

n some situations& it only ta"es a single volunteer to #rovide a #u!lic !enefit. A

dilemma arises if the #er+ca#ita value of this !enefit eceeds the #rivate cost of

volunteering& !ut each #erson would #refer that someone else incur this cost. ?or eam#le& each ma9or country on the U; 6ecurity Council may #refer that a

#ro#osal !y a small country is vetoed& !ut each would #refer to have another

country incur the #olitical cost of a veto that is un#o#ular with many small

mem!er nations. his dilemma raises interesting 'uestions& such as whether

volunteering is more or less li"ely in cases with large num!ers of #otential

volunteers. he volunteer s dilemma game #rovides data that can !e com#ared

with !oth intuition and theoretical #redictions. his setu# is im#lemented as the

we!game V& or it can !e run !y hand with #laying cards and the instructions

#rovided in the a##endi.

I. Sometimes It 5n%! akes 5ne 7ero

After seeing the actress eresa 6aldana in the film 'aging Bu## in ,2& a

cra<ed fan from 6cotland came to 1os Angeles and assaulted her as she was

leaving home for an audition. Her screams for hel# attracted a grou# of #eo#le&

!ut it was a man delivering !ottled water who instinctively charged in and ris"ed

in9ury or worse !y gra!!ing the assailant and holding him while the #olice and the

am!ulance arrived. 4s. 6aldana survived her "nife woundsB she has continued

acting and has !een active with victims su##ort grou#s. his is an eam#le of a

common situation where many #eo#le may want to see a situation corrected& !ut

all that is needed is for one #erson to incur a cost to correct it. Another eam#le isa case where several #oliticians may each #refer that someone else #ro#ose a

legislative #ay increase. 7ut if no!ody else is going to ma"e such a #ro#osal&

then each would rather ta"e the #olitical heat and ma"e it. hese are all cases

where it only ta"es one #erson s costly commitment to change an outcome for all

of the others& e.g. with a veto or the #rovision of information that can !e used !y

all. his ty#e of situation is called a vo#unteer9s di#emma& since #eo#le would

#refer that others volunteer& !ut they #refer to volunteer if no!ody else is going to

do so.

II. "(perimenta% "videncehis dilemma was first studied !y (ie"mann =,5& ,%>& where the

focus was on the effects of increasing the num!ers of #otential volunteers. 1et F

!e the num!er of #otential volunteers. *ach #erson must decide whether or not to



incur a cost of C and volunteer. *ach receives a high #ayoff value of : if at least

one #erson volunteers& and a lower #ayoff - if no!ody in the grou# volunteers.

hus there are three #ossi!le #ayoff outcomes& either you volunteer and earn a

certain return of : C & or you do not volunteer and earn either : or -& de#ending

on whether or not another #erson volunteers.

Consider a two+#erson setu#& with : C P -. his ine'uality ensuresthat it cannot !e a ;ash e'uili!rium for !oth #eo#le to refrain from volunteering&

nor can it !e an e'uili!rium for !oth to volunteer& since each can save C !y free

riding on the other s !ehavior. here are e'uili!ria in which one #erson

volunteers and the other does not& since each would #refer to volunteer if the other

is e#ected to refrain =see 'uestion % for an eam#le>. ?inally& there is also a

symmetric e'uili!rium in which each #erson volunteers with a #ro!a!ility pR&

which is a to#ic that we will consider su!se'uently. ntuitively& one would e#ect

that the e'uili!rium #ro!a!ility of volunteering would !e lower when there is a

larger num!er of #otential volunteers& since there is less ris" in relying on others

generosity in this case. his intuition is !orne out in an e#eriment re#orted !y

?ran<en =,5>& with : - J ,00 and C J 50. he volunteer rates =middlecolumn of a!le 2-.,> decline as grou# si<e is increased form 2 to -& and these

rates level off after that #oint. he third column indicates that the #ro!a!ility of

o!taining no volunteer is essentially 0 for larger grou#s.

a!le 2-., rou# 6i<e *ffects in Volunteer s (ilemma *#eriment =6ource:

?ran<en& ,5>

rou# 6i<e ndividual Volunteer

3ate

3ate of ;o+

Volunteer

Iutcome2 0.%5 0.,2

0.5 0.0-

5 0./ 0.0%

- 0.25 0.,

0.5 0.02

2, 0.0 0.00

5, 0.20 0.00

,0, 0.5 0.00

;et& consider the outcome for a classroom e#eriment conducted with

#ayoffs: : J 25.00& - J 0.00& and C J 5.00. he setu# involved ,2

#artici#ants who were randomly #aired in grou#s of si<e 2 for each of rounds&

followed !y rounds of random matching in grou#s of si<e /. =As with all



classroom e#eriments re#orted in this !oo"& #artici#ants were generally #airs of

students at a single com#uter& and one #artici#ant #air was selected e5 post at

random to !e #aid a small fraction of earnings.> he round+!y+round averages for

the 2+#erson treatment are generally in the 0.5 to 0.- range& as shown on the left

side of ?igure 2-.,. he increase in grou# si<e reduces the volunteer rate to a!out

0./.



00

01

02

03

04

05

06

07

08

09

10

[olunteer .ate

N 2 !ata

N 4 !ata

Nas, Pre!ictions



0 , 2 / 5 % - ,0 ,, ,2 , ,/ ,5 ,%

3ound

0ig$re ).'. o%$nteer9s Di%emma ith Gro$p SiOe Change @>-, "con 32, 0a%% ++'A

III. he Mi(ed?Strateg! "#$i%i&ri$m

4ost of the theoretical analysis of the volunteer s dilemma has !een

focused on the symmetric ;ash e'uili!rium with random strategies. he dashed

lines in ?igure 2-., indicate the ;ash mied+strategy e'uili!rium volunteer rates

for the two treatments. n such an e'uili!rium& each #erson must !e indifferent

!etween volunteering and refraining& since otherwise it would !e rational to

choose the #referred action. n order to characteri<e this indifference& we need to

calculate the e#ected #ayoffs for each decision. ?irst& the e#ected #ayoff from

volunteering is : C since this decision rules out the #ossi!ility of an -

outcome. A decision not to volunteer has an e#ected #ayoff that de#ends on theothers volunteer rate& p. ?or sim#licity& consider the case of only one other #erson

=grou# si<e 2>. n this case& the #ayoff for not volunteering is either V =with

#ro!a!ility p> or - =with #ro!a!ility , p>& so the e#ected #ayoff is:

=2-.,> *#ected 8ayoff =not volunteer> p: =, p -> =for F 2>.

o characteri<e indifference& we must e'uate this e#ected #ayoff with the #ayoff

from volunteering& : C & to o!tain an e'uation in p:

=2-.2> : C p: =, p -> =for F 2>&

which can !e solved for the e'uili!rium level of p:

=2-.> p ,R: C -

TUW =e'uili!rium

volunteer rate for F 2>. V

3ecall that : J 25.00& - J 0.00& and C J 5.00 for the #arameters used in the

first treatment in ?igure 2-.,. hus& the ;ash #rediction for this treatment is /5&

as indicated !y the hori<ontal dashed line with a height of 0.. he actual

volunteer rates in this e#eriment are not this etreme& !ut rather are closer to 0.%.

he formulas derived a!ove must !e modified for a larger grou# si<es& e.g.

F J / for the second treatment. he #ayoff for a volunteer decision is



inde#endent of what others do& so this #ayoff is : C regardless of grou# si<e.

f one does not volunteer& however& the #ayoff de#ends on what the F , others

do. he #ro!a!ility that one of them does not volunteer is , p& so the

#ro!a!ility that none of the F , others volunteer is =, p > F ,. =his calculation

is analogous to saying that if the #ro!a!ility of ails is ,2& then the #ro!a!ility of

o!serving two ails outcomes is =,2>2 J ,/& and the #ro!a!ility of having F +,

fli#s all turn out

ails is =,2> F ,.> o summari<e& when one does not volunteer& the low #ayoff of

- is o!tained when none of the others volunteer& which occurs with #ro!a!ility

=, p> F ,. t follows that the high #ayoff : is o!tained with a #ro!a!ility that is

calculated: , =, p> F , . hese o!servations #ermit us to e#ress the e#ected

#ayoff for not volunteering as the sum of terms on the right side of the following

e'uation:

=2-./> : C : K,=, p> F , LO -=, p> F ,.

he left side is the #ayoff for volunteering& so =2-./> characteri<es the indifference

!etween e#ected #ayoffs for the two decisions that must hold if a #erson is

willing to randomi<e. t is straightforward ='uestion -> to solve this e'uation for

the #ro!a!ility& , p& of not volunteering:

,

=2-.5> , p RV: C - TUW F ,

so the e'uili!rium volunteer rate is&

,

=2-.%> p ,RV: C -

TUW F , =e'uili!rium volunteer rate>.

;otice that =2-.%> reduces to =2-.> when F J 2. t is easily verified that the

volunteer rate is increasing in the value : o!tained when there is a volunteer and

decreasing in the value - o!tained when there is no volunteer. As would !ee#ected& the volunteer rate is also decreasing in the volunteer cost C . =hese

claims are fairly o!vious from the #ositions of C & : & and - in the ratio on the right

side of =2-.%>& !ut they can !e verified !y changing the #ayoff #arameters in the

s#readsheet #rogram that is created to answer 'uestion .>



;et& consider the effects of a change in the num!er of #otential volunteers& F .

?or the setu# in ?igure 2-.,& recall that : J 25.00& - J 0.00& C J

5.00& and therefore& the volunteer rate is 0. when F J 2. $hen F is increased to

/& the formula in =2-.%> yields ,=,5>,& which is a!out 0./, ='uestion >& as

shown !y the dashed hori<ontal line on the right side of ?igure 2-.,. his

increase in ; reduces the #redicted volunteer rate& which is consistent with the

'ualitative #attern in the data =although the F J2 #rediction is clearly too high

relative to the actual data>. t is straightforward to use alge!ra to show that the

formula for the volunteer rate in =2-.%> is a decreasing function of F . his

formula can also !e used to calculate the ;ash volunteer rates for the eight

different grou#+si<e treatments shown in a!le 2-., ='uestion >.

?inally& we will evaluate the chances that none of the F #artici#ants volunteer.

he #ro!a!ility that any one #erson does not volunteer& , p& is given in =2-.5>.

6ince the F decisions are inde#endent& li"e fli#s of a coin& the #ro!a!ility that all

F #artici#ants decide not to volunteer is o!tained !y raising the right side of

=2-.5> to the #ower F :

F

=2-.-> #ro!a!ility that no!ody volunteers =, p> F RV:C -

TUW F ,

he right side of =2-.-> is an increasing function of F =see 'uestion for a guide

to ma"ing s#readsheet calculations>. As F goes to infinity& the e#onent of the

e#ression on the far right side of =2-.-> goes to ,& and the #ro!a!ility of getting

no volunteer goes to C=: ->. he resulting #rediction =,5> is inconsistent with

the data from a!le 2-.,& where the #ro!a!ility of finding no volunteer is <ero for larger grou# si<es. $hat seems to !e ha##ening is that there is some randomness&

which causes occasional volunteers to !e o!served in any large grou#.

Ine interesting feature of the e'uili!rium volunteer rate in =2-.%> is that the

#ayoffs only matter to the etent that they affect the ratio& C =: ->. hus& a

treatment change that shifts !oth V and 1 down !y the same amount will not

affect the difference in the denominator of this ratio. his invariance is the !asis

for a treatment change designed !y a student who ran an e#eriment in class on

this to#ic using the Veconla! software. he student decided to lower these

#arameters !y ,5& from : J 25 and - J 0 to: : J ,0 and - J ,5. he

values of C and F were "e#t fied at 5 and /. a!le 2-.2 summari<es these

#ayoffs for alternative setu#s. All #ayoffs are #ositive in the sure gain treatment& !ut a large loss is #ossi!le in the loss treatment. his #air of treatments was

designed with the goal of detecting whether the #ossi!ility of large losses would



result in a higher volunteer rate in the second treatment. 6uch an effect could !e

due to loss aversion& which was discussed in cha#ter .

a!le 2-.2 Volunteer s (ilemma 8ayoffs for a ain1oss (esign

Iwn(ecision

;um!er of

Ither Volunteers

Iwn

8ayoff

6ure ain

reatment

1oss

reatment

Volunteer any num!er : C 20.00 5.00

;ot Volunteer ;one F 0.00 ,5.00

;ot Volunteer At least one : 25.00 ,0.00

he results for a single classroom e#eriment with these treatments are

summari<ed in ?igure 2-.2. he o!served volunteer rates closely a##roimate the

common ;ash #rediction for !oth treatments& so loss aversion seems to have had

no effect. his eam#le was included to illustrate an invariance feature of the

;ash #rediction and to show how students can use theory to come u# with clever

e#eriment designs. he actual results are not definitive& however& !ecause of the

constraints im#osed !y the classroom setu#: no re#lication& very limited

incentives& and no credi!le way of ma"ing a #artici#ant #ay actual losses.

0$rther Reading

Ither as#ects of the volunteer s dilemma are discussed in (ie"mann and 4itter

=,%> and (ie"mann =,>& who re#orts e#eriments for an asymmetric setu#.



05

06

07

08

09

10

[olunteer .ate

\ain Yrame

oss Yrame

Nas, Pre!iction



00

01

02

03

0 , 2 / 5 % - ,0 ,, ,2 , ,/ ,5 ,%

3ound

0ig$re ). . o%$nteer9s Di%emma 6ith Gain;Loss Change @>-, "con 32, 0a%% ++'A

oeree and Holt =,c> discuss the volunteer s dilemma and other

closely related games where individuals are faced with a !inary decision& e.g. to

enter a mar"et or not or to vote or not. As they note& the volunteer s dilemma is a

s#ecial case of a threshold #u!lic good in which each #erson has a !inary decision

of whether to contri!ute or not. he #u!lic !enefit is not availa!le unless the total

num!er of contri!utors reaches a s#ecified threshold& say M & and the volunteer s

dilemma is a s#ecial case where M J ,. oeree and Holt analy<e the e'uili!ria

for these ty#es of !inary+choice games when individual decisions are determined

!y #ro!a!ilistic choice rules of the ty#e discussed in cha#ters ,,+,. 7asically&

the effect of introducing noise into !ehavior is to #ull volunteer rates towards 0.5.his a##roach can e#lain why the #ro!a!ility of a no+volunteer outcome is not

o!served to !e an increasing function of grou# si<e =as #redicted in the ;ash

e'uili!rium>. he intuition for why large num!ers may decrease the chances of a

no+volunteer outcome is that noise ma"es it more li"ely that at least one #erson in

a large grou# will volunteer due to random causes. his intuition is consistent

with the data from ?ran<en =,5> and is inconsistent with the ;ash #rediction

that the #ro!a!ility of a no+volunteer outcome is an increasing function of grou#

si<e.

<$estions

,. Consider a volunteer s dilemma in which the #ayoff is 25 if at least one

#erson volunteers& minus the cost =if any> of volunteering& the #ayoff for

no volunteer is 0& and the cost of volunteering is ,. Consider a symmetric

situation in which the grou# si<e is 2 and each decides to volunteer with a

#ro!a!ility pN ?ind the e'uili!rium #ro!a!ility.

2. How does the answer to 'uestion , change if the grou# si<e is N

. he A##endi to this cha#ter #rovides instructions for a volunteer s

dilemma with grou# si<es of 2 and /. Calculate the ;ash e'uili!rium

volunteer rate in each case.

/. ?or the setu# in 'uestion & calculate the #ro!a!ility of o!taining novolunteer in a ;ash e'uili!rium& and show that this #ro!a!ility is ,%

times as large for grou# si<e / as it is for grou# si<e 2.

5. n deciding whether or not to volunteer& which decision involves more

ris"N (o you thin" a grou# of ris"+averse #eo#le would volunteer more



often in a ;ash =mied+strategy> e'uili!rium for this game than a grou#

of ris"+neutral #eo#leN *#lain your intuition.

%. he setu# in 'uestion , can !e written as a two+!y+two matri game

!etween a row and column #layer. $rite the #ayoff ta!le for this game&

with the row #ayoff listed first in each cell. *#lain why this volunteer s

dilemma is not a #risoner s dilemmaN Are there any ;ash e'uili!ria in

#ure =non+randomi<ed> strategies for this gameN

-. (erive e'uation =2-.5> from =2-./>& !y showing all intermediate ste#s.

. o calculate grou#+si<e effects for volunteer s dilemma e#eriments& set

u# a s#readsheet #rogram. he first ste# is to ma"e column headings for

the varia!les that you will !e using. 8ut the column headings& F & : & -& C &

C =: ->& and P in cells A,& 7,& C,& (,& *,& and ?, res#ectively. hen

enter the values for these varia!les =here !egin with the setu# from the

second treatment in ?igure 2-.,>: / in cell A2& 25 in cell 72& 0 in cell C2&

and 5 in cell (2. hen enter a formula: J(2=72 C2> in cell *2. ?inally

enter the formula for the e'uili!rium #ro!a!ility from e'uation =2-.%>

into cell ?2B the *cel code for this formula is: J, #ower=*2&,=A2,>>& which raises the ratio in cell *2 to the #ower ,= F ,>& where F

is ta"en from cell A2. f you have done this correctly& you should o!tain

the e'uili!rium volunteer rate mentioned in the cha#ter& which rounds off

to 0./,. *#eriment with some changes in : & -& and C to determine the

effects of changes in these #arameters on the volunteer rate. n order to

o!tain #redictions from the ?ran<en model& enter the values 2& ,00& 0& 50

in cells A& 7& C& and ( res#ectively& and co#y the formulas in cells

*2 and ?2 into cells * and ?. he resulting volunteer rate #rediction in

cell ? should !e 0.5. hen enter the other values for F vertically in the A

column: these values are & 5& -& & 2,& 5,& and ,0,. he #redictions for

these treatments can !e o!tained !y co#ying the values and formulas for

cells 7+? downward in the s#readsheet.

. o o!tain #redictions for the #ro!a!ility of getting no volunteers at all&

add a column heading in cell , = ;o Vol. > and enter the formula from

e'uation =2-.-> in ecel code into cell 2: J#ower=*2& A2=A2,>> and

co#y this formula down to the lower cells in the column. $hat does

this #rediction converge to as the num!er of #artici#ants !ecomes largeN

Hint: as F goes to infinity& the e#onent in e'uation =2-.-> converges to ,.

,0. *#lain the volunteer s dilemma to your roommates or friends and come

u# with at lease one eam#le of such a dilemma !ased on some common

e#erience from class& dorm living& a recent film& etc.



Cha#ter 2. *ternalities& Congestion& and

Common 8ool 3esources

4any #ersistent ur!an and environmental #ro!lems result from ecessive use or

e#loitation of a shared resource. ?or eam#le& an increase in fishing activity

may reduce the catch #er hour for all fishermen& or a commuter s decision to enter

a tunnel may slow down the #rogress of other commuters. *ach individual may

tend to ignore the negative im#act of their activity on other s harvests or travel

times. ndeed& #eo#le may not even !e aware of the negative effects of their

decisions if the effects are small and dis#ersed across many others. his negative

eternality is ty#ically not #riced in a mar"et& and overuse can result. his

cha#ter considers two somewhat styli<ed #aradigms of overuse or congestion.

n the Common 8ool 3esource game& individual efforts to secure more !enefits

from the resource have the effect of reducing the !enefits received !y others. ntechnical terms& the average and marginal #roducts of each #erson s effort is

decreasing in the total effort of all #artici#ants. he Veconla! game& C8& may !e

used to com#are !ehavior in alternate treatments.

n 4ar"et *ntry games& #artici#ants decide inde#endently whether or not to enter

a mar"et or activity for which the earnings #er entrant are a decreasing function of

the num!er of entrants. he outside o#tion earnings from not entering are fied.

here are ty#ically many ;ash e'uili!ria& including one involving randomi<ed

strategies. Dahneman once remar"ed on the tendency for #ayoffs to !e e'uali<ed

for the two decisions: o a #sychologist& it loo"s li"e magic. his im#ression

should change for those who have #artici#ated in a classroom entry game

e#eriment& which should !e run #rior to the class discussion. Although entry

rates are ty#ically distri!uted around the e'uili!rium #rediction& the rates are

varia!le and too high since each #erson ignores the negative effects of their

decisions on other entrants. he Veconla! 4* #rogram #rovides for some #olicy

o#tions = traffic re#ort information and tolls> that correct these #ro!lems.

I. ater ars

6ome of the most difficult resource management #ro!lems in !oth develo#ing

and advanced economies involve water. Consider a setu# where there are

u#stream and downstream users& and in the a!sence of norms or rules& thedownstream users are left with whatever is not ta"en u#stream. his is an

eam#le of a common #ool resource& where each #erson use may reduce the

!enefits o!tained !y others. Istrom and ardner =,> descri!e such a situation

involving farmers in ;a#al. f u#stream users tend to ta"e most of the water& then



their usage may !e inefficiently high from a social #oint of view& e.g. if the

downstream land is more fertile !ottom land. Here the inefficiency is due to the

fact that the u#stream farmers draw water down until the value of its marginal

#roduct is very low& even though more water downstream may !e used more

#roductively.

?or eam#le& the ham!esi system in ;e#al is one where the headenders have

esta!lished clear water rights over those downstream. he farmers located at the

head of each rotation unit ta"e all of the water they need !efore those lower in the

system. n #articular& the headenders grow water+intensive rice during the #re+

monsoon season& and those lower in the system cannot grow irrigated cro#s at this

time. f all farmers were to grow a less+water intensive cro# =wheat>& the area

under cultivation during the #re+monsoon season could !e e#anded dramatically&

nearly ten+fold =Moder& ,%& #. ,5>. he irrigation decisions made !y the

headenders raise their incomes& !ut at a large cost in terms of lost cro# value

downstream. A commonly used solution to this #ro!lem is to limit usage& and

such limits may !e enforced !y social norms or !y e#licit #enalties. n either

case& enforcement may !e #ro!lematic. n areas where farmers own mar"eta!leshares of the water system& these farmers have an incentive to sell them so that

water is diverted to its highest+value uses =Moder& ,%>.

7esides overuse& there is another #otential source of inefficiency if activities to

increase water flow have 9oint !enefits to !oth ty#es of users. ?or eam#le&

irrigation canals may have to !e maintained or renewed annually& and this wor"

can !e shared !y all users& regardless of location. n this case& the !enefit of

additional wor" is shared !y all users& so each #erson who contri!utes wor" will

only o!tain a #art of the !enefits& even though they incur the full cost of their own

effort& unless their contri!ution is somehow reim!ursed or otherwise rewarded.

;ote that the level of wor" on the 9oint #roduction of water flow should !e

increased as long as the additional !enefit& in terms of harvest value& is greater than the cost& in terms of lost earnings on other uses of the farmers time. hus

each #erson has an incentive to free ride on others efforts& and this #erverse

incentive may !e stronger for downstream users if their share of the remaining

water tends to !e low.

he #ossi!ility of under+#rovision can !e illustrated with a numerical eam#le.

6u##ose for sim#licity that there is one u#stream user and four downstream users&

and that each has u# to / units =e.g. days> of la!or to allocate to the esta!lishment

of the shared irrigation system. he value of the total water #roduced is a

function of the total effort& as shown in the to# two rows of a!le 2.,.

a!le 2., 8ayoffs from otal *ffort

8ota# 4ffort 1 3 & " * 7



otal 8ayoff

U#stream

(ownstream

(ownstream

(ownstream

(ownstream

%

2

,

,

,

,

/2

,/

-

-

-

-

-2

2/

,2

,2

,2

,2

%

2

,%

,%

,%

,%

,,/

,

,

,

,

,2%R

/2

2,

2,

2,

2,

,2

//

22

22

22

22

,2

//

22

22

22

22

n the a!sence of any restriction& the u#stream user is a!le to ca#ture , of the

value of the water& shown in the third row& and each of the four downstream users

are a!le to ca#ture only ,%. f the o##ortunity cost of effort is -& then the social

o#timum level of effort would !e % =as indicated !y the asteris" in the to# row>&

since an additional =-th> unit of effort only increases the value !y % =from ,2% to

,2>& which is less than the cost of -. f the u#stream user were forced to act

alone& that #erson would !e willing to #rovide all / availa!le units& !ut the total

value of water #roduced& %& would !e far !elow the social o#timum of ,2%. Met

none of the downstream users would have any incentive to 9oin in& since with /

units already #rovided& their share =,% each> would only increase to , if any one

of them contri!uted a unit of effort& a gain of that is less than the o##ortunity

cost of -. his is a ;ash e'uili!rium in which the downstream users free ride&

and the common resource is under#rovided. ndeed there are no ;ash e'uili!ria

where the downstream users contri!ute anything in this eam#le =see #ro!lem ,>.

o summari<e& there are two main sources of inefficiency associated with

common #ool resources& i.e. overuse and under#rovision. Iveruse can occur if

#eo#le do not consider the negative effects that their own use decisions have on

the !enefit of the resource that remains for others. Under+#rovision can occur if

the !enefits of #roviding the resource are shared !y all users& !ut each #erson

incurs the full cost of their own contri!ution to the grou# effort.

II. raffic

4any of the to# #olitical issues in large ur!an areas involve commuting and

traffic control. 1arge investments to #rovide fast freeways& tunnels& and !ridges

often are negated with a seemingly endless su##ly of eager commuters during

certain #ea" #eriods. he result may !e travel via the new freeways and tunnels

is no faster than the older alternative routes via a ma<e of surface roads. o

illustrate this #ro!lem& consider a styli<ed setu# in which there are F commuters

who must choose !etween a slow& relia!le surface route and a #otentially faster

route =freeway or tunnel>. However& if large num!ers decide to enter the faster

route& the resulting congestion will cause traffic to slow to a crawl& so entry can !ea ris"y decision. his setu# was im#lemented in a la!oratory e#eriment run !y

Anderson& Holt& and 3eiley =200/>. n each session& there were ,2 #artici#ants

who had to choose whether to enter a #otentially congested activity or eit to a



safe alternative. he #ayoffs for entrants were decreasing in the total num!er of

entrants in a given round& as shown in a!le 2.2.

a!le 2.2 ndividual 8ayoffs from *ntry

4ntrants 1 3 & " * 7 % 1 11 1

8ayoff /.00 .50 .00 2.50 2.00 ,.50 ,.00 0.50 0.00 0.50 ,.00 ,.50

Mou should thin" of the num!ers in the ta!le as the net monetary !enefits from

entry& which go down !y 0.50 as each additional entrant increases the time cost

associated with travel via the congested route. n contrast& the travel time

associated with the alternative route =the non+entry decision> is less varia!le. ?or

sim#licity& the net !enefit for each non+entrant is fied at 0.50& regardless of the

num!er who select this route. I!viously& entry is the !etter decision as long as

the num!er of entrants is not higher than & and the e'uili!rium num!er of

entrants is& therefore& . n this case& each #erson earns 0.50& and the total

earnings =net !enefit> for the ,2 #eo#le together is %.00. n contrast& if only /enter& they each earn 2.50 and the other earn 0.50& for total earnings of ,/. t

follows that an entry restriction will more than dou!le the social net !enefit. t is

straightforward to calculate the social !enefit associated with other entry levels

='uestion 2>& and the results are shown in a!le 2..

a!le 2. 6ocial 7enefit for all ,2 8artici#ants ogether

4ntrants 1 3 & " * 7 % 1 11 1

otal

8ayoff .50 ,2 ,.50 ,/ ,.50 ,2 .50 % ,.50 /.00 ,0.50 ,

n this setu#& it is reasona!le to e#ect that uncontrolled entry decisions would

lead to a##roimately entrants& the num!er that e'uali<es earnings from the two

alternative decisions. As a result& the total !enefit to all #artici#ants is

inefficiently low. he reason for this is that each additional #erson who decides

to enter lowers the net entry !enefit !y 0.50 for themselves and for a## other

entrants& and it is this eternal effect that may !e ignored !y each entrant. ?or

eam#le if there are currently / entrants& the addition of a fifth would reduce their

earnings !y 2 =/0.50> and only increase the entrant s earnings !y ,.50 =from

the non+entry #ayoff of 0.50 to the entry #ayoff of 2>& so total earnings decrease

!y 0.50. A sith entrant would reduce total earnings !y even more& ,.50 =a

reduction of 50.50 that is only #artly offset !y the , increase in the entrant s

earnings>& since there are more #eo#le to suffer the conse'uences. As even more

#eo#le enter& the earnings reduction caused !y an additional entrant !ecomes

higher& and therefore& the total #ayoff falls at an increasing rate. his is the



intuition !ehind the very high costs associated with dramatic traffic delays during

#ea" commuting #eriods. Any #olicy that #ushes some of the traffic to non+#ea"

#eriods will !e welfare im#roving& since the cost of increased congestion in those

#eriods is less than the !enefit from reducing #ea" congestion.

n la!oratory e#eriments& as on the o#en road& #eo#le ignore the negative

eternal effects that their earnings have on others earnings. he data #atterns

from one such e#eriment& shown in ?igure 2.,& illustrate this feature. he

average entry rate in the first ,0 rounds is 0.%& which is 'uite close to the

twothirds = out of ,2> rate that would occur in e'uili!rium. here is& however&

considera!le varia!ility& which in real traffic situations would !e am#lified !y

accidents and other unforseen traffic events.



0708

09

10

No :ntry Yee

:ntry Yee $2

:Auilibrium



00

01

02

03

04

05 n r y a e

0 , 2 / 5 % - ,0,,,2,,/,5,%,-,,20

3ound

0ig$re 2. '. "ntr! Game ith Imposition of "ntr! 0ee

So$rce: -nderson, 7o%t, and Rei%e! @++3A

After ,0 rounds& all entrants were re'uired to #ay an entry fee of 2& which

reduces all num!ers in the !ottom row of a!le 2.2 !y 2. n this case& the

e'uili!rium num!er of entrants is /& i.e. an entry rate of 0.& as shown !y the

dashed line on the right side of ?igure 2.,. he average entry rate for these final

,0 rounds was 'uite close to this level& at 0..

*ven though the fee reduces entry to =nearly> socially o#timal levels& the

individuals do not en9oy any !enefit. o see this& let s ignore the varia!ility and

assume that the fee reduces entry to the 0. rate. n e'uili!rium entrants earn

2.50 minus the 2 fee& and non+entrants earn 0.50& so everyone earns eactly

the same amount& 0.50& as if they did not enter at all. $here did the increase in

social !enefit goN he answer is in the collected fees of =/2>. his total feerevenue of is availa!le for other uses& e.g. ta reductions and road construction.

6o if the commuters in this model are going to !enefit from the reduction in

congestion& then they must get a share of the !enefits associated with the fee

collections.

A second series of sessions was run with a redistri!ution of fee revenues& with

each #erson receiving ,,2 of collections. n #articular& after ,0 rounds with no

fee& #artici#ants were allowed to vote on the level of the fee for rounds ,,+20& and

then for each ,0+round interval after that. he voting was carried out !y ma9ority

rule after a grou# discussion& with ties !eing decided !y the chair& who was

selected at random from the ,2 #artici#ants. n one session& the #artici#ants first

voted on a fee of ,& which increased their total earnings& and hence was selected

again after round 20 to hold in rounds 2,+0. After round 0& they achieved a

further reduction in entry and increase in total earnings !y voting to raise the fee

to 2 for the final ,0 rounds. n a second session with voting& the grou# reached

the o#timal fee of 2 in round 20& !ut the chair of the meeting "e#t arguing for a

lower fee& and these arguments resulted in reductions to ,.-5& and then to ,.50&

!efore it was raised to a near+o#timal level of ,.-5 for the final ,0 rounds.

nterestingly& the discussions in these meetings did not focus on maimi<ing total

earnings. nstead& many of the arguments were !ased on ma"ing the situation

!etter for entrants or for non+entrants =different #eo#le had different #oints of

view& !ased on what they tended to do>. n each of these two sessions& theim#osition of the entry fees resulted in an a##roimate dou!ling of earnings !y

the final ,0 rounds. *ven though the average #ayoffs for entry =#aying the fee>

and eit are each e'ual to 0.50 in e'uili!rium& for any fee& the fee of 2



maimi<ed the amount collected& and hence maimi<ed the total earnings when

these include a share of the collections.

3egardless of average entry rate& there is considera!le variation from round to

round& as shown in ?igure 2., and in the data gra#hs for the other sessions where

fees were im#osed& either !y the e#erimenter or as the result of a vote. his

varia!ility is detrimental at any entry rate a!ove one+third. ?or sim#licity&

su##ose that there is no fee& so that entry would tend to !ounce u# and down

around the e'uili!rium level of . t can !e seen from a!le 2. that an increase

in entry from to reduces total earnings =including the shared fee> from % to

,.50& whereas a reduction in entry from to - only raises earnings from % to

.50. n general& a !ounce u# in entry reduces earnings !y more than a !ounce

down !y an e'ual amount& so symmetric variation a!ound the e'uili!rium level of

would reduce total earnings !elow the amount& %& that could !e e#ected in

e'uili!rium with no variation. he intuition !ehind this asymmetry is that

additional entry causes more harm with larger num!ers of entrants.

*veryone who commutes !y car "nows that travel times can !e varia!le&

es#ecially on routes that can !ecome very congested. n this case& anything thatcan reduce varia!ility will tend to !e an im#rovement in terms of average

!enefits. n addition& most #eo#le disli"e varia!ility& and the resulting ris"

aversion& which has !een ignored u# to now& would ma"e a reduction in

varia!ility more desira!le. 6ometimes varia!ility can !e reduced !y !etter

information. n many ur!an areas& there is a news+oriented radio station that

issues traffic re#orts every ,0 minutes& e.g. on the eights at $I8 in $ashington&

(C. his information can hel# commuters avoid entry into already overcrowded

roads and freeways. A styli<ed ty#e of traffic information can !e im#lemented in

an e#eriment !y letting each #erson find out the num!er who have already

entered at any time u# to the time that they decide to enter. n this case& the order

of entry is determined !y the #artici#ants themselves& with some #eo#le ma"ing a'uic" decision& and others waiting. he effect of this "ind of information is to

!ring a!out a dramatic =!ut not total> reduction in varia!ility& as can !e seen for

the e#eriment shown in ?igure 2.2.

n a richer model& some commuters would have had higher time values than

others& and hence would have #laced a higher value on the reduction in

commuting time associated with reduced congestion. he fee would allow

commuters to sort themselves out in the value+of+time dimension& with high+value

commuters !eing willing to #ay the fee. he resulting social !enefit would !e

further enhanced !y the effects of using #rice =the entry fee> to allocate the good

=entry> to those who value it the most. his o!servation is similar to a result

re#orted in 8lott =,>& where the resale of licenses to trade& from those with low

values to those with high values& raised total earnings and mar"et efficiency.



III. Common Poo% Reso$rce "(periments

*ach #erson in the mar"et entry game has two decisions& enter or not. n

contrast& most common #ool resource situations are characteri<ed !y a range of

#ossi!le decisions& corres#onding to the intensity of use. ?or eam#le& a lo!ster

fisherman not only decides whether to fish on any given day& !ut also =if

#ermitted !y regulations and weather> how many days a year to go out and how

many hours to stay out each day. 1et the fishing effort for each #erson !e denoted

!y 5i& where the i su!scri#t indicates the #articular #erson. he total effort of all

F fishermen is denoted !y & which is 9ust the sum of the individual 5i for i J ,. ...

F . ?or sim#licity& assume that no #erson is more s"illed than any other& so it is

natural to assume that a #erson s fraction of the total harvest is e'ual to their

fraction of the total effort& 5i .



07

08

09

10

No :ntry Yee

:ntry Yee $2

:Auilibrium



00

01

02

03

04

05

40100 5 15 20 25 30 35

ntr y ate

3ound

0ig$re 2. . "ntr! Game 6ith Information -&o$t the 8$m&er of Prior "ntrants

So$rce: -nderson, 7o%t, and Rei%e! @++3A

n the e#eriment to !e discussed& each #erson selected an effort level

simultaneously& and the total harvest revenue& E & was a 'uadratic function of the

total effort:

=2.,> E A 7 2&

where A P 0& 7 P 0& and Q A7 =to ensure that the harvest is not negative>. his

function has the #ro#erty that the average revenue #roduct& E & is a decreasing

function of the total effort. his average revenue #roduct is calculated as the total

revenue from the harvest& divided !y the total effort: =A 7 2> J A 7 & which

is decreasing in since 7 is assumed to !e #ositive. he fact that average

revenue is decreasing in means that an increase in one #erson s effort willreduce the average revenue #roduct of effort for everyone else. his is the

negative eternality for this common #ool resource #ro!lem.

*ffort is costly& and this cost is assumed to !e #ro#ortional to effort& so the cost

for individual i would !e C 5i& where C P 0 re#resents the o##ortunity cost of the

fisherman s time. Assuming that the harvest in e'uation =2.,> is measured in

dollar value& then the earnings for #erson i would !e their share& 5i & of the total

harvest in e'uation =2.,> minus the cost of C 5i that is associated with their effort

choice:

=2.2> earnings =A 7 2

>= 5i >C 5i

=A 7 5> i

C & 5i

where right side of e'uation =2.2> is the average #roduct of effort times the

individual s effort& minus the cost of the #erson s effort decision. ;ote that this

earnings e#ression on the right side of e'uation =2.2> is analogous to that of a

firm in a Cournot mar"et& facing a demand =average revenue> curve of A 7 and

a constant cost of C #er unit.

A rational& !ut selfish& #erson in this contet would ta"e into account the fact that

an increase in their own effort reduces their own average revenue #roduct& !ut

they would not consider the negative effect of an increase in effort on the others

average revenue #roducts. his is the intuitive reason that unregulated

e#loitation of the resource would result in too much #roduction relative to whatis !est for the grou# in the aggregate =9ust as uncoordinated #roduction choices !y

firms in a Cournot mar"et results in a total out#ut that is too high relative to the

level that would maimi<e 9oint #rofits>.



n #articular& the ;ashCournot e'uili!rium for this setu# would involve each

#erson choosing the activity level& 5i& that maimi<es the earnings in e'uation

=2.2> ta"ing the others activities as given. 6u##ose that there are a total of ;

#artici#ants& and that the ; , others choose a common level of each. hus the

total of the others decisions is =; ,> & and the total usage is: X J 5i O =; ,> .

$ith this su!stitution& the earnings in e'uation =2.2> can !e e#ressed:

=2.> earnings A 5i B F =,> 5i B5i2 C & 5i

which can !e maimi<ed !y e'uating marginal revenue with marginal cost C. he

marginal revenue is 9ust the derivative of total revenue =as e#lained in Cha#ter

, and its a##endi>& and the marginal cost is the derivative of total cost& C 5i.

herefore& the earnings in e'uation =2.> can !e maimi<ed !y setting the

derivative of this e#ression with res#ect to 5i e'ual to <ero:

=2./> A 7= F ,> 27 5i C 0.

As discussed in Cha#ter ,& this e'uation can !e used to determine a common

decision& 5& !y re#lacing !oth and 5i with 5 and solving to o!tain:

=2.5> 5 A C

.

= F ,>7

he nature of this common #ool resource #ro!lem can !e illustrated in terms of

the s#ecific setu# used in a research e#eriment run with the Veconla! software !y u<i" =200/>& who at the time was an undergraduate at 4iddle!ury College.

He ran e#eriments with grou#s of 5 #artici#ants with A J 25& 7 J ,2& and C J ,.

he cost was im#lemented !y giving each #erson ,2 to"ens in a round. hese

could either !e "e#t& with earnings of , each& or used to etract the common #ool

resource. hus the o##ortunity cost& C& is , for each unit increase in 5i. $ith

these #arameters and ; J 5& e'uation =2.5> yields a common e'uili!rium

decision of 5 J . t can !e shown that the socially o#timal level of a##ro#riation

is /. ='uestion %>.

he default setting for the Veconla! software #resents the setu# with a mild

environmental inter#retation& i.e. in terms of a harvest from a fishery.

Ad9ustments to the software and the use of su##lemental instructions #ermittedu<i" to #resent the setu# using either environmental terminology& wor"#lace

terminology or a neutral = magic mar!les > terminology. n each of these three

treatments =environmental& wor"#lace& and mar!les> there were sessions& each

with 5 #artici#ants. *ach session was run for ,0 rounds with fied matchings.



he #ur#ose of the e#eriment was to see if the mode of #resentation or frame

would affect the degree to which #eo#le overused the common resource. here

were no clear framing effects: the average decision was essentially the same for

each treatment: -., =environmental>& .0/ =mar!les>& and -. =wor"#lace>. hese

small differences were not statistically significant& although there was a

significant increase in variance for the non+neutral frames =wor"#lace and

environmental>. Although the author was a!le to detect some small framing

effects after controlling for demogra#hic varia!les& the main im#lication for

e#erimental wor" is that if you are going to loo" at economic issues& then the

frame may not mater much in these e#eriments& although one should still hold

the frame constant and change the economic varia!les of interest.

he results re#orted !y u<i" =200/> are fairly ty#ical of those for common #ool

resource e#erimentsB the average decisions are usually close to the ;ash

#rediction& although there may !e considera!le variation across #eo#le and from

one round to the net. However& if #artici#ants are allowed to communicate

freely #rior to ma"ing their decisions& then the decisions are reduced toward

socially o#timal levels =Istrom and $al"er& ,,>.

I. "(tensions

Common #ool resource #ro!lems with fisheries are discussed in Hardin =,%>.

?or a wide+ranging coverage of issues related to the tragedy of the commons& see

Istrom& *.& 3. ardner& and F. D. $al"er =,/> and the s#ecial section of the ?all

, Kourna# of 4conomic Perspectives devoted to

4anagement of the 1ocal Commons. he first common #ool resource

e#eriments are re#orted in arnder& Istrom& and $al"er =,0>& who evaluate

the effects of su!9ect e#erience& and !y $al"er& ardner& and Istrom =,0>&

who consider the effects of changing the endowment of to"ens.?or results of a common #ool resource e#eriment done in the field& see

Cardenas& 6tranlund& and $illis =2000>& who conducted their e#eriment in rural

villages in Colum!ia. he main finding is that the a##lication of rules and

regulations that are im#erfectly monitored and outside of informal community

institutions tends to increase the effect of individualistic motives and ;ash+li"e

results. hey conclude ...modestly enforced government controls of local

environmental 'uality and natural resource use may #erform #oorly& es#ecially as

com#ared to informal local management =#. ,-2,>. 3egarding the self+governed

institutions& Cardenas =200> and Cardenas et.al =2002> re#ort also on a similar set

of e#eriments in the field aimed at e#loring the #otential of non+!inding faceto+

face communication which was #roven in Istrom& ardner and $al"er =,/> to

!e 'uite effective in solving the dilemma. heir findings suggest that actual

ine'ualities o!served in the field regarding the social status and the wealth

distances within and across grou#s of su!9ects limited the #ossi!ilities of



selfgovernance to solve Hardin s tragedy. rou#s of wealthier and more

heterogeneous villagers found it more difficult to coo#erate when allowed to have

communication. In the one hand& #oorer #artici#ants had in fact more e#erience

with actual commons dilemmas& were more familiar with the tas" and with other

#eers in the grou#& while wealthier villagers incomes were more de#endent on

their own assets and #ro!a!ly had less fre'uent interactions in the #ast in these

"inds of social echange situations.

<$estions

,. Consider the numerical eam#le in section & and su##ose that one of the

downstream users #rovides an effort of ,. *#lain what the !est res#onse

of the u#stream user would !e. f that !est res#onse were carried out&

would the downstream user still !e willing to su##ly a unit of effortN

2. Calculate the social !enefits associated with /& %& & ,0& and ,2 entrants

for the setu# in a!les 2.2 and 2. =show your wor">.

. Use the num!ers in a!le 2. to e#lain why un#redicta!ility in thenum!er of entrants is !ad& e.g. why a 5050 of either % entrants or ,0

entrants is not as good as the e#ected value = entrants for sure> in terms

of e#ected earnings.

/. *#lain in your own words why each increase in the num!er of entrants

a!ove / results in ever greater decreases in the social !enefit in a!le

2..

5. $hat would the e'uili!rium entry rate !e for the game descri!ed in

section if the entry fee were raised to #er entrantN How would the

total amount collected in fees change !y an increase from 2 to N How

would the total amount in collected fees change as the result of a fee

decrease from 2 to ,N%. =re'uires calculus> (erive a formula =in terms of A& 7& C& and F > for the

socially o#timal level of a##ro#riation of the common #ool resource

model in section . $hat is the socially o#timal level of a##ro#riation

#er #erson for the #arameters: A J 25& 7 J ,2. C J ,& and F J 5N



Cha#ter 2. 3ent 6ee"ing

Administrators and government officials often find themselves in a #osition of

having to distri!ute a limited num!er of #ri<ed items =locations& licenses& etc.>.

Contenders for these #ri<es may engage in lo!!ying or other costly activities thatincrease their chances for success. ;o single #erson would s#end more on

lo!!ying than the #ri<e is worth& !ut with a large num!er of contenders&

e#enditures on lo!!ying activities may !e considera!le. his raises the

distur!ing #ossi!ility that the total cost of lo!!ying !y all contenders may

dissi#ate a su!stantial fraction of the #ri<e value. n economics 9argon& the #ri<e

is an economic rent& and the lo!!ying activity is referred to as rent seeking . 3ent

see"ing is thought to !e more #revalent in develo#ing countries& where some

estimates are that such non+mar"et com#etitions consume a significant fraction of

national income =Drueger& ,-/>. Ine only has to serve as a (e#artment Chair in

a U.6. university& however& to e#erience the frustrations of rent see"ing. he

game considered in this cha#ter is one in which the #ro!a!ility of o!taining the #ri<e is e'ual to one s share of the total lo!!ying e#enditures of all contenders.

he game illustrates the #otential costs of administrative =non+mar"et> allocation

#rocedures. t can !e im#lemented with #laying cards& as indicated in the

a##endi& or it can !e run on the Veconla! software =game 36>.

I. Government 6ith a Smokestack on Its BackE

Ine of the most s#ectacular successes of recent government #olicy has !een the

auctioning off of !andwidth used to feed the e#loding growth in the use of cell

#hones and #agers. he U.6. ?ederal Communications Commission =?CC> hasallocated ma9or licenses with a series of auctions that raised !illions of dollars

without adverse conse'uences. 6ome *uro#ean countries followed with similarly

successful auctions& which also raised many more !illions than e#ected in some

cases. he use of auctions collects large num!ers of #otential com#etitors& and

the commodities can !e allocated 'uic"ly to those with the highest valuations. n

the U.6.& this !andwidth was originally reserved for the armed forces& !ut it was

underused at the end of the Cold $ar& and the transfer created a large increase in

economic wealth& without significant administrative costs. t could have easily

!een otherwise. 3adio and television !roadcasting licenses were traditionally

allocated via administrative #roceedings& which are sometimes called !eauty

contests. he successful contender would have to convince the regulatoryauthority that service would !e of high 'uality and that social values would !e

#rotected. his often re'uired esta!lishing a technical e#ertise and an effective

lo!!ying #resence. he lure of etremely high #otential #rofits was strong

enough to induce large e#enditures !y as#iring #roviders. hose who had



ac'uired licenses in this manner were strongly o##osed to mar"et+!ased

allocations that forced the reci#ients to #ay for the licenses. 6imilar economic

#ressures may e#lain why !eauty contest allocations continue to !e #revalent in

many other countries.

he first crac" in the door a##eared in the late ,0 s& when the ?CC decided to

s"i# the administrative #roceedings in the allocation of hundreds of regional cell

#hone licenses. he forces o##osed to the #ricing of licenses did manage to !loc"

an auction& and the licenses were allocated !y lottery instead. here were a!out

20&000 a##lications for %/ licenses. *ach a##lication involved significant

#a#er wor" =legal and accounting services>& and firms s#eciali<ing in #roviding

com#leted a##lications !egan offering this service for a!out %00 #er a##lication.

he resources used to #rovide this service have o##ortunity costs& and Ha<lett and

4ichaels =,> estimated the total cost of all su!mitted a##lications to !e a!out

/00&000. he lottery winners often sold their licenses to more efficient

#roviders& and resales were used to estimate that the total mar"et value of the

licenses at that time was a!out a million dollars. *ach individual lottery winner

earned very large #rofits on the difference !etween the license value and thea##lication cost& !ut the costs incurred !y others were lost& and the total cost of

the transfer of this #ro#erty was estimated to !e a!out fort percent of the mar"et

value of the licences. here are& of course& the indirect costs of su!se'uent

transfers of licenses to more efficient #roviders& a #rocess of consolidation that

may ta"e many years. n the meantime& inefficient #rovision of the cellular

services may have created ri##les of inefficiency in the economy. t is e#isodes

li"e this that once caused 4ilton and 3ose ?riedman =,>& in a discussion of the

unintended side effects of government #olicies& to remar": *very government

measure !ears& as it were& a smo"estac" on its !ac".

n a classic #a#er& ordon ulloc" =,%-> #ointed out that the real costs

associated with com#etitions for government granted rents may destroy or dissi#ate much of the value of those rents in the aggregate& even though the

winners in such contests may earn large #rofits. his destruction of value is often

invisi!le to those res#onsi!le& since the contestants #artici#ate willingly& and the

administrators may en9oy the #rocess of !eing lo!!ied. 4oreover& some of the

costs of activities li"e waiting in line and #ersonal lo!!ying are not directly #riced

in the mar"et. hese costs can !e 'uite a##arent in a la!oratory e#eriment of the

ty#e to !e descri!ed net.

II. Rent Seeking in the C%assroom La&orator!

n many administrative =non+mar"et> allocation #rocesses& the #ro!a!ility

of o!taining a #ri<e or mono#oly rent is an increasing function of the amount

s#ent in the com#etition. n the ?CC auctions& the chances of winning a license

were a##roimately e'ual to the a##licant s efforts as a #ro#ortion of the total



efforts of the other contestants. his #rovides a rationale for a standard

mathematical model of rentsee"ing with F contestants. he effort for #erson i is

denoted !y 5i for i J ,& F . he total cost of each effort is a cost c times the #erson

s own effort& i.e. c5i. n the sim#lest symmetric model& the value of the rent& : & is

the same for all& and each #erson s #ro!a!ility of winning the #ri<e is e'ual to

their effort as a fraction of the total effort of all contestants:

5i : c5i.

=2.,> e#ected #ayoff ] 5

,&... F

Here the e#ected #ayoff is the #ro!a!ility of success& which is the fraction of

total effort& times the #ri<e value : & minus the cost of effort. his latter cost is not

multi#lied !y any #ro!a!ility since it must !e #aid whether or not the #ri<e is

o!tained.

oeree and Holt =,> used the #ayoff function in =,.,> in a classroom

e#eriment with : J ,%&000& c J &000& and F J /. *ach of the four

com#etitors consisted of a team of 2+ students. *fforts were re'uired to !e

integer amounts. his re'uirement was enforced !y giving , #laying cards of

the same suit to each team. 3ent+see"ing effort was determined !y the num!er of

cards that the #erson #laced in an envelo#e& as descri!ed in the instructions to this

cha#ter. *ach team incurred a cost of &000 for each card #layed& regardless of

whether or not they o!tained the #ri<e. he cards #layed were collected& shuffled&

and one was drawn to determine which contender would win the ,%&000 #ri<e.

he num!er of cards #layed varied from team to team& !ut on average there were

slightly more than three cards #layed !y each team. hus a ty#ical team incurred&000 in e#enses& and the total lo!!ying cost for all four teams was over

%&000& all for a #ri<e worth only ,%&000.

6imilar results were o!tained in a se#arate classroom e#eriment using the

game 36 from the Veconla! setu#. he #arameters were the same =four

com#etitors& a ,%&000 value& and a &000 cost #er unit of effort>& and the ,2

teams were randomly #ut into grou#s of four com#etitors in a series of rounds.

he average num!er of lo!!ying effort units was in the first round& which was

reduced to a little over 2 in rounds / and 5. *ven with two units e#ended !y

each team& the total cost would !e 2 =num!er of effort units> times / =teams> times

the &000 cost of effort& for a total cost of 2/&000. his again resulted in

overdissi#ation of the rent& which was only ,%&000.



III. he 8ash "#$i%i&ri$m

A ;ash e'uili!rium for this e#eriment is a lo!!ying effort for each

com#etitor such that no!ody would want to alter their e#enditure given that of

the other com#etitors. ?irst& consider the case of four contenders& a ,%&000

value& and a &000 effort cost. ;ote that efforts of 0 cannot constitute an

e'uili!rium& since each #erson could deviate to an effort of , and o!tain the

,%&000 #ri<e for sure at a cost of only &000. ;et su##ose that each #erson is

#lanning to choose an effort of 2 at a cost of %&000. 6ince the total effort is & the

chances of winning are 2 J ,/ so the e#ected #ayoff is ,%&000/ %&000& which

is minus 2&000. his cannot !e a ;ash e'uili!rium& since each #erson would

have an incentive to deviate to 0 and earn nothing instead of losing 2&000. ;et&

consider the case where each #erson s strategy is to eert an effort of ,& which

#roduces an e#ected #ayoff of ,%&000/ &000 J ,&000. o verify that this is an

e'uili!rium& we have to chec" to !e sure that deviations are not #rofita!le. A

reduction to an effort of 0 with a #ayoff of 0 is o!viously !ad. A unilateral

increase to an effort of 2& when the others maintain efforts of ,& will result in a 25

chance of winning& for an e#ected #ayoff of ,%&000=25> %&000 J /00& which isalso worse than the #ayoff of ,&000 o!tained with an effort of ,. n the

symmetric e'uili!rium& the effort of , for each of the four contestants results in a

total cost of /=&000> J ,2&000& which dissi#ates three+fourths of the ,%&000 rent.

his raises the issue of whether a reduction in the cost of rent see"ing efforts

might reduce the etent of wasted resources. n the contet of the ?CC lotteries&

for eam#le& this could corres#ond to a re'uirement of less #a#er wor" for each

se#arate a##lication. 6u##ose that the resource cost of each a##lication is

reduced from &000 to ,&000. his cost reduction causes the ;ash e'uili!rium

level to rise from , to =see 'uestion ,>.

I. - Mathematica% Derivation of the "#$i%i&ri$m @5ptiona%A

he ;ash e'uili!rium calculations done thusfar were !ased on considering

a #articular level of rent+see"ing activity& common to each #erson& and showing

that a deviation !y one #erson alone would not increase that #erson s e#ected

#ayoff. his is a straightforward& !ut tedious& a##roach& and it has the additional

disadvantage of not e#laining how the candidate for a ;ash e'uili!rium was

found in the first #lace. A calculus derivation is relatively sim#le& and is offered

here as an o#tion for those who are familiar with !asic rules for ta"ing derivatives

=some of these rules are introduced in the a##endi to cha#ter ,>. ?irst& consider

the e#ected #ayoff function in =2.,>& modified to let the decisions of the ;,

others !e e'ual to a common level& 5R.

5iR : c5i.

=2.2> e#ected #ayoff



5i = F ,> 5

his will !e a hill+sha#ed =concave> function of the #erson s own rent+see"ing

activity& 5i & and the function is maimi<ed at the to# of the hill where the slo#e of

the function is <ero. o find this #oint& the first ste# is to ta"e the derivative and

set it e'ual to <ero. he resulting e'uation will determine #layer is !est res#onse

when the others are choosing 5R. n a symmetric e'uili!rium with e'ual rent+

see"ing activities& it must !e the case that 5i J 5R& which will yield an e'uation

that determines the e'uili!rium level of 5R. his analysis will consist of two

ste#s: finding the derivative of #layer is e#ected #ayoff and then using the

symmetry condition.

Step 1. n order to use the rule for ta"ing the derivative of a #ower function& it is

convenient to e#ress =2.2> as a #ower function:

=2.> e#ected #ayoff = 5i 5i = F ,> 5R>,: c5i.

he final term on the right side of =2.> is linear in 5i & so its derivative is c.

he first term on the right side of =2.> is the #roduct of 5i and a #ower function

that contains 5i & so the derivative is found with the #roduct rule =derivative of the

first function times the second& #lus the first function times the derivative of the

second>& which yields the first two terms on the left side of =2./>:

=2./> = 5i = F ,> 5R>,: = 5 5ii = F ,> 5R>2: c 0.

Step . ;et we use the fact that 5i J 5R in a symmetric e'uili!rium. 4a"ing thissu!stitution into =2./> yields a single e'uation in the common e'uili!rium level&

5R:

=2.5> = 5R = F ,> 5R>,: 5R= 5R = F ,> 5R>2: c 0&

which can !e sim#lified to:

, ,

=2.%> R : 2R : c 0.

F5 F 5

?inally& we can multi#ly !oth sides of =2.%> !y F 2 5R& which yields an e'uation

that is linear in 5R that reduces to:



=2.-> 5R = F 2,> : .

F c

t is straightforward to verify that 5R J , when : J ,%&000& c J &000& and F J

/. $hen the cost is reduced to ,&000& the e'uili!rium level of rent+see"ing

activity is raised to . ;otice that the #redicted effect of a cost reduction is that

the total amount s#ent on rent+see"ing activity is unchanged.

?inally& consider the total cost of all rent+see"ing activity& which will !e measured

!y the #roduct of the num!er of contenders& the effort #er contender& and the cost

#er unit of effort: F5Rc. t follows from =2.-> that this total cost is: = F +,> F

times the #ri<e value& : . hus the amount of the rent that is dissi#ated is a

fraction& = F +,> F & which is an increasing function of the num!er of contenders.

$ith two contenders& half of the value is dissi#ated in a ;ash e'uili!rium& and

this fraction increases towards , as the num!er of contenders gets large. n

general& rent dissipation is measured as the ratio of total e#enditures onrentsee"ing activity !y all contenders to the value of the #ri<e.

;otice that the fraction of rent dissi#ation is #redicted to !e inde#endent of the

cost of lo!!ying effort& c. ?or eam#le& recall the #revious eam#le with four

contenders& where the reduction in c from &000 to ,&000 raised the ;ash

e'uili!rium lo!!ying effort from , to & there!y maintaining a constant total level

of e#enditures.

hese #redictions were tested with a Veconla! e#eriment run on a single class

with a 22 design with high and low lo!!ying costs& and with high and low

num!ers of com#etitors. here were 20 rounds =5 for each treatment>. he #ri<e

value was ,%&000 in all rounds& and each #erson !egan with an initial cash

!alance of ,00&000. Ine #erson was selected at random e5 post to receive asmall #ercentage of earnings. he treatments involved a #er+unit cost of either

500 or ,&000& and either 2 or / contenders. he four treatment com!inations

are shown in the two columns on the left side of a!le 2.,. he ;ash #rediction

for a contender s lo!!ying effort& as calculated from =2.->& is shown in the third

column. 4ulti#lying this !y the num!er of contenders and then !y the cost #er

unit of effort yields the #redicted total cost of rent see"ing activities& which is

shown in the fourth column. hese total costs are ,2&000 with F J / =to# two

rows> and &000 for F J 2 =!ottom two rows>. hus the ;ash #rediction is that

an increase in the num!er of contenders ma"es the situation more com#etitive and

increases rent+see"ing activity and total costs. ?or each fied level of F & however&a reduction in the #er+unit lo!!ying cost will dou!le the effort& leaving the

#redicted total cost unchanged.



a!le 2.,. A Classroom 3ent+see"ing *#eriment with a 8ri<e Value of ,%&000

;um!er of

Contenders:

8er+unit Cost:

;ash 8redictions: I!served Averages:

F C 5R total cost = Fc5R> 5 total cost

= Fc5>

/ ,&000 ,2&000 % 2/&000/ 500 % ,2&000 ,%&0002 ,&000 / &000 % ,2&000

2 500 &000 ,0 ,0&000

he ;ash #redictions can !e com#ared with data from the classroom e#eriment&

shown in the final two columns. he average effort decisions have !een rounded

off to the nearest integer& to ma"e the ta!le easier to read. 6everal conclusions are

a##arent:

,> he total costs of rent see"ing activity are significant& and are greater than

50K of the #ri<e value in all treatments.2> he total costs of rent+see"ing activity are greater than the ;ash

#redictions.

> An increase in the num!er of contenders tends to increase the total costs of

rent see"ing.

/> A decrease in #er+unit effort costs does raise efforts& !ut not enough to

offset the cost reduction.


3ent see"ing models have !een used in the studying of #olitical lo!!ying

=Hillman and 6amet& ,->& which is a natural a##lication !ecause lo!!yinge#enditures often involve real resources. n fact& any ty#e of contest for a single

#ri<e may have similar strategic elements& e.g. #olitical cam#aigns or research and

develo#ment contests. here is a large and growing literature that uses the

rentsee"ing #aradigm to study non+mar"et allocation activities.

here have !een a num!er of research e#eriments using the #ayoff structure in

e'uation =2.,>. 4illner and 8ratt =,> re#ort that rent dissi#ation was

significantly higher than the ;ash #rediction. n a com#anion #a#er& 4illner and

8ratt =,,> used a lottery choice decision to se#arate #eo#le into grou#s

according to their ris" aversion. rou#s with more ris"+averse individuals tended

to e#end more on rent+see"ing activity.

Anderson& oeree& and Holt =,> #rovide a theoretical analysis of a model of

rent see"ing where the #ri<e goes to the #erson with the highest effort.

his is li"e a race or auction& where it only ta"es a small advantage to o!tain a

sure win =unli"e the model contained in this cha#ter where a higher effort



increases the #ro!a!ility of o!taining the #ri<e !ut does not ensure a win>. he

Anderson& oeree and Holt model is !ased on logit stochastic res#onse functions

that were introduced in cha#ter ,,. his incor#oration of randomness in the game

is used to derive a theoretical #rediction that rent+see"ing efforts will !e a!ove the

levels #redicted in a ;ash e'uili!rium.

<$estions

,. 6u##ose that the cost #er unit of effort is reduced from &000 to ,&000&

with four com#etitors and a #ri<e of ,%&000. 6how that a common effort

of is a ;ash e'uili!rium for the rent see"ing game with #ayoffs in

e'uation =2.,>. Hint: Chec" to !e sure that a unilateral deviation to an

effort of 2 or / would not !e #rofita!le& given that the other three #eo#le

"ee# choosing . If course& to !e com#lete& all #ossi!le deviations would

have to !e considered& !ut you can get the idea !y considering deviations

to 2 and /. An alternative is to use calculus.

2. ;ow su##ose that the effort cost is reduced to 500. 6how that a commoneffort of % is a ;ash e'uili!rium =consider unilateral deviations to / and

->.

. Calculate the total amount of money s#ent in rent+see"ing activities for the

setu# in 'uestion ,. (id the reduction in the #er+unit =marginal> cost of

rent see"ing from ,&000 to 500 reduce the #redicted tota# cost of rent

see"ing activity.

/. ;ow su##ose that the num!er of com#etitors for the setu# in 'uestion , is

reduced from / to 2. he #er+unit effort cost is ,000 and the #ri<e value

is ,%&000. 6how that a common effort of / is a ;ash e'uili!rium =for

sim#licity& only consider deviations to and 5& or use calculus>.

5. (id the reduction in com#etitors from / to 2 in 'uestion / reduce the #redicted etent of rent dissi#ationN =3ent dissi#ation is the total cost of

all rent+see"ing activity as a #ro#ortion of the #ri<e value.>

8art V. nformation and 1earning

nformation s#ecific to individuals is often uno!served !y others. 6uchinformation may !e conveyed at a cost& !ut misre#resentation and strategic

nonrevelation is sometimes a #ro!lem. nformational asymmetries yield rich

economic models that may have multi#le e'uili!ria and unusual #atterns of



!ehavior. Ine eam#le considered in an earlier #art is the case where individuals

may decide to rely on information inferred from others decisions and follow the

crowd. A diversity of investor information is also im#ortant in finance& where an

issue may !e the etent to which asset #rices aggregate diverse !its of

information. 4any other interesting issues may arise& since mar"et #rocesses may

fail when information cannot !e #riced.

Cha#ter 0 #rovides a closer loo" at eactly how new information is used

to revise #ro!a!ilistic !eliefs =7ayes rule>. Cha#ter , also #ertains to a se'uence

of decisions& !ut in this case& they are made !y different #eo#le. hose later in the

se'uence are a!le to learn from others earlier decisions. his raises the #ossi!ility

of a ty#e of !andwagon effect& which is "nown as an information cascade.

6tatistical discrimination can arise if the em#loyer uses #ast correlations

!etween the fied signal and wor"er #roductivity to ma"e 9o! assignment

decisions. 7iased e#ectations may !e confirmed in e'uili!rium as wor"ers of

one ty#e !ecome discouraged and sto# investing. he statistical discrimination

model #resented in Cha#ter 2 is im#lemented in the we!game 6( !y assigning a

color =#ur#le or green> to each #erson in the wor"er role and letting them ma"e anuno!served investment decision. hose in the em#loyer role must then ma"e 9o!

assignments on the !asis of an im#erfect #roductivity test. *#erience+!ased

discrimination is most li"ely to show u# when the test is inconclusive. his

eercise #rovides an ecellent o##ortunity for informed and non+emotional class

discussions of issues li"e race and gender discrimination. he e'uili!rium

analysis in this cha#ter ma"es use of 7ayes rule.



Cha#ter 0. 7ayes 3ule

Any careful study of human !ehavior has to deal with the issue of how #eo#le

learn from new information. ?or eam#le& an em#loyer may have !eliefs a!out a

#ros#ective em#loyee s value to the firm !ased on #rior e#erience with those of similar !ac"grounds. hen new information is #rovided !y a 9o! interview or a

test& and the issue is how to com!ine initial = #rior > !eliefs with new information

to form final = #osterior > !eliefs. his cha#ter uses a sim#le !all counting ruleof

thum! or heuristic to e#lain 7ayes rule& which is a mathematical formula for

forming #osterior !eliefs. Actual !ehavior will never !e #erfectly descri!ed !y a

mathematical formula& and some ty#ical #atterns of deviation are discussed. he

a##endi contains instructions for a game in which the #artici#ant sees draws of

colored !alls from one of two cu#s and has to form an o#inion a!out which of the

two cu#s is !eing used. his e#eriment can also !e run with the Veconla! 7ayes

3ule 8ro!a!ility *licitation ame& which is much 'uic"er than hand+run versions&

and it #rovides automatic calculations and a gra#h of average elicited #ro!a!ilitiesas a function of the 7ayes #rediction& under alternative scenarios =all data& data for

s#ecific #artici#ants& data from one draw or signal& etc.>.

I. Introd$ction

6u##ose that you have 9ust received a test result indicating that you have a

rare disease. Unfortunately& the disease is life+threatening& !ut you have some

ho#e !ecause the test is ca#a!le of #roducing false #ositives& and the disease is

rare. Mour doctor tells you that the test is fairly accurate& with a false #ositive rateof only , #ercent. he !ase rate or incidence of the disease for those in your

socio+economic grou# is only one #er ,0&000. $hat are your chances of having

the diseaseN =8lease write down a guess& so that you do not forget.>

Confronted with this #ro!lem& most #eo#le conclude will that it is more

li"ely than not that the #erson actually has the disease& !ut such a guess would !e

seriously incorrect. he , #ercent false #ositive rate means that testing ,0&000

randomly selected #eo#le will generate a!out ,00 #ositive results =,K>& !ut on

average only one #erson out of ,0&000 actually has the disease. hus the chances

of having the disease are only a!out one in a hundred& even after ou have tested

positive ;ith a test that is correct %% times out of 1. his eam#le illustrates the

dramatic effect of #rior information a!out the !ase rate of some attri!ute in the #o#ulation. his eam#le also indicates how one might set u# a sim#le

fre'uency+!ased counting rule that will #rovide a##roimately correct #ro!a!ility

calculations:



• 6elect a hy#othetical sam#le of #eo#le =say ,0&000>.

• Use the !ase rate to determine how many of those& on average&

would have the disease !y multi#lying the !ase rate and the sam#le

si<e =e.g. 0.000, times ,0&000 J ,>.

• ;et& net calculate the e#ected num!er of #ositive test results

that would come from someone who has the disease. =n the

eam#le& one #erson has the disease and the test would #ic" this

u#& so we have , #ositive from an infected #erson.>

• 6u!tract the num!er of infected #eo#le from the sam#le si<e to

determine the num!er that are not infected =,0&000 + , J &>.

• *stimate the num!er of #ositive test results that would come from

#eo#le who are not infected. =he false #ositive rate is 0.0, in the

eam#le& so the num!er of #ositives from #eo#le who are not

infected is .,00& which is a##roimately ,00.>

• Calculate the chances of having the disease given the #ositive test

result !y ta"ing the ratio of the num!er of #ositives from infected

#eo#le =,> to the num!ers of #ositives from those who are infected=,> #lus those who are not =,00>& which is a!out ,,0,& or a!out ,

#ercent.

Calculations such as those given a!ove are an eam#le of the use of

Baes9 'u#e. his cha#ter introduces this rule& which is an o#timal #rocedure for

using #rior information& li"e a #o#ulation !ase rate& together with new

information& li"e a test result. As the incorrect answers to the disease 'uestion

may suggest& #eo#les decisions and inferences in such situations may !e seriously

!iased. A related issue is the etent to which #eo#le are a!le to correct for

#otential !iases in mar"et situations where the incentives are high and one is a!le

to learn from #ast e#erience. And in some =!ut not all> situations& those who donot correct for !iases lose !usiness to those who do.

$hen ac'uiring new information& it is useful to thin" a!out the difference

!etween the initial !eliefs& the information o!tained& and the new !eliefs after

seeing the information. f the initial =#rior> !eliefs are very strongly held& then the

new =#osterior> !eliefs are not li"ely to change very much& unless the information

is very good. ?or eam#le& #assing a lie detector test will not eliminate sus#icion

if the investigator is almost #ositive that the sus#ect is guilty. In the other hand&

very good information may overwhelm #rior !eliefs& as would ha##en with the

discovery of (;A evidence that clears a #erson who has already !een convicted.

I!viously& the final =#osterior> !eliefs will de#end on the relative 'uality of the #rior information and on the relia!ility of the test result. 7ayes rule #rovides a

mathematical way of handling diverse sources of information. he 7ayesian

#ers#ective is useful !ecause it dictates how to evaluate different sources of

information& !ased on their relia!ility. his #ers#ective may even !e valua!le



when it is undesira!le or illegal to use some #rior information& e.g. in the use of

demogra#hic information a!out crime rates in a 9ury trial. n such cases& "nowing

how such #rior information would !e used =!y a 7ayesian> may ma"e it easier to

guard against a !ias derived from such information. ?inally& 7ayesian

calculations #rovide a !enchmar" or !aseline from which !iases can !e measured.

II. - Simp%e "(amp%e and a Co$nting 7e$ristic @-nderson and 7o%t,

'44)aA

he sim#lest informational #ro!lem is deciding which of two #ossi!le situations

or states of nature is relevant& e.g. guilty or innocent& infected or not& defective or

not& etc. o !e s#ecific& su##ose that there are two cu#s or urns that contain

e'ually si<ed Am!er =a> and 7lue =b> mar!les& as shown in a!le 0.,. Cu# A has

two Am!ers and one 7lue& and Cu# 7 has one Am!er and two 7lues. $e will use

the fli# of a fair coin to choose one of these cu#s. he cu# selection is hidden& so

your #rior information is that each cu# is e'ually li"ely. hen you are allowed to

see several draws of mar!les from the selected cu#. *ach time a draw is madeand shown& it is then returned to the cu#& so the draws are with re#lacement from

a cu# with contents that do not change.

a!le 0.,. A wo Cu# *am#le

Cu# A Cu# 7

a& a& b a& b& b

6u##ose the first draw is Am!er and you are as"ed to re#ort the #ro!a!ility that

cu# A is !eing used. An answer of ,2 is disa##ointingly common& sometimes

even among graduate students& since it can !e 9ustified !y the argument that each

urn was e'ually li"ely to !e selected. 7ut if each cu# was e'ually li"ely

!eforehand& what was learned from the drawN Another commonly re#orted

#ro!a!ility for cu# A following an Am!er draw is ,& since this cu# has a one

half chance of !eing used& and if used& there is a 2 chance of drawing an Am!er.

hen we multi#ly ,2 and 2 to get ,. his , #ro!a!ility is clearly wrong&

since the cu#s were e'ually li"ely e5 ante& and the Am!er draw is more li"ely

when cu# A is used& so the #ro!a!ility of cu# A should !e greater than ,2 after

seeing an Am!er draw. Another #ro!lem with this answer is that analogous

reasoning re'uires the #ro!a!ility of cu# 7 to !e ,2 times ,& or ,%. his yieldsan inconsistency& since if the #ro!a!ility of cu# A is , and the #ro!a!ility of cu#

7 is ,%& where does the rest of the #ro!a!ility goN hese #ro!a!ilities =, and

,%> sum to ,2& so we should dou!le them =to 2 and ,>& which are the correct

#ro!a!ilities for cu#s A and 7 after the draw of one Am!er mar!le. his "ind of



scaling u# of #ro!a!ilities will !e seen later as a #art of the mathematical formula

for 7ayes rule.

A close loo" at where the Am!er mar!les are in a!le 0., ma"es it clear why the

#ro!a!ility of cu# A !eing used is 2 after the draw of an Am!er mar!le. here

are two a mar!les on the left and one on the right. $hy does this suggest that the

right answer is 2N All si mar!les are e'ually li"ely to !e drawn !efore the die

is thrown to select one of the cu#s. 6o& no Am!er mar!le is more li"ely to !e

chosen than any other& and 2 of the Am!er mar!les are in cu# A. t follows that

the #osterior #ro!a!ility of cu# A given an Am!er draw is 2.

he calculations in the #revious #aragra#hs are a s#ecial case of 7ayes) rule with

e'ual #rior #ro!a!ilities for each cu#. ;ow consider what can !e done when the

#ro!a!ilities are not e'ual. n #articular& su##ose that the first mar!le drawn

=Am!er> is returned to the cu#& and the decision ma"er is told that a second draw

is to !e made from the same cu# originally selected !y the throw of the die.

Having already seen an Am!er& the #erson)s !eliefs !efore the second draw are

that the #ro!a!ility of cu# A is 2 and the #ro!a!ility of cu# 7 is ,& so the

#erson thin"s that it is twice as li"ely that cu# A is !eing used after o!serving onea draw. he net ste# is to figure out how the #revious #aragra#h s method of

counting !alls =when the two cu#s were initially e'ually li"ely> can !e modified

for the new situation where one cu# is twice as li"ely as the other one. n order to

create a situation that corres#onds to the new !eliefs& we want to somehow get

twice as many #ossi!le draws as coming from the cu# that is twice as li"ely.

*ven though the #hysical num!er of mar!les in each cu# has not changed& we can

re#resent these !eliefs !y thin"ing of cu# A as having twice as many mar!les as

cu# 7& ;ith each mar+#e in either cup having the same chance of +eing dra;n.

hese #osterior !eliefs are re#resented in a!le 0.2& where the #ro#ortions of

Am!er and 7lue mar!les are the same as they were in cu#s A and 7 res#ectively.

*ven though the #hysical num!er of mar!les is unchanged at si& the #rior corres#onds to a case in which the imagined mar!les in a!le 0.2 are num!ered

from one to nine& with one of the nine mar!les chosen randomly.

a!le 0.2. A 4ental 4odel after Ine Am!er (raw

=real mar!les in !old& imagined mar!les not !old>

Cu# A Cu# 7

a& a& b a&

a& +

a& b& b

$hen the #osterior !eliefs after an Am!er draw are re#resented in a!le 0.2& it

is clear that a 7lue on the second draw is e'ually li"ely to have come from either

urn& since each cu# contains two b mar!les. hus the #osterior #ro!a!ility for cu#



A after a 7lue on the second draw is ,2. his is consistent with intuition !ased

on symmetry& since the #rior #ro!a!ilities for each urn were initially ,2& and the

draws of an Am!er =first> and a 7lue =second> are !alanced. A mied sam#le in

the o##osite order =7lue first& then Am!er> would& of course& have the same effect.

6u##ose instead that the two draws were Am!er& with the two cu#s !eing e'ually

li"ely to !e used e5 ante. As !efore& the #osterior !elief after the first Am!er

draw can !e re#resented !y the cu#s in a!le 0.2. 6ince four of the five =real or

imagined> Am!er mar!les are in cu# A& the #osterior #ro!a!ility of cu# A after

seeing a second Am!er draw is /5. After two Am!er draws& cu# A is& therefore&

four times as li"ely as cu# 7& since /5 is four times as large as ,5. o re#resent

these #osterior !eliefs in terms of colored mar!les that are e'ually li"ely to !e

drawn& we need to have four times as many rows on the Cu# A side of a!le 0.2

as there are on the Cu# 7 side. hus we would need to add two imagined more

rows of three mar!les under cu# A in a!le 0.2& holding the #ro#ortions of

Am!er and 7lue mar!les fied. (oing this would #rovide the re#resentation in

a!le 0.. o test your understanding& you might consider the #ro!a!ility of cu#

A after seeing two Am!ers and a 7lue =eercise , at the end of the cha#ter>. U#to this #oint& the analysis has !een intuitive& !ut it is now time to !e a little more

analytical.

a!le 0.. A 4ental 4odel of the 6ituation

After wo Am!er (raws

Cu# A Cu# 7

a& a& b a&

a& + a,

a, + a,

a, +

a& b& b

III. Re%ating the Co$nting 7e$ristic to Ba!es R$%e

o ma"e the connection with 7ayes rule& we will need a little notation. 6u##ose

there are F mar!les in each cu#. he mar!les will !e Am!er or 7lue& and we will

use the letter C to re#resent a s#ecific color& so C can !e either Am!er or 7lue.

$hat we want to "now is the #ro!a!ility of cu# A given the draw of a mar!le of

color C. $hen C is Am!er and the contents are shown in a!le0.,&we already "now the answer =2>& !ut our goal here is to find a general

formula for the #ro!a!ility of cu# A given a draw of a color C mar!le. his

#ro!a!ility is denoted !y 8r=A]C>& which reads the #ro!a!ility of A given C. his



formula should !e general enough to allow for different #ro#ortions of colored

mar!les& and for differences in the #rior #ro!a!ility of each cu#.

Consider 8r=C]A>& which reverses the order of the A and the C from the

order used in the #revious #aragra#h. ;ote that 8r=C]A> is read as the #ro!a!ility

of color C given cu# A. hus 8r=C]A> denotes the fraction of mar!les in cu# A

that are of color C& where C is either Am!er or 7lue. 6imilarly& 8=C]7> is the

fraction of mar!les in cu# 7 that are of color C. ?or eam#le& if there are ,0

mar!les in cu# A and if 8r=C]A> J 0.%& then there must !e si mar!les of color C

in the cu# =calculated as 0.% times ,0>. n general& there are a total of 8=C]A> F

mar!les of color C in cu# A& and there are 8=C]7> F mar!les of color C in cu# 7.

f each cu# is e'ually li"ely to !e selected& then each of the 2 F mar!les in the two

cu#s is e'ually li"ely to !e drawn e5 ante =!efore the cu# is selected>. 6u##ose

the mar!le drawn is of color C. he #osterior #ro!a!ility that a mar!le of color C

was drawn from cu# A& denoted 8r=A]C>& is 9ust the ratio of the num!er of color C

mar!les in cu# A to the total num!er of mar!les of this color in !oth cu#s:

;um!er of Color C 4ar!les in Cu# A=0.,> 8r=A]C> &

;um!er of Color C 4ar!les in 7oth Cu#s

which can !e e#ressed:

8r=C]A> F

=0.2> 8r=A ]C> .

8r=C]A> F O 83=C]7> F

t is worth em#hasi<ing that this formula is only valid for the case of e'ual #rior

#ro!a!ilities and e'ual num!ers of mar!les in each urn. ;othing is changed if wedivide !oth numerator and denominator of the right side on =0.2> !y 2 F & which is

the total num!er of !alls in !oth cu#s& which yields a formula for calculating the

#osterior when the #riors are ,2:

8r=C]A>=,2>

=0.> 8r=A ] C> =for #riors of ,2>.

8r=C]A>=,2> O 83=C]7>=,2>

A #erson who has seen one or more draws may not have #rior #ro!a!ilities of

,2& so this formula must !e generali<ed. his involves re#lacing the =,2> terms

on the right side of the with the new #rior #ro!a!ilities& denoted 8r=A> and 8r=7>.his is 7ayes) rule:

8r=C]A> 8r=A>



123

21 11 1 3

=0./> 8r=A ] C> =7ayes) rule>.

8r=C]A> 8r=A> O 83=C]7> 8r=7>

?or the #revious eam#le with e'ual #riors& 8r=A> J ,2& 8r=a]A> J 2& and 8r=a]

7> J ,& so e'uation =0./> im#lies that the #osterior #ro!a!ility following anAm!er draw is:

2 ,

8r= ! a] > 2

.

2 2 2

6imilarly& the #ro!a!ility of cu# 7 is calculated: 8r=7]a> J =,%>=2% O ,%> J ,.

;otice that the denominators in !oth of the #revious calculations are ,2& so

dividing !y ,2 scales u# the numerator !y a factor of 2& which ma"es the

#ro!a!ilities add u# to one.

Having arrived at the 7ayes) rule formula =for two events> as it a##ears in the

tet!oo"s& it is im#ortant to #oint out that the argument was intuitive !ut not

rigorous. t hel#s& therefore& to relate the general formula for 7ayes) rule !ac" to

the counting heuristic. 3ecall that the two cu#s were initially e'ually li"ely& and

the first time we saw an a draw& the #ro!a!ility of cu# A was 2. so the ratio

8r=A>8r=7> was 2. hen we too" the F !alls in cu# A and imagined that there

were twice as many& i.e. 2 F !alls in cu# A = F real !alls and F imagined !alls>.

After seeing two a draws& the #ro!a!ility of Cu# A was /5& or four times as great

as the ,5 #ro!a!ility of Cu# 7& so we imagined that there were / F !alls in Cu# A

instead of the original F !alls. How can this a##roach !e generali<ed for the casewhere 8r=A>8r=7> is something other than 2 or /N I!viously& we calculate the

ratio 8r=A>8r=7> and imagine that there are F times 8r=A>8r=7> !alls in Cu# A&

holding the original #ro#ortions constant. o relate the 7ayes rule formula in

=0./> to this counting heuristic& we need to figure out how to get terms involving

; times 8r=A>8r=7> terms into =0./>. his can !e done !y dividing !oth

numerator and denominator of =0./> !y 8r=7> F & to o!tain:

8r=C]A> 8r=A>8r=7> F

=0.5> 8r=A ]C> 8r=A> .

8r=C]A> 8r=7> F O 83=C]7> F



f there are F mar!les in each cu#& this e'uation shows that the !asic 7ayes) rule

formula is e'uivalent to imagining that the F mar!les in cu# A are increased or

decreased to a num!er M & which is F times the #rior odds ratio& 8r=A>8r=7>.

a"e the re#resentation in a!le 0.2 for eam#le& where the #rior for cu# A is 2

after seeing an a draw. hen 8r=A> is now 2& 8r=7> is ,& and the odds ratio is

=2>=,> J 2. $ith F J mar!les in each cu#& the odds ratio times F is 2 times

& or %& which re#laces the odds ratio times F in the numerator and the left side of

the denominator of =0.5>. his is as if we imagine that there are % mar!les in

cu# A and only in cu# 7& the corres#onding to the F term on the right side of

the denominator in =0.5>.

o summari<e& if there is a #rior #ro!a!ility of ,2 that each cu# is used and if the

cu#s contain e'ual num!ers of colored mar!les& then the #osterior #ro!a!ilities

can !e calculated as ratios of num!ers of mar!les of the color drawn& as in

e'uation =0.,>. f the mar!le drawn is of color C& then the #osterior that the draw

was from cu# A is the num!er of color C mar!les in cu# A divided !y the total

num!er of color C mar!les in !oth cu#s. $hen the #rior #ro!a!ilities or num!ers

of mar!les in the cu#s are une'ual& then the ,2 terms in =0.> are re#laced !y the #rior #ro!a!ilities& as in 7ayes) rule =0./>. ?inally& the 7ayes) rule formula is

e'uivalent to imagining that the num!er ; of mar!les in cu# A is changed !y a

factor 8r=A>8r=7>& holding the #ro#ortion of color C mar!les fied at 8r=C]A>&

with the counting heuristic that each of the actual and imagined mar!les is e'ually

li"ely to !e chosen.

I. "(perimenta% Res$%ts

;o!ody would e#ect that something so noisy as the formation of !eliefs would

adhere strictly to a mathematical formula& and e#eriments have !een directed

towards finding the nature of systematic !iases. he disease eam#le mentionedearlier suggests that& in some contets& #eo#le may underweight #rior information

!ased on #o#ulation !ase rates.

his +ase rate +ias was the motivation !ehind some e#eriments re#orted

!y Dahneman and vers"y =,->& who gave su!9ects lists of !rief descri#tions of

#eo#le who were either lawyers or engineers. he su!9ects were told that the

descri#tions had !een selected at random from a sam#le that contained -0 #ercent

lawyers and 0 #ercent engineers. 6u!9ects were as"ed to re#ort the chances out

of ,00 that the descri#tion #ertained to a lawyer. A second grou# was given some

of the same descri#tions& !ut with the information that the descri#tions had !een

selected from a sam#le that contained 0 #ercent lawyers and -0 #ercentengineers. 3es#ondents had no trou!le with descri#tions that o!viously descri!ed

one occu#ation or another& !ut some were intentionally neutral with #hrases li"e

he is highly motivated or will !e successful in his career. he modal res#onse for

such neutral descri#tions involved #ro!a!ilities of near a half& regardless of the



res#ondent s treatment grou#. his !ehavior is insensitive to the #rior information

a!out the #ro#ortions of each occu#ation& a ty#e of !ase rate !ias.

rether =,0> #ointed out several #otential #rocedural #ro!lems with the

Dahneman and vers"y e#eriment. here was dece#tion to the etent that the

descri#tions had !een made u#& and even if #eo#le !ought into the contet& they

would have no incentive to thin" a!out the #ro!lem carefully. 4oreover& the

information conveyed in the descri#tions is hard to evaluate in terms of factors

that com#rise 7ayes rule formula. n other words& it hard to determine an

a##ro#riate guess a!out the #ro!a!ility of a #articular descri#tion conditional on

the occu#ation. rether !ased his e#eriments on cu#s with two ty#es of !alls as

descri!ed. Ine of the !iases that he considered is "nown as representativeness

+ias. n the two cu# eam#le discussed earlier& a sam#le of three draws that

yields two Am!ers and one 7lue has the same #ro#ortions as cu# A& and in this

sense the sam#le loo"s re#resentative of cu# A. $e saw that the #ro!a!ility of

cu# A after such a sam#le would !e 2& and a #erson who re#orts a higher

#ro!a!ility& say 0 #ercent& may !e doing so due to re#resentativeness !ias.

;otice that a sam#le of two Am!ers and one 7lue ma"es cu# A more li"ely. f you as" someone which cu# is more li"ely& an answer of A cannot distinguish

!etween 7ayesian !ehavior and a strong re#resentativeness !ias& which also

favors cu# A. rether cleverly got around this #ro!lem !y introducing some

asymmetries which ma"e it #ossi!le for re#resentativeness to indicate a cu# that

is less li"ely given 7ayes rule. ?or eam#le& if you lower the #rior #ro!a!ility of

cu# A to ,& then the #rior odds ratio on the right side of =0.5> will !e:

8r=A>8r=7> J =,>=-> J ,-& which will ma"e 8r=A]C> lower for each #attern

of draws re#resented !y C. n this manner& a sam#le of two Am!ers and one 7lue

would loo" li"e the contents of cu# A& !ut if the #rior #ro!a!ility of A is small

enough& the 7ayesian #ro!a!ility of cu# A would !e less than one half. n this

manner& re#resentativeness and 7ayes rule would have differing #redictions whena #erson is as"ed which cu# is more li"ely.

A !inary choice 'uestion a!out which cu# is more li"ely ma"es it easy to #rovide

incentives: sim#ly offer a cash #ri<e if the cu# actually used turns out to !e the

one the #erson said is more li"ely. his is the #rocedure that rether used& with a

,5 #ri<e for a correct #rediction and a 5 #ri<e otherwise. $hen

re#resentativeness and 7ayes rule gave different #redictions& su!9ects tended to

follow the 7ayesian #rediction more often than not& !ut a lot less often than when

the two criteria matched& as with the symmetric eam#le discussed in this cha#ter.

$hen re#resentativeness and 7ayesian calculations indicated the same answer&

su!9ects tended to give the correct answer a!out 0 #ercent of the time =with some

variation de#ending on the s#ecific sam#le>. his #ercentage fell to a!out %0

#ercent when re#resentativeness and 7ayesian calculations suggested different

answers.



. Ba!es9 R$%e 6ith "%icited Pro&a&i%ities

6ometimes it is useful to as" su!9ects to re#ort a #ro!a!ility instead of 9ust saying

which event is more li"ely. his can !e #hrased as a 'uestion a!out the chances

out of ,00 that the cu# used is A. he issue here is how to #rovide incentives for

#eo#le to thin" carefully. he instructions in the a##endi #rovide one a##roach&

which is com#licated& !ut which is !ased on a sim#le idea. 6u##ose that you send

your friends to a fruit stand& and they as" you whether you #refer red or yellow

tomatoes in case !oth are availa!le. Mou would have no incentive to lie a!out

your #reference& since telling the truth allows your friends to ma"e the !est

decision on your !ehalf. his section descri!es a method of eliciting #ro!a!ilities

that is !ased on this intuition. 8ro!a!ility elicitation is useful when we need

s#ecific #ro!a!ility num!ers to evaluate theoretical #redictions.

n #articular& su##ose that you have seen some draws =say Am!er and

7lue> and have concluded that the #ro!a!ility of cu# A is 0.5. f you are #romised

a ,&000 #ayment if cu# A is really used& then you essentially have a cu# A lottery

that #ays ,&000 with #ro!a!ility one half. he idea is to as" someone for the

#ro!a!ility =in chances out of ,00> that cu# A is used& and to give them theincentive to tell the truth. his incentive will !e #rovided !y having the #erson

running the e#eriment ma"e a choice for the su!9ect. n this contet& the

e#erimenter is li"e the friend going to the fruit standB the su!9ect needs to tell the

truth a!out their #references so that the e#erimenter will ma"e the !est choice on

the su!9ect s !ehalf. n order to set u# the right incentives to tell the truth& we will

have the e#erimenter choose !etween that cu# A lottery and another one that is

constructed randomly& !y using throws of ,0+sided dice to get a num!er& F &

!etween 0 and ,00& in a manner which ensures that this dice lottery #ays ,&000

with chances F out of ,00. f the #ro!a!ility of Cu# A is ,2& this dice lottery is

#referred if F P 50& and the cu# A lottery is #referred if F Q 50. he su!9ect

should answer that the chances of Cu# A are 50 out of ,00 so that thee#erimenter can ma"e the right choice. his is really li"e the red and yellow

tomato eam#le discussed a!ove& the e#erimenter is choosing !etween two

lotteries on the su!9ect s !ehalf& and therefore needs to "now the su!9ect s true

value of the cu# A lottery to ma"e the right decision.

o convince yourself that the su!9ect is motivated to tell the truth in this

situation& consider what might ha##en otherwise. 6u##ose that the su!9ect

incorrectly re#orts that the chances of cu# A are -5 out of ,00& when the #erson

really !elieves that each cu# is e'ually li"ely. hus& the cu# A lottery #rovides a

one+half chance of ,&000. f the e#erimenter then throws a - and a 0& then F J

-0& and the dice lottery would yield a -0K chance of ,&000& which is a much

!etter #ros#ect than the 50+50 chance !ased on the su!9ect s actual !eliefs that

each cu# is e'ually li"ely. 7ut since the su!9ect incorrectly re#orted the chances

for cu# A to !e -5 out of ,00& the e#erimenter would re9ect the dice lottery and

!ase the su!9ect s earnings on the cu# A lottery& which gives a 20 #ercent lower



chance of winning. A symmetric argument can !e made for why it is !ad to re#ort

that the chances of cu# A are less than would !e indicated !y the su!9ects true

!eliefs ='uestion >. ?or a mathematical derivation of the result that it is o#timal

to reveal one s true #ro!a!ility& see 'uestion %.

he advantage of direct #ro!a!ility elicitation is that you can often ma"e stronger

conclusions when s#ecific num!ers are availa!le& as o##osed to the 'ualitative

data o!tained !y as"ing which event is more li"ely. he disadvantage is that the

elicitation #rocess itself is not #erfect in the sense that the measurements may

contain more noise or measurement error than we get with !inary com#arisons.

a!le 0./. *licited 8ro!a!ilities for wo 6u!9ects in a 7ayes 3ule *#eriment

Cu# A: a& a& b Cu# 7: a& b& b

6u!9ect , 6u!9ect 2

3ound (raw

=7ayes>

*licited

8ro!a!ility

3ound (raw *licited

8ro!a!ility2, ;one

=0.50> 0./

2, ;one

=0.50> 0.5022 a

=0.%-> 0.%5

22 b

=0.> 0.0

2 bb

=0.20> 0.,

2 ba

=0.50> 0.%0

2/ bab

=0.> +.*

2/ aba

=0.%-> 0.-0

25 a

=0.%-> 0.%5

25 a

=0.%-> 0.%52% ab

=0.50> 0./

2% ab

=0.50> 0.0

2- bba

=0.> 0.

2- aab

=0.%-> +.2+

a!le 0./ shows some elicited #ro!a!ilities for a research e#eriment conducted

!y the author using University of Virginia su!9ects& and #ri<e amounts of ,.00

instead of ,&000. All earnings were #aid in cash. he two cu#s& A and 7& each

contained three mar!les with the contents as shown in a!le 0.,. he

e#eriment consisted of three #arts. he first #art was done largely to ac'uaint

#eo#le with the #rocedures& which are admittedly com#licated. hen there were

,0 rounds with asymmetric #ro!a!ilities =a two+thirds chance of using cu# A> and

,0 rounds with symmetric #ro!a!ilities =a one+half chance of using cu# A>. he

order of the symmetric and asymmetric treatments was reversed with different

grou#s of #eo#le.



he information and decisions for su!9ects , and 2 are shown in the ta!le&

for rounds 2,+2- where the #rior #ro!a!ility was ,2 for each cu#. ?irst consider

su!9ect , on the left. n round 2,& there were no draws& and the ;one =0.50> in the

(raw column indicates that the correct 7ayesian #ro!a!ility of cu# A is 0.50. he

su!9ect s res#onse in the *licited 8ro!a!ility column was 0./& as is also the case

for this #erson in round 2% where the draws& A7& cancelled each other out. he

other #redictions can !e derived using the !all counting heuristic or with 7ayes

rule& as descri!ed a!ove. n round 22& for eam#le& there was only one draw& a&

and the 7ayesian #ro!a!ility of cu# A is 0.%-& since two of the three a !alls from

!oth cu#s are located in cu# A. 6u!9ect , re#orts a #ro!a!ility 0.%5& which is

'uite accurate. his #erson was unusually accurate& with the largest deviation

from the theoretical #rediction !eing in round 2/& where the re#orted #ro!a!ility

is a little too low& 0.25 versus 0..

6u!9ect 2 is considera!ly less #redicta!le. he ab and ba sam#les& which leave

each cu# !eing e'ually li"ely& resulted in answers of 0.%0 and 0.0. he average

of these answers is not far off& !ut the dis#ersion is aty#ically large for such an

easy inference tas"& as com#ared with others in the sam#le. his #erson wasfairly accurate with single draws of b =round 22> and of a =round 25>& !ut the

three+draw sam#les show !ehavior that is consistent with re#resentativeness !ias.

n round 2-& for eam#le& the sam#le of aab loo"s li"e cu# A& and the elicited

#ro!a!ility of 0.0 for cu# A is much higher than the actual #ro!a!ility of 0.%-.

6u!9ect , also seems to fall #rey to this !ias in round 2/& where the sam#le& bab &

loo"s li"e cu# 7.

?igure 0., shows some aggregate results for the symmetric and asymmetric

treatments& for all 22 su!9ects in the study. he hori<ontal ais #lots the 7ayesian

#ro!a!ility of cu# A& which varies from 0.05 for a sam#le of &&&& in the

symmetric treatment =with e'ual #riors> to 0.- for a sam#le of aaaa in the

asymmetric treatment =where the #rior #ro!a!ility of cu# A was 2>. he elicited #ro!a!ilities are shown in the vertical dimension. he /5+degree dotted line

shows the 7ayes #rediction& the solid line connects the average of the elicited

#ro!a!ilities& and the dashed line connects the medians.



0

01

02

0304

05

06

07

08

09

1

:licite! Probability of +up @

ata @;era*eayes pre!ictionata e!ian

0 0., 0.2 0. 0./ 0.5 0.% 0.- 0. 0. ,

7ayesian 8rediction

0ig$re /+.'. "%icited Pro&a&i%ities ers$s Ba!es Predictions @So$rce: 7o%t, ++A

n the aggregate& 7ayes rule does 'uite well. A slight u#ward !ias on the left side

of the figure and a slight downward !ias on the right side might #ossi!ly !e due to

the fact that there is more room for random error in the u#ward direction on the

left and in the downward direction on the right. his con9ecture is motivated !y

the o!servation that the medians =solid dashed line> are generally closer to

7ayesian #redictions. ?or eam#le& one #erson got confused and re#orted a

#ro!a!ility of 0.0, for cu# A after o!serving draws of aa in the symmetric

treatment& which should lead to a #osterior of 0.. =he draws were of light and

dar" mar!les& and were coded on the decision sheet as 11.> In the right side of the figure& there is more room for etreme errors in the downward direction. o

see this& consider a vertical line a!ove the #oint 0. on the hori<ontal ais of

?igure 0.,. f you imagine some #eo#le with no clue who #ut mar"s more or

less e'ually s#aced on this vertical line& more of the mar"s will !e !elow the /5+



degree dashed line& since there is more room !elow. 3andom errors of this ty#e

will tend to #ull average re#orted #ro!a!ilities down !elow the /5degee line on

the right side of the figure& and the reverse effect =u#ward !ias> would occur on

the left side. Averages are much more sensitive to etreme errors than are

medians& which may e#lain why the averages show more of a deviation from the

/5 degree dashed line.

he averages gra#hed in the figure mas" some interesting #atterns of deviation.

n the symmetric treatment& for eam#le& there are several #osterior #ro!a!ility

levels that can result from different num!ers of draws. A #osterior of 0.%- can !e

achieved !y draws of either a or aab. he average #osterior for the a draw alone

is 0.%,& which a little too low relative to the 7ayes #rediction of 0.%-& and the

re#resentativeness #attern of aab is actually closer to the mar" at 0.%%. 6imilarly&

the #osterior of 0. can !e reached !y draw #atterns of b or bba. he b #attern

yielded an average of 0./2& whereas the bba #attern resulted in a lower average of

0.,. 3e#resentativeness would im#ly that the elicited #ro!a!ility should !e lower

than what is o!served for the bba sam#le and higher than what is o!served for the

aab sam#le. t may !e after several draws& some #eo#le are ignoring the #rior information and re#orting a #ro!a!ility that matches the sam#le average& which

would !e a natural ty#e of heuristic.

I. "(tensions

t is well "nown that elicited !eliefs may deviate dramatically from 7ayes s rule

#redictions& es#ecially in etreme cases li"e the disease eam#le discussed in

section . nstruction in the mathematics of conditional #ro!a!ility calculations

may not hel# much& and such s"ills are #ro!a!ly 'uic"ly forgotten. In the other

hand& the counting heruistics& once learned& may hel# one ma"e good #ro!a!ility

assessments in other situations. ?rom a modeling #ers#ective& the main issue iswhere to !egin& i.e. whether to throw out 7ayes rule and !egin fresh with

something that has a more !ehavioral and less mathematical foundation. he

alternative to !egin !y ma"ing 7ayesian calculations and then trying to assess

#ossi!le sources of !ias.

Although there seem to !e some systematic deviations from the #redictions of

7ayes rule& there is no widely acce#ted alternative model of how information is

actually #rocessed !y #eo#le in these situations. 6ome !iases seem to !e due to

the use of heuristics& and some of the !ias may !e due to asymmetries in the way

the measurements are made. At #resent& economists tend to use 7ayes rule to

derive #redictions& although there is some renewed interest in non7ayesian

models li"e that of reinforcement learning discussed in the #revious cha#ter.

?inally& it is useful to "now that 7ayes rule can !e used to derive the #revious

cha#ter s model of !elief learning !y ma"ing a s#ecific assum#tion a!out the

nature of the #rior !eliefs. his derivation is !eyond the sco#e of this !oo" =see



(eroot& ,-0>& !ut it is not very im#ortant since most #eo#le who use these

models start ma"ing additional changes to the !elief learning model that cannot

!e 9ustified from 7ayes rule.

<$estions

,. $hat would the mental model with imagined !alls in a!le 0. loo" li"e

after seeing two Am!er draws and one 7lue drawN $hat does this modelim#ly that the #osterior #ro!a!ility for cu# A would !eN

2. Answer 'uestion , for a sam#le of three Am!er draws& and chec" your

answer using 7ayes rule. =Hint: the #ro!a!ility of getting three Am!er

draws from cu# A is =2>=2>=2> J 2-. Mou will also need to calculate

the #ro!a!ility of getting three Am!er draws from cu# 7.>

. 6u##ose we are using the elicitation scheme descri!ed in section V& and

that the su!9ect s !eliefs are that cu#s A and 7 are e'ually li"ely. 6how

why it would !e !ad for the #erson to re#ort that the chances for cu# A are

25 out of ,00.

/. n the asymmetric treatment discussed in 6ection V& the #rior #ro!a!ilityof cu# A is 2. Use the !all counting heuristic to calculate the #ro!a!ility

of cu# A after seeing a sam#le of: b. $hat is the #ro!a!ility of cu# A after

a sam#le of: aaaaN

5. wenty+one University of Virginia students in an undergraduate

*#erimental *conomics =6#ring 2002> #artici#ated in the 7ayes rule

elicitation e#eriment& using the instructions a##endi for this cha#ter =!ut

with only - rounds>. he e#eriment lasted for a!out 20 minutes& and cash

#ayments amounted to a!out +/ #er #erson& with all #ayments made in

cash& des#ite the fact that this was a classroom e#eriment. *ach cu# was

e'ually li"ely to !e used. 8artici#ants claimed that they did have time to

do mathematical calculations. he draw se'uences and the averageelicited #ro!a!ilities are shown in the ta!le !elow. A> Calculate the 7ayes

#osterior for each case& and write it in the relevant !o. 7> How would

you summari<e the deviations from 7ayesian #redictionsN s there

evidence for the re#resentativeness !ias in this contetN (iscuss !riefly.

4ean and 4edian *licited 8ro!a!ilities for Cu# A

Cu# A J ^1& 1& (_ Cu# 7 J ^1& (& (_

(raw: no

draws

1 ( (( 1( 1(( (((

4ean 0.5 0.% 0. 0.2/ 0./- 0.2 0.,

4edian

7ayes

0.5 0.%- 0. 0.2 0.50 0.0 0.,,



%. =Advanced and edious> he incentives for the method of eliciting

#ro!a!ilities discussed in 6ection V can !e evaluated with calculus. 1et P

denote the #erson s true #ro!a!ility of Cu# A& i.e. the #ro!a!ility that

re#resents their !eliefs after seeing the draws of colored mar!les. 1et '

denote the re#orted #ro!a!ility. =o sim#lify notation& !oth ' and P are

fractions !etween 0 and ,& not num!ers !etween 0 and ,00 as re'uired for

the chances out of ,00 discussion in the tet.> aA *#lain why the

following statement is true for the elicitation mechanism discussed in

6ection V: f the #erson re#orts '& then there is an ' #ro!a!ility of ending

u# with the Cu# A lottery& which #ays , with #ro!a!ility P . &A *#lain

why the following statement is true: 6imilarly& there is a ,+ ' #ro!a!ility of

ending u# with the dice lottery& which #ays , with a #ro!a!ility F ,00&

where F is the outcome of the throw of the ten+sided die twice. cA $e

"now that F ,00 is greater than '& since the dice lottery is only relevant if

its #ro!a!ility of #aying , is greater then the re#orted #ro!a!ility of Cu#

A. Use these o!servations to e#ress the e#ected #ayoff as: ,2 O P'

'2

2. dA hen show that this 'uadratic e#ression is maimi<ed when ' J P & i.e. when the re#orted #ro!a!ility e'uals the #ro!a!ility that re#resents

the #erson s !eliefs. Aint : 6ince the dice lottery is only relevant if F ,00

P ' and F ,00 Q ,& this dice lottery has an e#ected value that is halfway

!etween 3 and ,& i.e. =,O '>2. Comment$ 8ro!a!ility elicitation methods

of this ty#e are sometimes called scoring rules. he #articular method

!eing discussed is a 'uadratic scoring rule since the function !eing

maimi<ed& ,2 O P' '22& is 'uadratic.

-. =?un for a Change + his 'uestion was #rovided !y 8rofessor 7rent

Dreider.> 6u##ose that a coo" #roduces three #anca"es& one with one

!urnt side& one with two !urnt sides& and one with no !urnt sides. he

coo" throws a % sided die and chooses the #anca"e at random& with eachone having e'ual #ro!a!ility of the one !eing served. hen the coo" fli#s

the #anca"e high so that each side is e'ually li"ely to !e the one that

shows. All you "now& !esides the way the #anca"e was selected& is that

the one on your #late is showing a !urnt side on to#. $hat is the

#ro!a!ility that you have the #anca"e with only one !urnt sideN *#lain

with 7ayes rule or with the counting heuristic.

. A test for !reast cancer is given and comes !ac" #ositive. he rate of

#reviously undiagnosed !reast cancer in the female #o#ulation for her age

grou# is 0.0,. he test is accurate in the sense that it #roduces virtually no

false negative results& so if a #erson has !reast cancer& the test will always

!e #ositive. he rate of false #ositives on the test is ,& or a!out 0.,,.

$hat is the #ro!a!ility that the #atient actually has the cancerN



Cha#ter ,. nformation Cascades

6u##ose that individuals may receive different information a!out some un"nown

event& li"e whether or not a newly #atented drug will !e effective. his

information is then used in ma"ing a decision& li"e whether or not to invest in the

com#any that develo#ed the new drug. f decisions are made in se'uence& thenthe second and su!se'uent decision ma"ers can o!serve and learn from earlier

decisions. he dilemma occurs when one s #rivate information suggests a

decision that is different from what others have done !efore. An information

cascade forms when #eo#le follow the consensus decision regardless of their own

#rivate information. his game could !e im#lemented with draws from cu#s and

throws of dice& !ut the Veconla! nformation Cascade game is easier to administer

in a manner that controls for unwanted informational signals.

I. To do exactly as your neighbors do is the on%! sensi&%e r$%e.E

Conformity is a common occurrence in social situations& as the section titleim#lies =from *mily 8ost& ,2-& Cha#ter >. 8eo#le may follow others decisions

!ecause they value conformity and fear social sanctions. ?or eam#le& an

economic forecaster may #refer the ris" of !eing wrong along with others to the

ris" of having a deviant forecast that turns out to !e inaccurate. 6imilarly& Deynes

=,%> remar"ed: $orldly wisdom teaches that it is !etter for re#utation to fail

conventionally than to succeed unconventionally. 6ome e#erimenters have

suggested that there is an irrational #reference for the current state of affairs& i.e. a

status Huo +ias. n some hy#othetical choice e#eriments& 6amuelson and

Tec"hauser =,> showed su!9ects two #ortfolios that could !e used to invest

money inherited from a rich uncle. here was a strong #reference for the

#ortfolio that was indicated to !e the current investment #rofile& and a switch inthe status Huo designation caused a tendency to switch in #reference =in a

!etween+su!9ects design>.

n some situations& #eo#le may !e tem#ted to follow others decisions

!ecause of the !elief that there is some wisdom or e#erience im#licit in an

esta!lished #attern. 4ay!e su!9ects in the status Huo !ias e#eriments concluded

that the deceased uncle was wealthy !ecause he had selected a good #ortfolio.

he #ossi!le effects of collective wisdom may !e am#lified in a large grou#. ?or

eam#le& someone may #refer to !uy a Honda or oyota sedan thin"ing that the

large mar"et shares for those models signal a lot of customer satisfaction. his

raises the #ossi!ility that a choice #attern started !y a few individuals may set a

#recedent that results in a string of incorrect decisions. At any given moment& for eam#le& there are certain classes of stoc"s that are to !e considered good

investments& and herd effects can !e am#lified !y efforts of some to antici#ate



where the net fads will lead. Anyone who #urchased a wide range of tech stoc"s

several years ago can attest to the dangers of following the !ulls.

A #articularly interesting ty#e of !andwagon effect can develo# when individuals

ma"e decisions in a se'uence and can o!serve others #rior decisions. ?or

eam#le& su##ose that a #erson is a##lying for a 9o! in an industry with a few

em#loyers who "now each other well. f the a##licant ma"es a !ad im#ression in

the first cou#le of interviews and is not hired& then the third em#loyer who is

a##roached may hesitate even if the a##licant ma"es a good im#ression on the

third try. his third em#loyer may reason: anyone may have an off day& !ut

wonder what the other two #eo#le saw that missed. f the 9oint information

im#lied !y the two #revious decisions is deemed to !e more informative than one

s own information& it may !e rational to follow the #attern set !y others decisions&

in s#ite of contradictory evidence. A chain reaction started in this manner may

ta"e on a life of its own as su!se'uent em#loyers hesitate even more. his is why

first im#ressions can !e im#ortant in the wor"#lace. he effect of information

inferred from a se'uential #attern of conforming decisions is referred to as an

information cascade =7i"hchandani& Hirschleifer& and $elch& ,2>.6ince first im#ressions can !e wrong& the interviews or tests that determine

initial decisions may start an incorrect cascade that im#lies false information to

those who follow. his #ossi!ility was raised in an article in the 4conomist a!out

the #o#ular drug 8ro<ac: Can ,0 4illion 8eo#le !e $rongN Fohn (ryden is

somewhat more #oetic: ;or is the #eo#le s 9udgment always true& the most may

err as grossly as the few. his cha#ter considers how cascades& incorrect or

otherwise& may form even if individuals are good 7ayesians in the way that they

#rocess information.

II. - Mode% of Rationa% Learning from 5thers9 Decisions?ollowing Anderson and Holt =,->& the discussion of cascades will !e

!ased on a very styli<ed model in which the two events are referred to as Cu# A

with contents a& a& b& and Cu# 7 with contents a& b& b. hin" of these cu#s as

containing am!er or !lue mar!les& with the #ro#ortions !eing correlated with the

cu# la!el. *ach cu# is e'ually li"ely to !e selected& with its contents then !eing

em#tied into an o#a'ue container from which draws are made #rivately. *ach

#erson sees one randomly drawn mar!le from the selected cu#& and then must

guess which cu# is !eing used. (ecisions are made in a #re+s#ecified se'uence&

so that the first #erson has nothing to go on !ut the color of the mar!le drawn.

(raws are made with re#lacement& so the second #erson sees a draw from the

un"nown cu#& and must ma"e a decision !ased on two things: the first decision

and the second =own> draw. 1i"e the em#loyers who cannot sit in on others

interviews& each #erson can see #rior decisions !ut not the #rivate information

that may have affected those decisions. here is no eternal incentive to conform



in the sense that one s #ayoff de#ends only on guessing the cu# !eing usedB there

is no !enefit in conforming to others =#rior or su!se'uent> decisions.

he first #erson in the se'uence has only the o!servation& a or b& and the

#rior information that each cu# is e5 ante e'ually li"ely. 6ince two of the three a

mar!les are in cu# A& the #erson should assess the #ro!a!ility of cu# A to !e 2 if

an a is o!served& and similarly for cu# 7 =cf. Cha#ter 0>. hus& the first decision

should reveal that #erson s information. n the e#eriments discussed !elow&

a!out 5K of the #eo#le made the decision that corres#onded to their information

when they were first in the se'uence.

he second #erson then sees a #rivate draw& a or b& and faces an easy

choice if the draw matches the #revious choice. A conflict is more difficult to

analy<e. ?or eam#le& if the first #erson #redicts cu# A& the second #erson who

sees a b draw may reason: the cu#s are now e'ually li"ely& since that was the

initial situation and the a that thin" the first #erson o!served cancels out the b

that 9ust saw. n such cases& the second #erson should !e indifferent& and might

choose each cu# with e'ual #ro!a!ility. In the other hand& the second #erson

may !e a little cautious& !eing more sure a!out what they 9ust saw than a!out whatthey are inferring a!out what the other #erson saw. *ven a slight chance that the

other #erson made a mista"e =deli!erate or not> would cause the second #erson to

go with their own information in the event of a conflict. n the e#eriments& a!out

5K of the second decision ma"ers !ehaved in this manner in the e#eriment

descri!ed !elow& and we will !ase su!se'uent discussion on the assum#tion that

this is the case.

a!le ,.,. 8ossi!le nferences 4ade !y the hird (ecision 4a"er

8rior (ecisions Iwn (raw nferred 8r=cu# A> (ecision

A& A a 0. A =no dilemma>A& A b 0.%- A =start cascade>A& 7 a 0.%- A

A& 7 b 0. 7

;ow the third #erson will have o!served two decisions and one draw.

here is no loss of generality in letting the first decision !e la!eled A& so the

various #ossi!ilities are listed in a!le ,.,. =An analogous ta!le could !e

constructed when the first decision is 7& and the same conclusions would a##ly.>

n the to# row of the ta!le& all three #ieces of information line u#& and the standard

7ayesian calculations would indicate that the #ro!a!ility of cu# A is 0.& ma"ing

A the !est choice =see 'uestion ,>. he case in the second row is more interesting&

since the #erson s o;n draw is at odds with the information inferred from other s

decisions. $hen there are two others& who each receive inde#endent draw signals



that are 9ust as informative as the #erson s own draw& it is rational to go with the

decision im#lied !y the others decisions. =3ecall from cha#ter 0 that an

im!alance of one draw in favor of cu# A will raise the #ro!a!ility of cu# A to

0.%-.> he decision in the final two rows is less difficult !ecause the

#re#onderance of information favors the decision that corres#onds to one s own

information.

he #attern of following others !ehavior =seen in the to# two rows of a!le ,.,>

may have a domino effect: the net #erson would see three A decisions and would

!e tem#ted to follow the crowd regardless of their own draw. his logic a##lies

to all su!se'uent #eo#le& so an initial #air of matching decisions =AA or 77> can

start an information cascade that will !e followed !y all others& regardless of their

information. his logic also a##lies when the first two decisions cancel each

other out =A7 or 7A> and the net two form an im!alance that causes the fifth

#erson to decide to follow the ma9ority. An eam#le of such a situation would !e:

A7AA& in which case the fifth #erson should choose A even if the draw o!served

is b. Again& the intuition is that the first two decisions cancel each other and that

the net two matching decisions are more informative than the #erson s owndraw.

;otice that there is nothing in this discussion that im#lies that the first two

#eo#le will guess correctly. here is a , chance that the first #erson will see the

odd mar!le drawn from the selected cu#& and will guess incorrectly. here is a

, chance that the same thing will ha##en to the second #erson& so in theory&

there is a =,>=,> J , chance that !oth of the first two #eo#le will guess

incorrectly and s#ar" an incorrect cascade. If course& cascades =incorrect or not>

may initially fail to form and then may later form when the im!alance of draws is

2 or more in one direction or the other. n either case& the aggregate information

inferred from others decisions is greater than the information inherent in any

single #erson s draw.

III. "(perimenta% "vidence

Anderson and Holt =,-> used this setu# in a la!oratory e#eriment in which

#eo#le earned 2 for a correct guess& nothing otherwise. 6u!9ects were in isolated

!ooths& so that they could not see others draws. (ecisions =!ut not su!9ect (

num!ers> were announced !y a third #erson to avoid having confidence or dou!ts

communicated !y the decision+ma"er s tone of voice. he mar!les were light or

dar"& with two lights and a dar" in cu# A& and with two dar"s and a light in cu# 7.

he cu# to !e selected was determined !y the throw of a si+sided die& with a ,& 2&

or determining cu# A. he die was thrown !y a monitor selected at random

from among the #artici#ants at the start of the session. here were si other

su!9ects& so each #rediction se'uence consisted of si #rivate draws and si #u!lic

#redictions. ?or each grou# of #artici#ants& there were ,5 #rediction se'uences.



he monitor used a random device to determine the order in which individuals

saw their draws and made their #redictions. he monitor announced the cu# that

had actually !een used at the end of the se'uence& and all #artici#ants who had

guessed correctly added 2 to their cumulative earnings. he monitor received a

fied #ayment& and all others were #aid their earnings in cash. his !all and urn

setu# was designed to reduce or eliminate #references for conformity that were

not !ased on informational considerations. t is #ossi!le that an im!alance of

signals does not develo#& e.g. alternating a and + draws& ma"ing a cascade

unli"ely. An im!alance did occur in a!out half of the #rediction se'uences& and

cascades formed a!out -0K of the time in such cases. A ty#ical cascade se'uence

is:

a!le ,.2. A Cascade

6u!9ect: 5 5- 5 55 5% %0

(raw: b b A b a a8rediction: 7 7 B 7 B B

6ometimes #eo#le do deviate from the !ehavior that would !e im#lied !y

a 7ayesian analysis. ?or eam#le& consider the se'uence:

a!le ,.. A y#ical *rror

6u!9ect: ,2 ,0 ,, -

(raw: a a B a b a

8rediction: A A B A - A

he decision of su!9ect ,2 in the third #osition is the most commonly o!served

ty#e of error& i.e. !asing a decision on one s own information even when it

conflicts with the information im#lied !y others #rior decisions. his ty#e of

error occurred in a!out a fourth of the cases in which it was #ossi!le. ;otice that

the cascade starts later with su!9ect ,, in the fifth #osition.

Ince a cascade !egins& the su!se'uent decisions convey no information

a!out their signals& since these #eo#le are 9ust following the crowd. n this sense&

#atterns of conformity are !ased on a few draws& and conse'uently& cascades may

!e very fragile. n #articular& one #erson who !rea"s the #attern !y revealing their

own information may alter the decisions of those who follow.

?inally& a num!er of incorrect cascades were o!served. Cu# 7 was actually !eing

used for the se'uence shown in a!le ,./& !ut the first two individuals were

unfortunate and received misleading a signals. heir matching #redictions caused



the others to follow with a string of incorrect A #redictions& which would have

!een frustrating given that these individuals had seen #rivate draws that indicated

the correct cu#.

a!le ,./. An ncorrect Cascade =Cu# 7 $as Used>

6u!9ect: ,, ,2 - ,0

(raw: a a B b b b

8rediction: A A - - - -

o summari<e& the general tendency was for individuals to let an emerging

#attern of others decisions override their own #rivate information. he resulting

information cascades were common& !ut were sometimes !ro"en !y later deviant

decisions.

I. "(tensions and 0$rther Reading

he model discussed in this cha#ter is !ased on a model #resented in

7i"hchandani& Hirschleifer& and $elch =,2>& which contains a rich array of

eam#les and a##lications. Hung and 8lott =200,> discuss cascades in voting

situations& e.g.& when the #ayoff de#ends on whether the ma9ority decision is

correct. 6ome of the more interesting a##lications are in the area of finance.

Deynes =,%> com#ared investment decisions with #eo#le in a guessing game

who must #redict which !eauty contestant will receive the most votes. *ach

#layer in this game& therefore& must try to guess who is viewed as !eing attractive

to the others& and on a dee#er level& who the others will thin" that others will find

more attractive. 6imilarly& investment decisions in the stoc" mar"et may involve !oth an analysis of fundamentals and an attem#t to guess what stoc"s will attract

attention from other investors& and the result may !e herd effects that may cause

surges in #rices and later corrections in the other direction. 6ome of these #rice

movements may !e due to #sychological considerations& which Deynes com#ared

with animal s#irits& !ut herd effects may also result from attem#ts to infer

information from others decisions. n such situations& it may not !e irrational to

follow others decisions during u#swings and downswings in #rices =Christie and

Huang& ,5>. 7anner9ee s =,2> model of herd !ehavior is motivated !y an

investment eam#le. his model has !een evaluated in the contet of a la!oratory

e#eriment re#orted !y Also## and Hey =2000>. Ither a##lications to finance are

discussed in (evenow and $elch =,%> and $elch =,2>.

t is very difficult to sort out all of the #ossi!le factors that affect #rices in a stoc"

mar"et& so again la!oratory e#eriments may !e useful. 8lott& $it& and Mang

=,-> ran some e#eriments with a #arimutuel !etting structure li"e that of a



horse race where the #urse is divided among those who !et on the winner& with

#ayouts in #ro#ortion to the amounts of the winning !ets. here were si

alternative assets& and only one of them would have a #ositive #ayout. he

#ayout was divided e'ually among the investors in that asset. *ach su!9ect

received a #rivate signal a!out asset #rofita!ility& !ut the order of decision ma"ing

was not eogenous as in the cascade e#eriments. ndividuals could o!serve

others decisions as they were made. n this manner& the information dis#ersed

among the investors could !ecome aggregated and incor#orated into the #rices of

the si assets. n most cases& the asset #rices ended u# signaling which asset

would actually #roduce a #ositive #ayout. n some cases& however& an initial

surge of #urchases for a #articular asset would stimulate others to follow& and the

result was that the mar"et #rice rose for an asset that had no #ayout in the end& as

would !e the case with an incorrect cascade.

<$estions

,. Use the !all counting heuristic from cha#ter 0 to verify the 7ayesian

#ro!a!ilities in the right+hand column of a!le ,., under the assum#tion

that the first two decisions correctly reveal the draws seen.

2. =Advanced> he discussion in this cha#ter was !ased on the assum#tion

that the second #erson in a se'uence would ma"e a decision that reveals

their own information& even when this information differs from the draw

inferred from the first decision. he result is that the third #erson will

always follow the #attern set !y two matching decisions& !ecause the

information im#lied !y the first two decisions is greater than the

informational content of the third #erson s draw. n this 'uestion&

consider what ha##ens if we alter the assum#tion that the second #ersonalways ma"es a decision that reveals the second draw. 6u##ose instead

that the second #erson chooses randomly =with #ro!a!ilities of ,2> when

the second draw does not match the draw inferred from the first decision.

Use 7ayes rule to show that two matching decisions =AA or 77> should

start a cascade even if the third draw does not match. =he intuition for

this result is that the information im#lied !y the first decision is 9ust as

good as the information im#lied !y the third draw& so the second draw

can !rea" a tie if it contains at least some information.>

. n the Anderson and Holt e#eriments& the mar!les were not la!eled a

and +& !ut rather& were referred to as light and dar"& since they were

either translucent or dar". $hat are the advantages of each la!eling

methodN $hich would you #refer to use in a research e#eriment& and

whyN



Cha#ter 2. 6tatistical (iscrimination

*conomists and other social scientists have long s#eculated that discrimination

!ased on o!serva!le traits =race& gender& etc.> could !ecome self+#er#etuating in a

cycle of low e#ectations and low achievement !y wor"ers of one ty#e. n such acase& the em#loyers may !e reacting rationally& without !ias& to !ad em#loyment

e#eriences with one ty#e of wor"er. $or"ers of that disadvantaged ty#e& in turn&

may correctly #erceive that 9o! o##ortunities are diminished& and may reduce

their investments in human ca#ital. he result may !e a ty#e of statistical

discrimination that is !ased on e#erience& not on any underlying !ias. he game

im#lemented !y the Veconla! #rogram 6( sets u# this "ind of situation& with

each wor"er !eing assigned a color& 8ur#le or reen. he game can !e used to

stimulate a class discussion of to#ics that might !e too sensitive for many

students.

I. Bro6n?e!ed Peop%e -re More Civi%iOedE

As Fane *lliott a##roached her 3iceville& owa classroom one ?riday in

A#ril ,%& she was going over the #rovocative e#eriment that she had stayed u#

late #lanning. 4artin 1uther Ding had !een murdered in 4em#his the day !efore&

and she antici#ated a lot of 'uestions and confusion. 6he was des#erate to do

something that might ma"e a difference. Her ideas had !egun to ta"e sha#e when

she recalled an argument a!out racial #re9udice with her father that had caused his

ha<el eyes to flare. 6he remem!ered commenting to her roommate that her father

would !e in trou!le if ha<el eyes ever went out of favor.

As soon as her third grade class was seated& 4s. *lliott divided them intotwo grou#s !ased on eye color& as descri!ed in ! C#ass (ivided =8eters& ,-,>. At

first& !rown+eyed #eo#le were designated as !eing su#erior& with the

understanding that the roles would !e reversed the net day. 6he !egan: $hat

mean is that !rown+eyed #eo#le are !etter than !lue+eyed #eo#le. hey are cleaner

more civili<ed . And they are smarter than !lue+eyed #eo#le. As the !rown+eyed

#eo#le were seated in the front of the room and allowed to drin" from the water

fountain without using #a#er cu#s& the !lue+eyed students slum#ed in their chairs

and ehi!ited other su!missive ty#es of !ehavior. I!9ections and com#laints

were transformed into rhetorical 'uestions a!out whether !lue+eyed children were

im#olite& etc.& which resulted in a chorus of enthusiastic re#lies from the !rown+

eyed #eo#le. $hen one student 'uestioned 4s. *lliott a!out her own eye color =!lue> and she tried to defend her intelligence& the re#ly was that she was not as

smart as the !rown+eyed teachers. $hat !egan as a role+#laying eercise !egan to

ta"e on an erie reality of its own. hese #atterns were reversed when !lue+eyed

#eo#le were designated as su#erior on the following day.



8sychologists soon !egan conducting e#eriments under more controlled

conditions =e.g.& a9fel& ,-0B Vaughan& a9fel& and $illiams& ,,>. A ty#ical

setu# involved a tas" of s"ill li"e a trivia 'ui< or estimation of the lengths of some

lines. he su!9ects would !e told that they were !eing divided into grou#s on the

!asis of their s"ill& !ut in fact the assignments were random. hen the su!9ects

would !e as"ed to #erform a tas" li"e division of money or candy that might

indicate differential status =see the review in Anderson& ?ryer& and Holt& 2002>.

6uch effects were often manifested as lower allocations to those with lower status.

4oreover& #eo#le sometimes offered #referential treatment to #eo#le of their own

grou#& which is "nown as an in+grou# !ias.

*conomists are ty#ically interested in whether grou# and status effects

will carry over into mar"et settings. ?or eam#le& 7all et al. =200,> eamined the

effects of status assignments in dou!le auctions where the demand and su##ly

functions were !o sha#es with a large vertical overla#& which #roduced a range

of mar"et+clearing #rices. n the earned+status treatment& traders on one side of

the mar"et =e.g. !uyers> were told that they o!tained a gold star as a result of their

#erformance on a trivia 'ui<. he random+assignment treatment !egan with some #eo#le !eing selected to receive stars that were awarded in a s#ecial ceremony.

he su!9ects were not told that the star reci#ients were selected at random. n

each treatment& the #eo#le with stars were all on one side of the mar"et =all !uyers

or all sellers>. his seemingly trivial status #ermitted them to earn more of the

total sur#lus& whether or not they were !uyers or sellers.

rou# differences were largely eogenous in the status e#eriments& !ut

economists have long worried a!out the #ossi!ility that differences in

endogenously ac'uired s"ills may develo# and #ersist in a vicious circle of

selfconfirming differential e#ectations. ?ormal models of e#erienced+!ased

economic discrimination were develo#ed inde#endently !y Arrow =,-> and

8hel#s =,-2>& and have !een refined !y others. he intuition is that if somehistorical differences in o##ortunity cause em#loyment o##ortunities for one

grou# =race& gender& etc.> to !e less attractive& then mem!ers of that grou# may

rationally choose not to invest as much in human ca#ital. n res#onse& em#loyers

may form lower e#ectations for mem!ers of that grou#. *#ectations are !ased

on statistics from #ast e#erience& so these have !een called models of statistical

discrimination.

he o!vious 'uestion is why a mem!er of the disadvantaged grou# could

not invest and !rea" out of the cycle. his strategy may not wor" if the em#loyer

attri!utes a good im#ression made in an interview to random factors& and then

!ases a decision not to hire on #ast e#erience and the fear that the interview did

not reveal #otential #ro!lems. =hin" of the human ca#ital !eing discussed as any

as#ect of #roductivity that is learned and that cannot !e o!served without error in

the 9o! a##lication #rocess. ?or eam#le& the em#loyer may !e a!le to ascertain

that a #erson attended a #articular software class& !ut not the etent to which the



#erson mastered the details of wor"ing with s#readsheets.> *ven if investment !y

mem!ers of the disadvantaged grou# results in good 9o! #lacement some of the

time& a lower success rate for #eo#le in this grou# will nevertheless result in a

lower investment rate. After all& investment is costly in terms of lost income and

lost o##ortunities to have children& etc. hese o!servations raise the distur!ing

#ossi!ility that two grou#s with e5 ante identical a!ilities will !e treated

differently !ecause of differential rates of investment in ac'uired s"ills. n such a

situation& the em#loyers need not !e #re9udiced in any way other than reacting

rationally to their own e#eriences. t is& of course& easy to imagine how real

#re9udice might arise =or continue> and accentuate the economic forces that

#er#etuate discrimination.

II. Being P$rp%e or Green

he focus in this cha#ter is on e#eriments in which some grou# as#ects

are eogenous =color> and some are endogenous =investments>. he #artici#ants

are assigned a role& em#loyer or wor"er. $or"ers are also assigned a color& 8ur#leor reen. $or"ers are given a chance to ma"e a costly investment in s"ills that

would !e valua!le to an em#loyer& !ut only if they were hired. he em#loyers

can o!serve wor"ers colors and the results of an im#erfect #reem#loyment test

!efore ma"ing the hiring decision. nvestment is costly for the wor"er& !ut it

increases the chances of avoiding a !ad result on the #reem#loyment test& and

hence& of getting hired. n this setu#& the em#loyers #refer to hire wor"ers who

invested& and not to hire those who did not.

a!le 2.,. *#eriment 8arameters

$or"er s #ayoffs:,.50 if hired

0.00 if not hired

,.50 if the hired wor"er invested

*m#loyer s #ayoffs: ,.50 if the hired wor"er did not invest

0.50 if the wor"er is not hired

his e#erimental setu#& used !y ?ryer& oeree& and Holt =2002>& matches

a theoretical model =Coate and 1oury& ,> in which statistical discrimination is

a #ossi!le outcome. he e#eriment consisted of a num!er of rounds with

random #airings of em#loyers and wor"ers. *ach round !egan with wor"ers

finding out their own =randomly determined> costs of ma"ing an investment in

some s"ill. nvestment costs are uniform on the range form 0.00 to ,.00& ece#t

as noted !elow. he em#loyer could o!serve the wor"er s color& !ut not the

investment decision. A test given to each wor"er #rovided noisy information



a!out the wor"er s investment& and on the !asis of this information the em#loyer

decided whether or not to hire the wor"er. he #ayoff num!ers are summari<ed in

a!le 2.,.

$or"ers #refer to !e hired& which yields a #ayoff of ,.50 #er #eriod& as

o##osed to the <ero #ayoff for not !eing hired. he investment cost =if any> is

deducted from this #ayoff& whether or not the wor"er is hired. nvestment is good

in that it increases the #ro!a!ility of getting hired at ,.50. hus& a ris"+neutral

wor"er would want to invest if the investment cost is less than the wage times the

increase in the hire #ro!a!ility that results from investing. f the investment cost

is C & then the decision rule is:

orker decision: invest if C Q ,.50 times the increase in hire

#ro!a!ility.

*m#loyers #refer to hire a wor"er who invested& which yields a #ayoff of

,.50. Hiring someone who did not invest yields a #ayoff of ,.50& which is

worse than the 0.50 #ayoff the em#loyer receives if no wor"er is hired. =hin" of the fifty cents as what the manager can earn without any com#etent hel#.> he

manager wishes to hire a wor"er as long as the #ro!a!ility of investment& p& is

such that the e#ected #ayoff from hiring the wor"er& p=,.50> O =, p>= ,.50>& is

greater than the 0.50 earnings from not hiring. t is straightforward to show that

p=,.50> O =, p>= ,.50> P 0.50 when p P 2:

"mp%o!er decision: hire when the #ro!a!ility of investment P 2.

he decision rules 9ust derived& are of course& incom#lete& since the #ro!a!ilities

of investment and !eing hired are determined !y the interaction of wor"er and

em#loyer decisions.

he em#loyer s !eliefs a!out the #ro!a!ility that a wor"er invested are affected

!y a #re+em#loyment test& which #rovides an informative !ut im#erfect indication

of the wor"er s decision. f the wor"er invests& then the em#loyer sees two

inde#endent draws& with re#lacement& from the invest cu# with 7lue mar!les

and 3ed mar!les. f the wor"er does not invest& the em#loyer sees two draws

with re#lacement from a cu# with only , 7lue and 5 3eds. hus the cu#s are:

nvest Cu# ;o+nvest Cu#

B& B& B& R

B& R & R & R & R & R



I!viously& the nvest Cu# #rovides three times as great a chance that each draw

will !e 7lue =7>& so 3ed =3> is considered a !ad signal. f the decision is to invest

=nv>& the #ro!a!ilities of the draw com!inations are 9ust #roducts: 8r=77 ]nv> J

=,2>=,2> J ,/& 8r=33 ]nv> J =,2>=,2> J ,/& and therefore& the residual

#ro!a!ility of a mied signal =37 or 73> is ,2. 6imilarly& the #ro!a!ilities of the

signal com!inations for no investment =;o> are: 8r=77 ];o> J =,%>=,%> J ,%&

8r=33 ];o> J =5%>=5%> J 25%& and 8r=37 or 73> J ,0%& which is the residual.

I. -n >nfair "#$i%i&ri$m

he signal com!ination #ro!a!ilities 9ust calculated can !e used to determine

whether or not it is worthwhile for a wor"er to invest& !ut we must "now how

em#loyers react to the signals. he discussion will #ertain to an asymmetric

e'uili!rium outcome in which one color gets #referential treatment in the hiring

#rocess. =6ymmetric e'uili!ria without discrimination will !e discussed

afterwards.> he theoretical 'uestion is whether asymmetric e'uili!ria =with

discriminatory hiring decisions> can eist& even if wor"ers of each color have thesame investment cost o##ortunities and #ayoffs. Consider the discriminatory

hiring strategy:

6ignal BB: Hire !oth colors.

=2.,> 6ignal BR or R B: Hire reen& not hire 8ur#le.

6ignal RR : (o not hire either color.

here are two ste#s to com#lete the analysis of this e'uil!rium: ,> to figure out

the ranges of investment costs for which wor"ers of each color will invest& and 2>

to verify that the con9ectured hiring strategy given a!ove is o#timal for

em#loyers.

Step 1$ =orker Investment (ecisions

3ecall that the wor"er will invest if the cost is less than the wage ,.50 times the

increase in the #ro!a!ility of !eing hired due to a decision to invest. he 8ur#le

wor"er is only hired if !oth draws are !lue& which ha##ens with #ro!a!ility =,2>

=,2> J ,/ following investment and =,%>=,%> J ,% following no investment&

so the increase in the #ro!a!ility of !eing hired is ,/ ,% J % ,% J

% J 2. Hence =2>,.50 J 0. is the e#ected net gain from investment&

which is the o#timal decision if the investment cost is less than 0.. n

contrast& the favored reen wor"ers are hired whenever the draw com!ination isnot 33. 6o investment results in a hire with #ro!a!ility , =,2>=,2> J /.

6imilarly& the #ro!a!ility of a reen wor"er !eing hired without investing is the

#ro!a!ility of not getting two 3 draws from the ;o nvest Cu# =733333>& which

is , =5%>=5%> J ,,%. he increase in the chances of getting a 9o! due to



investment is / ,,% J 2-% ,,% J ,%% J /. Hence the e#ected

net gain from investment !y a reen wor"er is =/>,.50 J 0.%-. o

summari<e& given the hiring strategy that discriminates in favor of reen wor"ers&

the reens will invest whenever their investment costs are less than 0.%-& and the

8ur#le wor"ers will invest half as often& i.e. when their costs are less than 0..

hus the discriminatory hiring rates induce more investment !y the favored color.

Step $ 4mp#oer Airing (ecisions

3ecall that the em#loyer s #ayoffs in a!le 2., are such that it is !etter to hire

whenever the #ro!a!ility that the wor"er invested is greater than 2. 6ince

reens invest twice as often as 8ur#les =a #ro!a!ility of 2 versus ,>& the

em#loyer s #osterior #ro!a!ility that a wor"er invested will !e higher for reen&

regardless of the com!ination of draws. hese #osterior #ro!a!ilities are

calculated with 7ayes rule and are shown in a!le 2.2.

a!le 2.2. 8ro!a!ilities that the $or"er nvested Conditional on est and Color:

(ecision is to Hire if the 8ro!a!ility P 2

Green $or"ers 8ur#le $or"ers

8r=wor"er invested ] BB> ,, =hire reen> ,, =hire 8ur#le>

8r=wor"er invested ] BR or R B> ,2 =hire reen> , =not hire 8ur#le>

8r=wor"er invested ] RR > ,/ =not hire reen> 5 =not hire 8ur#le>

?or eam#le& consider the to# row of the reen column& which shows the

#ro!a!ility that a reen wor"er with a 77 signal invested. his #ro!a!ility&

,,& is calculated !y ta"ing the ratio& with the reens who invested and

received the 77 signal in the numerator and all reens =who invested or not> who

received the 77 singal in the denominator. 6ince two+thirds of reens invest andinvestment #roduces the 77 signal with #ro!a!ility =,2>=,2> J ,/& the

numerator is =2>=,/>. 6ince one+third of the reens do not invest and not

investment #roduces 77 signal with #ro!a!ility =,%>=,%> J ,%& the

denominator also includes a term that is the #roduct: =,>=,%>. t follows that

the #ro!a!ility of investment for a reen with the 77 signal is: =2>=,/> divided

!y =2>=,/> O =,>=,%>& which reduces to ,,& as shown in the a!le. he

em#loyer s o#timal decision is to hire when the investment #ro!a!ility is greater

than 2& so the reen wor"er with a 77 test result should !e hired. he other

#ro!a!ilities and hire decisions in the ta!le are calculated in a similar manner

='uestion ,>. A com#arison of the middle and right columns of the ta!le show thatthese decisions discriminate against 8ur#le in a manner that matches the original

s#ecification in =2.,>.

t can !e shown that there is also a symmetric e'uili!rium where !oth

colors invest when the cost is less than 0.%- and !oth colors are treated li"e the



em#loyer treated reens in =2.,>& i.e. always hire a wor"er of either color unless

the test outcome is 33 ='uestion 2>.

I. Data

he advantage of running e#eriments is o!vious in this case& since the model

has !oth asymmetric and symmetric e'uili!ria& and since eogenous sources of !ias can !e controlled. 4oreover& the setu# is sufficiently com#le that !ehavior

may not !e drawn to any of the e'uili!ria. ?ryer& oeree& and Holt =2002> have

run a num!er of sessions using the statistical discrimination game. n some of the

sessions there is little distinction !etween the way wor"ers with different color

designations are treated. his is not too sur#rising& given the symmetry of the

model and the #resence of a symmetric e'uili!rium.

4any cases where une'ual treatment is li"ely to #ersist have evolved from

social situations where discrimination was either directly sanctioned or indirectly

su!sidi<ed !y legal rules and social norms. herefore& we !egan some of the

sessions with ,0 rounds of une'ual investment cost distri!utions. n #articular&one color =e.g. reen> might !e favored initially !y drawing investment costs

from a distri!ution that is uniform on 0.00& 0.50& and 8ur#les may draw from

a distri!ution that is uniform on 0.50& ,.00. he vertical dashed line at round

,0 in ?igure 2., shows the round were these une'ual cost o##ortunities ended.

6u!9ects were not told of this cost differenceB they were 9ust told that all cost

draws would !e !etween 0.00 and ,.00& which was the case. All draws after

round ,0 were from a common uniform distri!ution on 0.00& ,.00.

?igure 2., shows five+#eriod average investment and hiring rates. ;otice

that the initial cost asymmetry wor"s in the right directionB reens start out

investing at a!out twice the rate as that for 8ur#les& as can !e seen from the left

#anel. here is a slight crossover in rounds ,0+,5& which seems to have !eencaused !y a relative cost effect& i.e. the tendency for 8ur#les to invest a lot when

encountering costs that are lower than initially high levels that they were used to

seeing. 6imilarly& investment rates for reens fell !riefly after #eriod ,0 when

they !egan drawing from a higher cost distri!ution. As can !e seen from the right

#anel in the figure& the em#loyers seemed to notice the lower initial investment

rates for 8ur#les& and they continued hiring reens at a higher rate. he inertia of

this #reference for reen eventually caused the investment rates for reen to rise

a!ove those for 8ur#le again& and se#aration continued.

he color+!ased hiring #reference was clear for a ma9ority of the

em#loyers& although the se#aration was not as uniform as that #redicted in the

#revious section s e'uili!rium analysis. After an em#loyer saw mied signals

=73 or 37>& reens were hired more fre'uently =%K versus 5K in the final 20

rounds>. n fact& reens were even hired more fre'uently after a negative 33

signal =5K versus 0K in the final 20 rounds>.



00

01

020304

05060708

0910

00

01

020304

050607

08

0910

nvestment 3ates Hire 3ates

0 5 ,0 ,5 20 25 0 5 /0 /5 50 0 5 ,0 ,5 20 25 0 5 /0 /5 50

3ound 3ound

0ig$re /.'. - Separation "ffect: 0ive?Period -verage Investment and 7ire Rates &! Co%or

@Green is So%id, P$rp%e is DashedA 6ith Statistica% Discrimination Ind$ced &! an Initia%

Investment Cost -s!mmetr! @So$rce: 0r!er, Goeree, and 7o%t, ++A

he individual em#loyer decisions for the final ,5 rounds of this session

are shown in a!le 2.. 4ost of the em#loyers discriminate to some etent&

either in the case of a mied =37 or 73> signal& or in the case of a negative =33>

signal. *m#loyers ,& 2& and hired all reen wor"ers encountered& even when

the test result was 33. *m#loyers / and 5 tended to discriminate after mied

signals& and em#loyers 2 and tended to discriminate after the negative signal.

*m#loyer % is essentially color+!lind in the way wor"ers are treated.

he e#erience of em#loyer 2 indicated the #ro!lems facing 8ur#le

wor"ers. All 8ur#les encountered !y this em#loyer after round /0 had invested&

!ut the em#loyer was una!le to s#ot this trend since no 8ur#les were hired after round /0. =he investment decisions of the wor"ers are shown in dar" !old letters

=Inv or 8o> when the wor"er was hired& and the uno!served investment decisions

are shown in light gray when the wor"er was not hired.> 6uch differential

treatment& when it occurs& is #articularly interesting when the e#eriment is

conducted in class for teaching #ur#oses. n classroom e#eriments& one #erson

once remar"ed 8ur#le wor"ers 9ust can t !e trusted they won t invest. Another

student remar"ed: invested every time& even when costs were high& !ecause felt

confident that would get the 9o! !ecause I am Green.

a!le 2.. *m#loyer nformation and (ecisions for the ?inal ,5 3oundsDey: Inv J investment o!served e5 post !y em#loyer.

nv J investment not o!served e5 post !y em#loyer.



3ound

*m#loyer , *m#loyer 2

*m#loyer *m#loyer / *m#loyer 5 *m#loyer %

%

P$rp%e BB

Hire Inv

Green BR

Hire Inv Green BR

Hire Inv P$rp%e BR

Hire Inv

P$rp%e BR

Hire Inv

Green BB

Hire Inv

-

P$rp%e BR

Hire Inv

Green RR

Hire Inv

Green BB

Hire Inv

Green BR

;o Hire nv

P$rp%e BR

;o Hire nv

P$rp%e BB

Hire Inv

Green BR

Hire 8o

P$rp%e BB

Hire Inv

P$rp%e RR

;o Hire ;o

Green RR ;o

Hire nv

Green RR

;o Hire nv

P$rp%e BB

Hire Inv

P$rp%e RR ;o

Hire ;o

Green RR

Hire 8o

P$rp%e RR

;o Hire nv

Green RR ;o

Hire nv Green BR

Hire Inv P$rp%e BR

Hire Inv

/0

P$rp%e BR ;o

Hire nv

P$rp%e BB

Hire Inv

P$rp%e BB

Hire Inv

Green BR

Hire Inv Green BR

Hire Inv Green BR

Hire Inv

/,

P$rp%e RR ;o

Hire nv

Green RR

Hire Inv

Green BR

Hire 8o Green BR

Hire Inv P$rp%e RR

;o Hire nv

P$rp%e BR

Hire 8o

/2

P$rp%e BB

Hire Inv

P$rp%e BR

;o Hire nv

Green RR

Hire Inv P$rp%e BR

;o Hire nv

Green BB

Hire Inv Green BB

Hire Inv

/P$rp%e BR

Hire Inv P$rp%e RR

;o Hire nvGreen RRHire Inv

P$rp%e BR ;o Hire nv

Green BRHire Inv

Green BBHire Inv

//

P$rp%e BB

Hire Inv

Green BB

Hire Inv

P$rp%e BR

Hire Inv

P$rp%e BR

;o Hire nv

Green BR

Hire Inv Green RR

;o Hire nv

/5

Green BR

Hire Inv

Green BB

Hire Inv

P$rp%e BR

Hire Inv

P$rp%e BR

;o Hire ;o

Green BR

Hire Inv P$rp%e RR

;o Hire ;o

/%

P$rp%e RR ;o

Hire nv

P$rp%e BR

;o Hire nv

Green RR

Hire Inv Green BR

Hire Inv Green RR

;o Hire nv P$rp%e BR

Hire Inv

/-

P$rp%e BB

Hire 8o

Green RR

Hire Inv

Green BR

Hire Inv P$rp%e BB

Hire Inv

P$rp%e RR

;o Hire ;o

Green BR

Hire Inv

/

P$rp%e RR ;o

Hire ;o

Green BB

Hire Inv

Green BB

Hire Inv

Green RR ;o

Hire nv

P$rp%e RR

;o Hire nv

P$rp%e BR

Hire Inv

/

Green BR

Hire Inv

Green RR

Hire Inv

P$rp%e BB

Hire Inv

P$rp%e RR

;o Hire ;o

P$rp%e BR

Hire Inv

Green BB

Hire Inv

50

P$rp%e RR ;o

Hire ;o

Green BB

Hire Inv

Green RR

Hire 8o P$rp%e BR

Hire Inv

Green BR

Hire Inv P$rp%e RR

;o Hire ;o

he initial cost differences #roduced the same initial #atterns in a second

e#eriment& shown in ?igure 2.2. he vertical dashed line again indicates the

#oint after which the cost distri!utions !ecame symmetric. ;otice that the surge

caused !y the relative cost effect after ,0 rounds carries over& causing afascinating reverse se#aration where the initially disadvantaged grou# =8ur#le>

ends u# investing more for the remainder of the session. his differential

investment rate seems to cause the hiring rates to diverge in the final ten #eriods.



0001

0203

0405

0607080910

0001

02

0304

05

0607

08

0910

A similar crossover effect was o!served in a classroom e#eriment with 5 rounds

of asymmetric costs followed !y ,5 rounds of symmetric costs.

nvestment 3ates Hire 3ates

0 5 ,0 ,5 20 25 0 5 /0 /5 50 0 5 ,0 ,5 20 25 0 5 /0 /5 50

8eriod 8eriod

0ig$re /.. - Cross?5ver "ffect: 0ive?Period -verage Investment and 7ire Rates &! Co%or

@Green is So%id, P$rp%e is DashedA 6ith Reverse Statistica% Discrimination Ind$ced &! an

Initia% Investment Cost -s!mmetr! @So$rce: 0r!er, Goeree, and 7o%t, ++A


here are a num!er of related theoretical models of statistical discrimination.

hese models are #resented with a uniform notation in ?ryer =200,>. he

classroom e#eriments mentioned in the #revious section are discussed in more

detail in ?ryer& oeree& and Holt =2005>. Anderson and Hau#ert =,> descri!e a

classroom e#eriment with eogenously determined wor"er s"ill levels. (avis

=,-> #rovides an e#erimental test of the idea that #erce#tions a!out a grou#

may !e influenced !y the highest s"ill level encountered in the #ast. n this case&

a ma9ority grou# #roduces more wor"ers& so statistically s#ea"ing& the highest

s"ill level from that grou# should !e higher than from the other grou#.

<$estions,. Verify the 7ayes rule #ro!a!ility calculations for 8ur#le wor"ers in the

right column of a!le 2.2.

2. Consider a symmetric e'uili!rium where em#loyers hire a wor"er

regardless of color if the test result is 77& 73& or 37& and not



otherwise. o do this& first show that wor"ers of either color will

invest as long as the cost is less than 0.%-. hen calculate the

#ro!a!ilities of investment conditional on the test results& as in a!le

2.2. ?inally& show that the em#loyer s !est res#onse is to hire unless

the test result is 33.



A##endices: nstructions for Class *#eriments

8it 4ar"et nstructions =Cha#ter 2>$e are going to set u# a mar"et in which the #eo#le on my right will !e !uyers&

and the #eo#le on my left will !e sellers. here will !e e'ual num!ers of !uyers

and sellers& which is the reason that some of you had to switch to the other side of

the room. 6everal assistants have !een selected to hel# record #rices. will now

give each !uyer and seller a num!ered #laying card. 6ome cards have !een

removed from the dec"=s>& and all remaining cards have a num!er. 8lease hold

your card so that others do not see the num!er. he !uyers) cards are red =Hearts

or (iamonds>& and the sellers) cards are !lac" =Clu!s or 6#ades>. *ach card

re#resents a ZunitZ of an uns#ecified commodity that can !e !ought !y !uyers or

sold !y sellers.

8rading : 7uyers and sellers will meet in the center of the room =or other

designated area> and negotiate during a 5 minute trading #eriod. $hen a !uyer

and a seller agree on a #rice& they will come together to the front of the room to

re#ort the #rice& which will !e announced to all. hen the !uyer and the seller

will turn in their cards& return to their original seats& and wait for the trading

#eriod to end. here will !e several mar"et #eriods.

Se##ers: Mou can each sell a single unit of the commodity during a trading #eriod.

he num!er on your card is the dollar cost that you incur if you ma"e a sale& andyou will not !e allowed to sell !elow this cost. Mour earnings on the sale are

calculated as the difference !etween the #rice that you negotiate and the cost

num!er on the card. f you do not ma"e a sale& you do not earn anything or incur

any cost in that #eriod. hin" of it this way: its as if you "new someone who

would sell you the commodity for a #rice that e'uals your cost num!er& so you

can "ee# the difference if you are a!le to resell the commodity for a #rice that is

a!ove the ac'uisition cost. 6u##ose that your card is a 2 of Clu!s and you

negotiate a sale #rice of . hen you would earn: + 2 J ,. Mou would not !e

allowed to sell at a #rice !elow 2 with this card =2 of Clu!s>. f you mista"enly

agree to a #rice that is !elow your cost& then the trade will !e invalidated when

you come to the front des"B your card will !e returned and you can resumenegotiations.

Buers: Mou can each !uy a single unit of the commodity during a trading #eriod.

he num!er on your card is the dollar value that you receive if you ma"e a



#urchase& and you will not !e allowed to !uy at a #rice a!ove this value. Mour

earnings on the #urchase are calculated as the difference !etween the value

num!er on the card and the #rice that you negotiate. f you do not ma"e a

#urchase& you do not earn anything in the #eriod. hin" of it this way: its as if

you "new someone who would later !uy the unit from you at a #rice that e'uals

your value num!er& so you can "ee# the difference if you are a!le to !uy the unit

at a #rice that is !elow the resale value. 6u##ose that your card is a of

(iamonds and you negotiate a #urchase #rice of /. hen you would earn: + /

J 5. Mou would not !e allowed to !uy at a #rice a!ove with this card = of

(iamonds>. f you mista"enly agree to a #rice that is a!ove your value& then the

trade will !e invalidated when you come to the front des"B your card will !e

returned and you can resume negotiations.

'ecording 4arnings: 6ome !uyers and sellers may not !e a!le to negotiate a

trade& !ut do not !e discouraged since new cards will !e #assed out at the

!eginning of the net #eriod. 3emem!er that earnings are <ero for any unit not

!ought or sold =sellers incur no cost and !uyers receive no value>. $hen the #eriod ends& will collect cards for the units not traded& and you can calculate

your earnings while shuffle and redistri!ute the cards. Mour total earnings e'ual

the sum of earnings for units traded in all #eriods& and you can use the *arnings

3ecord ?orm on the net #age to "ee# trac" of your earnings. 6ellers use the left

side of the *arnings 3ecord ?orm& and !uyers use the right side. At this time&

#lease draw a diagonal line through the side that you will not use. All earnings

are hy#othetical. 8lease do not tal" with each other until the trading #eriod

!egins. Are there any 'uestionsN

6ina# /+servations: $hen a !uyer and a seller agree on a #rice& !oth should

immediate# come to the front ta!le to turn in their cards together& so that we canverify that the #rice is neither lower than the seller)s cost nor higher than the

!uyer)s value. f there is a line& #lease wait together. After the #rice is verified&

the assistant at the !oard will write the #rice and announce it loudly. hen those

two traders can return to their seats to calculate their earnings. he assistants

should come to their #ositions in the front of the room. 7uyers and sellers& #lease

come to the central trading area ;I$& and !egin calling out #rices at which you

are willing to !uy or sell. he mar"et is o#en& and there are 5 minutes remaining.

Mour ;ame: SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS

6eller *arnings 7uyer *arnings

=sellers use this side> =!uyers use this side>



J ?irst J

8eriod

=#rice> =cost> =earnings> =value> =#rice> =earnings>

J 6econd J

8eriod


J hird J

8eriod


J ?ourth J

8eriod


J ?ifth

8eriod

J


J 6ith

8eriod

J


total earnings& for

all #eriods:

total earnings&

for all #eriods:



ame nstructions =Cha#ter >

$e are going to #lay a card game in which every!ody will !e matched with

someone on the o##osite side of the room. will now give each of you a #air of

#laying cards& one red card =Hearts or (iamonds> and one !lac" card =Clu!s or

6#ades>. he num!ers or faces on the cards will not matter& 9ust the color. Mouwill !e as"ed to #lay one of these cards !y holding it to your chest =so we can see

that you have made your decision& !ut not what that decision is>. Mour earnings

are determined !y the card that you #lay and !y the card #layed !y the #erson

who is matched with you. f you #lay your red card& then your earnings in dollars

will increase !y 2& and the earnings of the #erson matched with you will not

change. f you #lay your !lac" card& your earnings do not change and the dollar

earnings of the #erson matched with you go u# !y . n other words& thin" of

there !eing some dollar !ills on the ta!le !etween you and the other #erson. Mou

can either Z#ullZ 2 to yourself !y #laying the red card& or you can Z#ushZ to

the other #erson !y #laying the !lac" card. f you each #lay your red card& you

will each earn 2. f you each #lay the !lac" card& you will each earn . f you

#lay your !lac" card and the other #erson #lays a red card& then you earn <ero and

the other #erson earns the 5. f you #layl red and the other #erson #lays !lac"&

then you earn the 5& and the other #erson earns <ero. ;either of you will !e a!le

to see what the other does =#ush or #ull> until !oth decisions have !een made. All

earnings are hy#othetical& ece#t as noted !elow.

After you choose which card to #lay& hold it to your chest. $e then tell you who

you are matched with& and you can each reveal the card that you #layed. 3ecord

your earnings on the net #age. =I#tion: After we finish all #eriods& will #ic"

one #erson with a random throw of dice and #ay that #erson ,0K of his or her

total earnings& in cash. All earnings for everyone else are hy#othetical. o ma"ethis easier& #lease write your identification num!er that will now give each of

you at the to# of the #age. Afterwards& will throw a ,0+sided die twice& with the

first throw determining the ZtensZ digit& until o!tain the ( num!er of one of

you& who will then !e #aid ,0K of his or her total earnings in cash.> Any

'uestionsN

o !egin: will #roceed row !y row& with each of the #eo#le on one side of the

row !eing matched with someone on the other side. $ould the #eo#le in the row

that designate #lease choose which card to #lay and write the color =3 for 3ed&

or 7 for 7lac"> in the first column. 6how that you have made your decision !y

#ic"ing u# the card you want to #lay and holding it to your chest. Has everyone

finishedN ;ow& will #air you with another #erson& as" you to reveal your choice&and calculate your earnings. 3emem!er to "ee# trac" of earnings in the s#ace

#rovided !elow. ?inally& #lease note that you will !e matched with a different

#erson in round 2& and #ayoffs will change. n round you will !e matched with



a different #erson and #ayoffs change again& !ut you will !e #aired with this

#erson in #eriods +5.

3ound 8ayoffs Mour card=3 or 7>

Ither s card=3 or 7>

Mour *arnings

, 3ed: #ull 2

7lac": #ush

2 3ed: #ull 2

7lac": #ush

3ed: #ull 2

7lac": #ush

/ 3ed: #ull 2

7lac": #ush 5 3ed: #ull 2

7lac": #ush



1ottery Choice nstructions =Cha#ter />

Mour decision sheet on the net #age shows ten decisions listed in the left

column. *ach decision is a #aired choice !etween ZI#tion AZ and ZI#tion 7.Z

Mou will ma"e ten choices and record these in the far right column& !ut only oneof them will !e used in the end to determine your earnings. 7efore you start

ma"ing your ten decisions& #lease let me e#lain how these choices will affect

your earnings& which will !e hy#othetical unless otherwise indicated.

Here is a ten+sided die that will !e used to determine #ayoffsB the faces are

num!ered from , to ,0 =the Z0Z face of the die will serve as ,0.> After you have

made all of your choices& we will throw this die twice& once to select one of the

ten decisions to !e used& and a second time to determine what your #ayoff is for

the o#tion you chose& A or 7& for the #articular decision selected. *ven though

you will ma"e ten decisions& only one of these will end u# affecting your

earnings& !ut you will not "now in advance which decision will !e used.

I!viously& each decision has an e'ual chance of !eing used in the end.

;ow& #lease loo" at (ecision , at the to#. I#tion A #ays 2.00 if the throw of

the ten sided die is ,& and it #ays ,.%0 if the throw is 2+,0. I#tion 7 yields

.5 if the throw of the die is ,& and it #ays 0.,0 if the throw is 2+,0. he other

(ecisions are similar& ece#t that as you move down the ta!le& the chances of the

higher #ayoff for each o#tion increase. n fact& for (ecision ,0 in the !ottom row&

the die will not !e needed since each o#tion #ays the highest #ayoff for sure& so

your choice here is !etween 2.00 and .5.

o summari<e& you will ma"e ten choices: for each decision row you will have to

choose !etween I#tion A and I#tion 7. Mou may choose A for some decision

rows and 7 for other rows& and you may and ma"e your decisions in any order.$hen you are finished& we will come to your des" and throw the ten+sided die to

select which of the ten (ecisions will !e used& i.e. which row in the ta!le will !e

relevant. hen we will throw the die again to determine your money earnings for

the I#tion you chose for that (ecision. *arnings for this choice will !e added to

your #revious earnings =if any>.

6o now #lease loo" at the em#ty !oes on the right side of the record sheet. Mou

will have to write a decision& A or 7 in each of these !oes& and then the die throw

will determine which one is going to count. $e will loo" at the decision that you

made for the choice that counts& and circle it& !efore throwing the die again to

determine your earnings for this #art. hen you will write your earnings at the

!ottom of the #age. Are there any 'uestionsN



I#tion A I#tion 7 Mour

Choice

A or 7

(ecision , 2.00 if throw of die is ,

,.%0 if throw of die is 2+,0

.5 if throw of die is ,

0.,0 if throw of die is 2+,0

(ecision 2 2.00 if throw of die is ,+2

,.%0 if throw of die is +,0

.5 if throw of die is ,+2

0.,0 if throw of die is +,0

(ecision 2.00 if throw of die is ,+

,.%0 if throw of die is /+,0

.5 if throw of die is ,+

0.,0 if throw of die is /+,0

(ecision / 2.00 if throw of die is ,+/

,.%0 if throw of die is 5+,0

.5 if throw of die is ,+/

0.,0 if throw of die is 5+,0

(ecision 5 2.00 if throw of die is ,+5

,.%0 if throw of die is %+,0

.5 if throw of die is ,+5

0.,0 if throw of die is %+,0

(ecision % 2.00 if throw of die is ,+%

,.%0 if throw of die is -+,0

.5 if throw of die is ,+%

0.,0 if throw of die is -+,0

(ecision - 2.00 if throw of die is ,+-


.5 if throw of die is ,+-







,.%0 if throw of die is ,0


0.,0 if throw of die is ,0

(ecision ,0 2.00 if throw of die is ,+,0 .5 if throw of die is ,+,0

76 ame nstructions =Cha#ter 5>

$e are going to #lay a card game in which every!ody will !e matched with

someone on the o##osite side of the room. will now give each of you a #air of

#laying cards& one 3ed card =Hearts or (iamonds> and one 7lac" card =Clu!s or

6#ades>. =nstead& these may !e inde cards with num!ers written in red or

!lac">. he #eo#le on the left side of the room have a red and a !lac" 2&

whereas the #eo#le on the right side of the room have a red 2 and a !lac" .



1eft 6ide 3ight 6ide

3ed & 7lac" 2 3ed 2& 7lac"

Mou will !e as"ed to #lay one of these cards !y holding it to your chest =so we can

see that you have made your decision& !ut not what that decision is>.Mour earnings are determined !y the card that you #lay and !y the card

#layed !y the #erson who is matched with you:

If the co#ors of the cards do not match 0'ed and B#ack2, ou each earn nothing.

If the co#ors match, earnings in do##ars are eHua# to the num+er on our card$

After you choose which card to #lay& hold it to your chest. $e then tell you who

you are matched with& and you can each reveal the card that you #layed. 3ecord

your earnings on the net #age. All earnings are hy#othetical& ece#t as noted

!elow. =I#tion: After we finish all rounds& will #ic" one #erson with a random

throw of dice and #ay that #erson SSSK of his or her total earnings& in cash. All

earnings for everyone else are hy#othetical. o ma"e this easier& #lease write the

identification num!er that will now give each of you at the to# of the record

#age. Afterwards& will throw a ,0+sided die twice& with the first throw

determining the ZtensZ digit& until o!tain the ( num!er of one of you& who will

then !e #aid ,0K of his or her total earnings in cash.> Any 'uestionsN

o !egin: will #roceed !y row& matching the each #erson on one side of a row

with someone on the other side. $ould the #eo#le in the row that designate

#lease choose which card to #lay and write the color =3 for red or 7 for !lac"> in

the Mour Card column. 6how that you have made your decision !y #ic"ing u# thecard you want to #lay and holding it to your chest. *veryone finishedN ;ow&

will #air you with another #erson& as" you to reveal your choice& and calculate

your earnings. 3emem!er to "ee# trac" of the other s card and of your earnings

on the net #age. ?inally& #lease note that in round 2 you will !e matched with a

different #erson& and #ayoffs change. n round you will !e matched with a

different #erson and #ayoffs change again& !ut you will !e #aired ;ith the same

person in a## three su+seHuent rounds.

BS Game Pa!offs:

Color "ey: 3 for 3ed& 7 for 7lac".



f the cards match in color =33 or 77>& you earn a dollar amount that is e'ual to

the num!er on the card you #layed.

f the cards to not match in color =37 or 73>& you do not earn anything.

nitial #ayoffs for round ,:

Mou #lay 3& other #lays 3& you earn SSSS.

Mou #lay 7& other #lays 7& you earn SSSS.

Mou #lay 3& other #lays 7& you earn 0.00.

Mou #lay 7& other #lays 3& you earn 0.00.

3ound 8ayoffs Mour card

=3 or 7>

Ither s card

=3 or 7>

Mour

*arnings

, 3 card: SSSS

7 card: SSSS2 3 card: SSSS

7 card: SSSS

3 card: SSSS 7

card: SSSS

/ 3 card: SSSS 7

card: SSSS

5 3 card: SSSS 7

card: SSSS

% 3 card: SSSS 7card: SSSS

- 3 card: SSSS 7

card: SSSS

3 card: SSSS 7

card: SSSS

7inary 8rediction ame =Cha#ter %>

his is an eercise in which you will !e as"ed which of two random events will

occur. he events will !e called 1 =left> and 3 =right>. $e will use the throw of

dice to determine which event will occur in each #eriod. 8lease loo" at the



decision sheet !elow. he num!er of the round is on the left& and you will use the

second column to record your #rediction& 1 or 3. Mou !egin !y recording your

#rediction for round 1 on#& leaving all remaining rows !lan" After every!ody

has written their #rediction for round , into the to# row& we will throw the dice in

the front of the room to determine the event& 1 or 3. his throwing of the dice

will !e done !ehind a screen so that you cannot see what method is !eing used.

he main thing to remem!er here is that the #erson throwing the dice will not

"now your #redictions& so these decisions cannot affect the chances that one event

or the other will !e o!served in future rounds. $hen we announce the event =1 or

3> for the current #eriod& #lease write in your earnings: O50 cents if you were

correct& and +50 cents if you were incorrect. hen you can #roceed to record

your decision for the following round. Mou can "ee# trac" of your cumulative

earnings in the final column. Mour instructor will ma"e an announcement a!out

whether or not the earnings will actually !e #aid in cash. o summari<e& each

#eriod consists of ,> the #rediction stage& 2> the throwing of the dice to determine

the event& > the announcement of the event and the calculation of results. Are

there any 'uestionsN

3ound Mour 8rediction

=1 or 3>

I!served *vent

=1 or 3>

Mour

*arnings

otal

*arnings

,

2

/

5

%

-

,0



1ottery Choice nstructions =Cha#ter ->

Mour decision sheet shows ten decisions listed on the left& which are la!eled 0& ,&

2& ..& . *ach decision is a #aired choice !etween a randomly determined #ayoff

descri!ed on the left and another one descri!ed on the right. Mou will ma"e tenchoices and record these in the final column& !ut only one of them will !e used in

the end to determine your earnings. 7efore you start ma"ing your ten choices&

#lease let me e#lain how these choices will affect your earnings. Unless

otherwise indicated& all earnings are hy#othetical.

Here is a ten+sided die that will !e used to determine #ayoffsB the faces are

num!ered from 0 to . After you have made all of your choices& we will throw

this die three times& once to select one of the ten decisions to !e used& and then

two more times to determine what your #ayoff is for the o#tion you chose =on the

left side or on the right side> for the #articular decision selected. *ven though you

will ma"e ten decisions& only one of these will end u# affecting your earnings& !ut

you will not "now in advance which decision will !e used. I!viously& each

decision has an e'ual chance of !eing used in the end.

;ow& #lease loo" at (ecision 0 at the to#. f this were the decision that we ended

u# using& we would throw the ten+sided die two more times. he two throws will

determine a num!er from 0 to & with the first throw determining the tens digit

and the second one determining the ones digit. he o#tion on the left side #ays

%.00 if the throw of the ten+sided die is 0+& and it #ays 0.00 otherwise. 6ince

all throws are !etween 0 and & this o#tion #rovides a sure %.00. he o#tion on

the right #ays .00 if the throw of the die is 0+-& and it #ays 0.00 otherwise.

hus the o#tion on the right side of (ecision 0 #rovides 0 chances out of ,00 =a

four+fifths #ro!a!ility> of getting .00. he left and right o#tions for the other decision rows are similar& !ut with differing #ayoffs and chances of getting each

#ayoff. n addition& you will receive ,0.00 for #artici#ating& so any earnings will

!e added to this amount. 6ome of the decisions involve losses& indicated !y

minus signs. f the decision selected ends u# with a loss& this loss will !e

su!tracted from the initial ,0.00 #ayment to determine your final earnings.

o summari<e& you will ma"e ten choices: for each decision row you will have to

choose !etween the o#tion on the left and the o#tion on the right. 4a"e your

choice !y #utting a chec" !y the o#tion you #refer. f you change your mind&

cross out the chec" and #ut it on the other side. hus there should !e one chec"

mar" in each row. Mou may ma"e your decisions in any order. $hen you are

finished& mar" your final choices& 1 for left or 3 for right& in the far+right column&and we will come to your des" and throw the ten+sided die to select which of the

ten (ecisions will !e used. $e will circle that decision !efore throwing the die

again to determine your money earnings for the I#tion you chose for that

(ecision.



7efore you !egin& let me mention that different #eo#le may ma"e different

choices for the same decision& in the same manner that one #erson may #urchase a

sweater that differs from that #urchased !y someone else. $e are interested in

our #references& i.e. which o#tion you #refer to have& so #lease thin" carefully

a!out each decision& and #lease do not tal" with others in the room. Are there any

'uestionsN

1eft 6ide 3ight 6ide Mour

Choice

1 or 3

(ecision 0 %.00 if throw of die is 0+ .00 if throw of die is 0+-

0.00 if throw of die is 0+

(ecision , 2.00 if throw of die is 0+ /.00 if throw of die is 0+-

/.00 if throw of die is 0+

(ecision 2 %.00 if throw of die is 0+2/0.00 if throw of die is 25+

.00 if throw of die is 0+,0.00 if throw of die is 20+

(ecision 0.00 if throw of die is 0+ /0.00 if throw of die is 0+-0.00 if throw of die is 0+

(ecision / /.00 if throw of die is 0+/

.20 if throw of die is 50+

-.-0 if throw of die is ,+/


(ecision 5 %.00 if throw of die is 0+ .00 if throw of die is 0+-


(ecision % 2.00 if throw of die is 0+/

,.20 if throw of die is 50+

5.-0 if throw of die is ,+/

,.0 if throw of die is 50+

(ecision - %.00 if throw of die is 0+2/


.00 if throw of die is 0+,


(ecision /.00 if throw of die is 0+/

.20 if throw of die is 50+

-.-0 if throw of die is ,+/


(ecision 0.00 if throw of die is 0+2/


/0.00 if throw of die is 0+,




6earch nstructions =Cha#ter >

n this game& we will use throws of dice to determine a series of money amounts&

and you have to choose which amount to acce#t& with the understanding that each

additional throw entails a cost that will !e deducted from the money amount that

you finally acce#t.8lease loo" at this ,0+sided die. he sides are mar"ed from 0 to & so if

throw it twice and use the first throw for the tens digit& will determine a num!er

from 0 to cents. 7y ignoring the first draw if it is a =and throwing again for a

new first num!er& will o!tain a num!er from !etween =and including> 0 and .

A num!er of #ennies determined in this way will !e yours to "ee#& !ut there is a

cost of o!taining this amount.

n #articular& if you #ay a cost of 5 cents& will throw the dice for you& and

you can either "ee# the amount determined& or you can decide to #ay another 5

cents and )ll throw the dice again to get a second num!er in the range from 0 to

cents. =f the throw for the tens digit is a & will throw the die again.> Mou

can do this as many times as you wish =with no limit>& !ut you have to #ay 5 cents

for each new num!er. $hen you decide to sto#& you can use the highest 2+digit

num!er that you have received u# to that #oint& and your earnings will !e that

num!er minus the total cost of the search #rocess& which is 5 cents times the

num!er of times that you as"ed me to throw the dice. here is no limit on the

num!er of times you can #ay the 5+cent cost in search of a higher num!er.

Use the ta!le on the net #age to "ee# records& and you can use another sheet if

you need additional s#ace. *ach new 2+digit num!er will !e called an ZofferZ. ?or

the ,st offer that you #ay to o!tain& loo" at the Z,stZ column. After throw the

dice& write the num!er in the to# row for Zvalue of current offerZ. he highest

offer received u# to now is written in the second row. =At first& the current offer isalso the highest offer.> 4oving down to the net row& the total search cost is the

num!er of draws times 5 cents. he Zearnings if you sto#Z are calculated !y

su!tracting the total search cost from the highest draw thusfar. After calculating

these earnings& you can decide whether to sto# or to #ay and search again. $rite

Z6Z for sto# or ZCZ for continue. $hen you sto#& #lease circle your earnings at

that #oint and sto# recording the new num!ers. will "ee# throwing the dice and

announcing num!ers until everyone had decided to sto#.

hen the whole #rocess will !e re#eated& for another Zsearch se'uenceZ& etc . All

earnings are hy#othetical unless otherwise indicated. Are there any 'uestions

!efore we !eginN

Mour ;ame: SSSSSSSSSSSSSSSSSSS

3ecord of the ?irst 6earch 6e'uence



order of offer ,st 2nd rd /th 5th %th -th th th ,0t

h

value of current offer

=in cents>

highest offer thusfar

total search cost

=5 for each search>

5 ,0 ,5 20 25 0 5 /0 /5 50

earnings if you sto#

=use + for losses>

will you sto# nowN

=6to#Continue>

3ecord of the 6econd 6earch 6e'uence

order of offer ,st 2nd rd /th 5th %th -th th th ,0th


=in cents>

highest offer thusfar

total search cost

=5 for each search>

5 ,0 ,5 20 25 0 5 /0 /5 50


=use + for losses>

will you sto# nowN

=6to#Continue>

P%ease note: the cost of search for the ne(t se#$ence has increased to + cents.

?inally& will do this again for a search cost of + cents& !ut there is a maimum

of three offers& which are e'ually li"ely to !e any amount !etween 0 and cents.

Mou can sto# !efore the first draw and earn 0& or you can #ay 20 cents for the first

offer& and sto# or #ay 20 more for a second& and sto# with the highest of the two

or #ay 20 more for a third& and ta"e the highest of the three. 7efore doing this&

#lease decide how high the first offer must !e for you to sto# after only one& and

how high the !est of the two initial offers must !e for you to sto# after only two



times. Mou may want to discuss this with your neigh!or. 8lease record your

decisions !elow.

In the first offer& will sto# if it is at least SSSSSSS .

In the 2nd draw& will sto# if the highest of the first two offers is at least

SSSSSSS .

;ow will throw the dice for the ,st offer& etc.& and you can record your earnings

!elow:

order of offer ,st 2nd rd /th 5th %th -th th th ,0t

h


=in cents>

Highest offer thusfar

otal search cost =20

for each search>

20 /0 %0 0 ,00 ,20 ,/0 ,%0 ,0 200


=use for losses>

will you sto# nowN

=6to#Continue>

nstructions: raveler s (ilemma =Cha#ter ,,>

will now distri!ute one record sheet to each #air =or small grou#> of #eo#le in

the class. n each #eriod& your grou# will !e randomly matched with another

grou# of students. he decisions made you grou# and !y the other grou# will

determine the amounts earned !y each grou#.

At the !eginning of each #eriod& you will choose a num!er or Zclaim.Z Claims

will !e made !y writing the claim the record sheet. Mour claim amount may !e

any amount !etween and including 0 and 200 cents. he other grou# will also

ma"e a claim !etween and including 0 and 200 cents. f the claims are e'ual&

then your grou# and the other grou# each receive the amount claimed. f the

claims are not e'ual& then each of you receives the lower of the two claims. naddition& the grou# which ma"es the lower claim earns a reward of ,0 cents& and

the grou# with the higher claim #ays a #enalty of ,0 cents. hus your grou# will

earn an amount that e'uals the lower of the two claims& #lus a ,0 cent reward if



you made the lower claim& or minus a ,0 cent #enalty if you made the higher

claim. here is no #enalty or reward if the two claims are eactly e'ual& in which

case each grou# receives what they claimed.

*am#le: 6u##ose that your claim is X and the other claim is M.

f X J M& you get X& and the other gets M.

f X P M& you get M minus ,0 cents& and the other gets M #lus ,0. f X Q M&

you get X #lus ,0 cents& and the other gets X minus ,0.

8lease loo" at your 3ecord 6heet and write your name or assigned ( num!er in

the u##er right corner. oing from left to right& you will see columns for the

#eriod& your claim& other claim& minimum claim& #enalty or reward =if any>& and

your earnings. Mour grou# !egins !y writing down your own claim in the first

column& for the current period on#. As mentioned a!ove& this claim must !e

greater than or e'ual to 0 cents and less than or e'ual to 200 cents& and the claim

may !e any amount in this range. 8lease do not use decimals& e.g. write 5

instead of 0. 5 to indicate 5 cents.After you record your claim& hold your #a#er u# to your chest& and will

designate two grou#s and as" them to announce their decisions. $hen you find

out the other grou# s claim& #lease write it in the third column& followed !y the

minimum& the #enalty or reward =if the claims are not e'ual>& and your earnings.

hen leave your #a#er on your des" so that will "now which #eo#le have not

!een matched yet. his same #rocess is re#eated.

Record Sheet for Minim$m C%aim Game Mour (: SSSSSSSSSSS

#eriod your claim other claim minimum #enaltyreward earnings

,

2

/

5

%

-



,0



nstructions: Coordination ame =Cha#ter ,2>

will now distri!ute one record sheet to each of you. 8lease write your name or

( num!er in the u##er right+hand corner. n each round of this e#eriment& you

will !egin !y choosing a num!er or ZeffortZ that can !e ,& 2& & /& 5& %& or -.*fforts will !e made !y writing the num!er you choose in the second column of

your record sheet& for the current round on#. $hen everyone is finished& the

sheets will !e collected. hen will call out the num!ers and as" one of you to

write them on the !oard.

Mour #ayoff will de#end on your effort choice and on the minimum of all efforts&

including your own. After we write all efforts on the !lac"!oard& the lowest one

will !e circled. he second lowest effort will !e mar"ed with an asteris". Mour

#ayoff will !e determined in one of two ways:

• f your effort is not the lowest& then the circled num!er will determine the

relevant column of the #ayoff ta!le !elow& and your effort determines the

row. he num!er in the cell that corres#onds to the intersection of this

row and column will !e your earnings in cents.

• f your effort is the lowest& then the lowest of the other efforts is the

num!er mar"ed with an asteris"& and that num!er determines the column&

while your num!er determines the row as !efore.

Minim$m of 5ther "fforts

' / 3 * 1 )

) ,0 0 50 -0 0 ,,0 ,01 20 /0 %0 0 ,00 ,20 ,20

Qo$r * 0 50 -0 0 ,,0 ,,0 ,,0

"ffort 3 /0 %0 0 ,00 ,00 ,00 ,00 / 50 -0 0 0 0 0 0

%0 0 0 0 0 0 0

' -0 -0 -0 -0 -0 -0 -0

Ince you "now the minimum of the other efforts and have determined your

earnings& #lease write them in the relevant columns& and we will !egin the net

round.

Record Sheet for "ffort Game Mour ;ame(: SSSSSSSSSSSSSS



3ound Mour *ffort 4inimum of

Ither *fforts

Mour

*arnings

,

2

/

5

%

-

,0

nstructions: 4ar"et ame =Cha#ter ,>

will now distri!ute one record sheet to each #air =or small grou#> of #eo#le in

the class. n each mar"et #eriod =after the first one>& your grou# will !e matched

with the same other grou# of students. he decisions made !y your grou# and !y

the other grou# will determine the amounts earned !y each grou#. =n the first

#eriod& your grou# will !e the only seller in your mar"et.>

At the !eginning of each #eriod& your grou# will choose a 'uantity to #roduce

and sell. (ecisions will !e made !y writing the 'uantity on your record sheet.

Mour 'uantity may !e any amount !etween and including , and , units. he

other grou# will also select a 'uantity on this interval. *ach unit that you #roduce

will cost your grou# ,& and similarly for the other grou#. he #rice at which you

can sell all units is determined !y the total 'uantity& that is& the sum of your

'uantity and that of the other grou#& as shown in the ta!le at the to# of your



record sheet. ?or eam#le& if you each select a 'uantity of ,& the total is 2 and the

#rice will !e ,,. f you each select ,& then the total will !e 2%& and the #rice will

!e 0& as will !e the case whenever the total is greater than ,2.

*am#le: 6u##ose that your 'uantity is 2 and the other grou# selects ,.

he #rice will !e SSSS.

Mour total sales revenue =#rice times your 'uantity> will !e SSSSS.

Mour total cost =, times your 'uantity> will !e SSSSS.

Mour #rofit =total revenue total cost> will !e SSSSS.

Mou should have an answer of , in #rofit.

;ow #lease write your name or assigned ( num!er in the u##er right corner of

your record sheet. oing from left to right& you will see columns for the 8eriod&

Mour Luantity& Ither s Luantity& 4ar"et 8rice& otal 3evenue& Mour Cost& and

Mour *arnings. Mour grou# !egins !y writing down your own 'uantity in the

a##ro#riate column& for the current period on#. As mentioned a!ove& this

'uantity must !e !etween =and including> 0 and ,2. he units are indivisi!le& so #lease use only integer amounts =no fractions>. After you find out the other

grou# s 'uantity& you fill out the information for the current #eriod 9ust as you did

in the eam#le. here will !e a num!er of rounds& as determined !y your

instructor.

At this time& ma"e your decision for the first #eriod. ;ote that the other grou# s

'uantity has !een set to 0& i.e. your grou# will !e the only seller in this first

#eriod. After that& you will !e matched with the same other grou# in all

remaining #eriods.

Record Sheet for Market Game Mour ;ame(: SSSSSSSSSSSSS

Determination of Price as a 0$nction of ota% <$antit!

< , 2 / 5 % - ,0 ,, ,2 ,O

Price ,2 ,, ,0 - % 5 / 2 , 0

8eriod Mour

Luantity

Ither s

Luantity

4ar"et

8rice

otal

3evenue

Mour Cost Mour

*arnings

, +

2



/

5

%

-

,0

nstructions: 8riceLuality 4ar"et =Cha#ter

,->=hese instructions are !ased on Holt and 6herman& ,.>

his is a mar"et with / !uyers and sellers& and each of you will !e assigned to a

grou# that will have one of these roles. he sellers will !egin !y choosing a #rice

and a 'uality Zgrade.Z $e will collect these decisions and write them on the

!lac"!oard. hen we will give !uyers the chance to #urchase from one of the

sellers at the grade and #rice listed. he grade can !e any num!er from , to .

he grade is li"e a 'uality attri!uteB a higher grade costs more to #roduce and is

worth more to !uyers. he ta!le on your record sheet shows your costs of

different grades if you are a seller& and it shows your money values of different

grades if you are a !uyer.

*ach !uyer can !uy only , ZunitZ of the commodity during a #eriod. *ach seller

can sell u# to 2 units in a #eriod& !ut the 2nd unit costs , more to #roduce. f

you are a seller& the to# row of the ta!le a!ove shows the cost of the ,st unit that

you actually sell in a #eriod =for the grade you choose>& the 2nd unit costs , more

than the ,st unit. Unsold units are not #roduced and hence incur no cost.7uyers earn money !y ma"ing a #urchase at a #rice that is !elow the value. he

value to the !uyer de#ends only on the grade& not on whether it is the seller)s ,st

or 2nd unit in the #eriod. A !uyer)s earnings are calculated as the difference



!etween the value and the #urchase #rice. f a !uyer does not ma"e a #urchase&

the !uyer earns 0 for the #eriod.

6ellers earn money !y ma"ing one or more sales at a #rice that is a!ove the cost

of the unit =determined from the ta!le a!ove>. A seller)s earnings are calculated

as the sum of the earnings on the units actually sold& and such earnings are

calculated as the difference !etween the sale #rice and the cost of the grade

#roduced. A seller who does not ma"e a sale in a #eriod will earn 0.

$hen all sellers have finished choosing their #rices and grades for the #eriod& we

will collect these sheets and write the #rices and grades on the !lac"!oard under

the seller num!ers. hen will draw randomly select a !uyer num!er& and that

!uyer can #urchase a unit from one of the sellers or choose not to #urchase. he

remaining !uyers are selected in orderB if !uyer 2 goes first& then !uyer is

second& ... and !uyer , is last. Ince a seller has sold a unit& the 2nd unit costs ,

more& so the seller will !e as"ed whether he or she wishes to sell a 2nd unit at the

advertised #rice and grade. f a 2nd unit is sold& it must !e at the same #rice and

grade as the ,st unit. f a seller refuses to sell or sells !oth units in a #eriod& will

draw a line through that seller)s #rice.Mou can use the ta!le !elow to calculate =hy#othetical> earnings. Any 'uestionsN

$e will !egin !y having each seller choose a #rice and a grade for #eriod ,&

which you should write in the to# two rows of your record sheet.



Se%%er Record Sheet 6eller ;um!er: SSSS

grade , grade 2 grade

seller cost of ,st unit ,./0 /.%0 ,,.00

seller cost of 2nd unit 2./0 5.%0 ,2.00

#d., #d.2 #d. #d./ #d.5

,> grade for current #eriod SSSSS SSSSS SSSSS SSSSS SSSSS

2> #rice for current #eriod SSSSS SSSSS SSSSS SSSSS SSSSS

> sales #rice on ,st unit SSSSS SSSSS SSSSS SSSSS SSSSS

/> cost of ,st unit SSSSS SSSSS SSSSS SSSSS SSSSS

5> #rofit on ,st unit: => =/> SSSSS SSSSS SSSSS SSSSS SSSSS

%> sales #rice on 2nd unit SSSSS SSSSS SSSSS SSSSS SSSSS

-> cost of 2nd unit SSSSS SSSSS SSSSS SSSSS SSSSS

> #rofit on 2nd unit: =%> =-> SSSSS SSSSS SSSSS SSSSS SSSSS

> total #rofit: =5> O => SSSSS SSSSS SSSSS SSSSS SSSSS

,0> cumulative #rofit SSSSS SSSSS SSSSS SSSSS SSSSS

B$!er Record Sheet 7uyer ;um!er SSSSSS

rade , grade 2 grade !uyer

value /.00 .0 ,.%0



#d., #d.2 #d. #d./ #d.5

,> ( of seller of #roduct SSSSS SSSSS SSSSS SSSSS SSSSS

2> grade of #roduct SSSSS SSSSS SSSSS SSSSS SSSSS

> value to you =from ta!le> SSSSS SSSSS SSSSS SSSSS SSSSS

/> #urchase #rice SSSSS SSSSS SSSSS SSSSS SSSSS

5> earnings: => =/> SSSSS SSSSS SSSSS SSSSS SSSSS

%> cumulative earnings SSSSS SSSSS SSSSS SSSSS SSSSS

8rivate Value Auction =Cha#ter ,>

$e will conduct a series of auctions in which the highest !idder o!tains the

Z#ri<eZ. n each auction& you will !e #aired with another randomly selected

mem!er of the class. Mou and the #erson you are #aired u# with will each see half

of the #ri<e value. he value of the #ri<e will !e determined !y throws of ,0

sided dice. 8rior to the auction& we will come to each of you and throw the ,0

sided die two times: the first throw determines the dimes digit& and the final throw

determines the #ennies digit. hese two throws determine the #ri<e value to you

if you win. he other #erson will also see two throws of the dice that will

determine hisher #ri<e value& so the #ri<e will generally !e worth different

amounts to each of you. Mou will ma"e your !id decision after seeing your

value& !ut without "nowledge of the other #erson)s value or the other #erson)s !id.

he higher !idder wins the #ri<e and earns the difference !etween their own #ri<e

value and their own !id amount. he low !idder earns nothing. Mou will !e

#aired with another #erson in the classroom in each #eriod. his other #erson will

!e selected at random& unless your instructor indicates otherwise.he 3ecord 6heet should !e used to "ee# trac" of your decisions. he ( num!er

will !e used to match you with another !idder each #eriod. he #eriod num!er is

shown on the left side of each row. At the !eginning of the #eriod& will come to

your des" to throw the dice that determine your #ri<e value& which can !e



recorded in column =,>. After recording this num!er& you should decide on a !id

for the #eriod& which will !e entered in column =2>. (o this when you ma"e your

!id& !efore you hear what the other #erson)s !id was. hen will match you with

someone else in the class& and you will each call out your !ids. Mou should use

column => to record the other #erson)s !id. f you had the higher !id& your

earnings will !e the value =,> minus your !id in =2>. f you had the lower !id&

your earnings will !e <ero for the #eriod. *arnings are entered in column =/>.

Any 'uestionsN

Record Sheet for Private a%$e -$ction Mour (: SSSSSSS

8eriod

=,>

Mour

Value

=2>

Mour

7id

=>

Ither

7id

=/>

Mour

*arnings

=5>

Cumulative*arnings

,

2

/

5

Instr$ctions for Part B:

his #art will !e the same as !efore& with values determined !y the throws of ,0

sided dice and with the high !idder !eing the winner. he only difference is thatthe winning !idder only has to #ay the secondhighest +id price. he high !idder

earns the difference !etween their own value and the other #erson s !id& and the

low !idder earns nothing. 6o& for eam#le& if you value is 2 and you !id and

the other #erson !ids E & then you win with a !id of !ut you only have to #ay E &

so you earn 2 E .

Record Sheet for Second Price Private a%$e -$ction

8eriod

=,>

Mour

Value

=2>

Mour

7id

=>

Ither

7id

=/>

Mour

*arnings

=5>

Cumulative*arnings

%



-

,0

nstructions for a"eover ame =Cha#ter 20>

n a minute will divide the class into 2+#erson teams. Half of the teams will !e

!uyers and half will !e sellers in a mar"et. will choose teams and indicate your

role: !uyer or seller. *ach team should have a single co#y of these instructions&

with the role assignment& !uyer or seller& written net to the #lace for your names

!elow.

*ach seller team is the owner of a !usiness. he money value of the !usiness to

the seller is only "nown !y the seller. *ach of the !uyers will !e matched withone of the sellers. he !uyer will ma"e a single !id to !uy the !usiness. he

!uyer does not "now the value of the !usiness to the seller& !ut the !uyer is a

!etter manager and "nows that he or she can increase the #rofits to ,.5 times the

current level. After receiving the !uyer)s !id& the seller must decide whether or

not to acce#t it. he seller will earn the value of the !usiness if it is not sold& and

the seller will earn the amount of the acce#ted !id if the !usiness is sold. he

!uyer will earn nothing if the !id is re9ected. f the !uyer)s !id is acce#ted& the

!uyer will earn ,.5 times the seller)s value& minus the !id amount.

A ten+sided die will !e thrown twice to determine the value to the seller& in

thousands of dollars. his will !e done for each seller individually. he die is

num!ered from 0 to B the first throw determines the tens digit and the second

determines the ones digit& so the seller value is e'ually li"ely to !e any integer

num!er of thousands of dollars& from 0 to thousand dollars. f the firm is

#urchased !y the !uyer& it will !e worth ,.5 times the seller value& so the value to

the !uyer will !e !etween 0 and ,/.5 thousand dollars.

he form on the net #age can !e used to record the outcomes. ?or sim#licity&

records will !e in thousands of dollarsB i.e. a 50 means 50 thousand. will !egin

!y coming to each seller)s des" to throw the ,0+sided die twice& and you can

record the resulting seller value in column =,>. hen each !uyer& not "nowing the

seller)s value& will decide on a !id and record it in column =2>. will then match

the !uyers and sellers randomly& and each !uyer will communicate the !id to thecorres#onding seller& who will say yes =acce#t> or no =re9ect>. he seller s

decision is recorded in column =>. f the !id is acce#ted& then the seller will

communicate the seller)s value to the !uyer& who will multi#ly it !y ,.5 and enter

the sum in column =/>. ?inally& !uyers and sellers calculate their earnings in the



a##ro#riate column =5> or =%>. he !uyer earns either 0 =if no #urchase> or the

difference !etween the !uyer value =/> and the acce#ted !id. he seller either

earns the seller value =if no sale> or the amount of the acce#ted !id. *ach of you

will !egin with an initial cash !alance of 500 thousand dollarsB gains are added to

this amount and losses are su!tracted. Mou can "ee# trac" of your cumulative

cash !alance in column =->. Are there any 'uestionsN

;ow we will come around to the des"s of sellers and throw the dice to determine

the seller values. $hile we are doing this& those of you who are !uyers should

now decide on a !id and enter it in column =2>. ?or those of you who are !uyers&

the seller value column =,> is !lan". Mou only find out the seller value after you

announce your !id. hose of you who are sellers have no decision to ma"e at this

time. $hen you hear the !uyer s !id& you must choose !etween "ee#ing the value

of the firm =determined !y the dice throws> or giving it u# in echange for the

!uyer !id amount. 8lease only write in the first row of the ta!le at this time.

=After everyone has !een matched and has calculated their earnings for the first

set of decisions& you will switch roles& with !uyers !ecoming sellers& and vice

versa. he second set of decisions will !e recorded in the !ottom row.>

Record Sheet for akeover Game

Mour 3ole: SSSSSSSSSSSSSSSS Mour ;ame: SSSSSSSSSSSSSSSSSSSSSS

=,> =2> => =/> =5> =%> =->

3ound 6eller s

Value

7uyer s

7id

6eller s

(ecision

=acce#t or

re9ect>

Value to

7uyer

,.5R=,>

6eller s

*arnings

=,> or =2>

7uyer s

*arnings

=/>=2>

or 0

Cash

7alance

500

=thousand>

,

2

Common Value Auction =Cha#ter 2,>

$e will conduct a series of auctions in which the highest !idder o!tains the

Z#ri<eZ. n each auction& you will !e #aired with another randomly selected

mem!er of the class. Mou and the #erson you are #aired with will each see half of

the #ri<e value. he value of the #ri<e will !e determined !y throws of ,0 sided



dice. 8rior to the auction& we will come to each of you and throw the ,0 sided die

two times: the first throw determines the dimes digit& and the final throw

determines the #ennies digit. hese two throws determine your value com#onent&

which is e'ually li"ely to !e any #enny amount from 0.00 to 0.. he other

#erson will see two throws of the dice that will determine hisher value

com#onent in eactly the same manner. he value of the #ri<e is 9ust the sum of

the two value com#onents& so the #ri<e value will !e no less than 0.00 and no

greater than ,.. Mou will ma"e your !id decision after seeing your value

com#onent& !ut without "nowledge of the other #erson)s value com#onent or the

other #erson)s !id. he higher !idder wins the #ri<e and earns the difference

!etween the #ri<e value =sum of the two value com#onents> and the !id amount.

his will !e a gain if the !id is lower than the #ri<e value& and this will !e a loss if

the !id is higher than the #ri<e value. he low !idder earns nothing. Mou will !e

randomly #aired with another #erson in the classroom in each #eriod& so you only

!id against one other #erson& !ut that #erson will generally !e different in each

successive auction Z#eriodZ.

he 3ecord 6heet should !e used to "ee# trac" of your decisions. he ( num!er will !e used to match you with another !idder each #eriod. he #eriod num!er is

shown on the left side of each row. At the !eginning of the #eriod& will come to

your des" to throw the dice to determine your value com#onent& which can !e

recorded in column =,>. After recording this num!er& you should decide on a !id

for the #eriod& which will !e entered in column =2>. (o this when you ma"e your

!id& !efore you hear what the other #erson)s !id was. hen will randomly match

you with someone else in the class& and you will each call out your !ids. Mou

should use column => to record the other #erson)s !id. At this time& you will

"now whether you won the auction or not& !ut you will not yet "now the #ri<e

value. ?inally& each #erson will announce their value com#onents& so that you can

record the other #erson)s value com#onent in column =/> and calculate the sum of the com#onents in column =5>. f you had the higher !id& your earnings will !e

the value sum in =5> minus your !id in =2>. f you had the lower !id& your

earnings will !e <ero for the #eriod. *arnings are entered in column =%>. Mou

!egin #eriod , with an initial cash !alance of 5.00B gains are added to this and

losses are su!tracted. Mou can "ee# trac" of your cumulative cash !alance in

column =->. Any 'uestionsN

Record Sheet for Common a%$e -$ction Mour (: SSSSSSS

8eriod

=,>Mour

Value

Com#onent

=2>Mour

7id

=>Ither

7id

=/>Ither s

Value

Com#onent

=5>6um of

Value

Com#onents

=%>Mour

*arnings

=->Cumulative*arnings

*.++

,



2

/



4ulti+Unit Auction =Cha#ter 22>

=hese instructions are loosely ada#ted from a #ractice #rocedure in the

instructions used for some of the e#eriments re#orted in Cummings& Holt& and

1aury =200,>. he num!er of units to !e #urchased in the auction is announced inadvance. An alternative to the fied #urchase 'uantity is for the instructor to

announce a !udget amount that will !e s#ent& which corres#onds more closely to

the irrigation reduction auction. $e use a fied 'uantity here to clarify the

incentives in the uniform #rice auction. he instructor may wish to give the

discriminatory auction instructions to half of the class& and to give the uniform

#rice instructions that follow to the other half.>

! (iscriminator !uction

*ach #erson has !een given a single colored #en. his is yours to "ee# when you

leave today& unless you decide to sell it !ac" to us. Here have an amount of

money that is sufficient to #urchase SSSSSSS of these #ens. $hat will do is let

each of you write down an offer to sell your #en& using the form:

Mour ;ame:

6ale offer #rice:=!etween 0.00 and ,.00>

8lease write your name and offer =!etween 0.00 and ,.00> in the two !oes.

After you have turned in your offer& all offers will !e ran"ed from low to high&

regardless of #en color& and the SSSSS lowest offers will !e acce#ted. *ach

#erson with an acce#ted offer will receive an amount that e'uals that person9s

o;n offer amount& and they must& in turn& give u# their #ens. 8eo#le with re9ectedoffers will "ee# their #ens.

?or eam#le& if the 'uantity goal were two #ens and the !ids were ,0

cents& , cents& ,5 cents& and 20 cents& then two !ids =,0 and ,> would !e

acce#ted. his is !ecause we start with the lowest !id and sto# when the 'uantity



goal is reached. he #eo#le with acce#ted offers get reim!ursed an amount that

e'uals their own offer& which means that one #erson receives ,0 cents and the

other receives , cents. *ach of these two #eo#le must then give u# the #en. he

others "ee# their #ens and receive no #ayment. n the actual auction& the 'uantity

goal will !e SSSSS #ens instead of the two #ens used in this eam#le.



! >niform Price !uction

*ach #erson has !een given a single colored #en. his is yours to "ee# when you

leave today& unless you decide to sell it !ac" to us. Here have an amount of

money that is sufficient to #urchase SSSSSSS of these #ens. $hat will do is let

each of you write down an offer to sell your #en& using the form:

Mour ;ame:

6ale offer #rice:

=!etween 0.00 and ,.00>

8lease write your name and offer =!etween 0.00 and ,.00> in the two !oes.

After you have turned in your offer& all offers will !e ran"ed from low to high&

regardless of #en color& and the SSSSS lowest offers will !e acce#ted. *ach

#erson with an acce#ted offer will receive an amount that e'uals the #o;est

reected offer & and they must& in turn& give u# their #ens. 8eo#le with re9ected

offers will "ee# their #ens.

?or eam#le& if the 'uantity goal were two #ens and the !ids were ,0cents& , cents& ,5 cents& and 20 cents& then two !ids =,0 and ,> would !e

acce#ted. his is !ecause we start with the lowest !id and sto# when the 'uantity

goal is reached. he #eo#le with acce#ted offers get reim!ursed an amount that

e'uals the lowest re9ected !id& which is ,5 cents in this case. hese #eo#le must

then turn each give u# the #en. he others "ee# their #ens and receive no

#ayment. n the actual auction& the 'uantity goal will !e SSSSS #ens instead of

the two #ens used in this eam#le.

7argaining =Cha#ter 2>

8ro#oser ;um!er: SSSSS



have ,0 to s#lit. hose in the front of the room are res#onders& and those in the

!ac" are #ro#osers. here are ,, ways that the ,0 can !e divided !etween two

#eo#le in even dollar amounts =see the ta!le !elow>. he #ro#oser must suggest

one of these !y circling one =and only one> of the listed o#tions. he #ro#oser

sheets will then !e collected and shuffled. Ine of the sheets will !e given to each

res#onder& who must either acce#t or re9ect. f the res#onder acce#ts& then the

#ro#osal is enacted. f the res#onder re9ects& then the money is not divided and

each #erson& #ro#oser and res#onder& will earn nothing. hen will collect the

sheets and use the throw of a ten+sided die to select one of the #ro#oser num!ers&

and the earnings =if any> determined !y the decisions on that sheet will actually !e

#aid to the #ro#oser and res#onder who made those decisions. his #ayment will

!e in cash.

0 for the #ro#oser& ,0 for the res#onder

, for the #ro#oser& for the res#onder

2 for the #ro#oser& for the res#onder

for the #ro#oser& - for the res#onder

/ for the #ro#oser& % for the res#onder

5 for the #ro#oser& 5 for the res#onder

% for the #ro#oser& / for the res#onder

- for the #ro#oser& for the res#onder

for the #ro#oser& 2 for the res#onder for the #ro#oser& , for the res#onder

,0 for the #ro#oser& 0 for the res#onder

3es#onder ;um!er: SSSSS

SSSSSSS acce#t& and earnings will !e determined !y the #ro#osal.

SSSSSSS re9ect& and !oth of us will earn nothing.

8lay+or+Dee# ame =Cha#ter 2%>=hese instructions are ada#ted from those in Holt and 1aury& ,-.>



*ach of you will now !e given four #laying cards& two of which are red

=hearts or diamonds>& and two of which are !lac" =clu!s or s#ades>. All of your

cards will !e the same num!er.

he eercise will consist of a num!er of rounds. At the start of a round&

will come to each of you in order& and you will #lay t;o of your four cards !y

#lacing these two cards face down on to# of the stac" in my hand.

Mour earnings in dollars are determined !y what you do with your red

cards. ?or each red card that you "ee# in a round you will earn four dollars for

the round& and for each !lac" card that you "ee# you will earn nothing. 3ed cards

that are #laced on the stac" affect everyone)s earnings in the following manner.

will count u# the total num!er of red cards in the stac"& and everyone will earn

this num!er of dollars. 7lac" cards #laced on the stac" have no effect on the

count. $hen the cards are counted& will not reveal who made which decisions.

o summari<e& your earnings for the round will !e calculated:

Mour *arnings J / times the ` of red cards you "e#t

O , times the total ` of red cards collect.

At the end of the round& will return your own cards to you !y coming to

each of you in reverse order and giving you the to# two cards& face down& off of

the stac" in my hand. hus you !egin the net round with two cards of each

color& regardless of which cards you 9ust #layed.

After round 5& will announce a change in the earnings for each red card

you "ee#. *ven though the value of red cards "e#t will change& red cards #laced

on the stac" will always earn one dollar for each #erson.

Use the s#ace !elow to record your decisions& your earnings& and your

cumulative earnings. =I#tional: At the end of the game& one #erson will !e

selected at random and will !e #aid SSSS K of his or her actual earnings& in cash.>All earnings are hy#othetical for everyone else. Are there any 'uestionsN

3ecord 6heet: *arnings from *ach 3ed Card Collected J ,.00

3ound ;um!er of

3ed Cards

De#t

Value of

*ach 3ed

Card De#t

*arningsfrom 3ed

Cards De#t

*arningsfrom 3ed

CardsCollected

otal

*arnings

Cumulative

*arnings



, /

2 /

/

/ /

5 /

% 2

- 2

2

2

,0 2

Volunteer s (ilemma ame =Cha#ter 2->=he terminology in these instructions is designed for classroom use.>

*ach of you will now !e given two #laying cards& one red =hearts or

diamonds>& and one !lac" =clu!s or s#ades>. At the start of a round& will as" each of you to #lay one of your cards !y #ic"ing it u# and holding it against your

chest with the color hidden until as" everyone to reveal which card they #layed.

After the card #lay decisions have !een made& !ut !efore they have !een

revealed& will select one or more others to !e in your grou# and will as" all of

you to reveal your cards at the same time. Mour earnings in dollars are determined

!y the card that you #lay and !y the cards #layed !y the others in your grou#.

hin" of #laying a red card as a decision to volunteer to #erform some tas".

Volunteering has a cost for you =0.25>& !ut everyone in the grou# will !enefit =!y

receiving 2.00> if at least one #erson volunteers. here is no additional !enefit if

there is more than one volunteer in your grou#. he #ayoffs are:

8ayoff if there is no volunteer: 0.00

8ayoff if there is at least one volunteer and you (I ;I volunteer: 2.00

8ayoff if there is at least one volunteer and you (I volunteer: ,.-5



$e will !egin with grou#s of si<e 2 for the first 5 rounds& so when #oint

to you and to another randomly selected #erson in the room& show your cards&

return your card to your des"& and calculate your earnings. he #eo#le who have

not yet !een matched should "ee# holding their cards to their chests so that can

continue to match #eo#le until no!ody remains to !e matched. hen the net

round will !egin. *ach #erson will decide which card to #lay for that round !y

holding that card against their chest until they are matched. After round 5& will

divide #eo#le into randomly selected grou#s of si<e / in each round. 7efore

!eginning& will select one #erson to record all decisions.

All earnings are hy#othetical. =I#tional: At the end of the game& one

#erson will !e selected at random and will !e #aid SSSS K of his or her actual

earnings& in cash.> Are there any 'uestionsN

3ecord 6heet for Volunteer s (ilemma ame

3ound rou# 6i<e

Card 8layed

=3 or 7>

;um!er of

3ed Cards

8layed in

Mour rou#

Mour

*arnings

Cumulative

*arnings

, 2

2 2

2

/ 2

5 2

% /

- /



/

/

,0 /

8ayoff if there is no volunteer: 0.00

8ayoff if there is at least one volunteer and you (I ;I volunteer: 2.00

8ayoff if there is at least one volunteer and you (I volunteer: ,.-5

1o!!ying ame nstructions =Cha#ter 2>6ource: hese instructions are ada#ted from oeree and Holt =,>.

his is a sim#le card game. *ach of you has !een assigned to a team of investors

!idding for a local government communications license that is worth ,%&000.

he government will allocate the license !y choosing randomly from the

a##lications received. he #a#erwor" and legal fees associated with each

a##lication will cost your team &000& regardless of whether you o!tain the

license or not. =hin" of this &000 as the o##ortunity cost of the time and

materials used in com#leting the re'uired #a#erwor".> *ach team is #ermitted to

su!mit any num!er of a##lications& u# to a limit of thirteen #er team. *ach team

!egins with a wor"ing ca#ital of ,00&000.

here will !e four teams com#eting for each license& each of which is #rovided

with thirteen #laying cards of the same suit. Mour team will #lay an num!er of

these cards !y #lacing them in an envelo#e #rovided. *ach card you #lay is li"e alottery tic"et in a drawing for a #ri<e of ,%&000. All cards that are #layed !y

your team and the other three teams will !e #laced on a stac" and shuffled. hen

one card will !e drawn from the dec". f that card is one of your suit& your team

will win ,%&000. Itherwise you receive nothing from the lottery. $hether or

not you win& your earnings will decrease !y &000 for each card that you #lay.

o summari<e& your earnings are calculated:

*arnings J ,%&000 if you win the lottery

+ &000 times the num!er of cards you #lay.

*arnings are negative for the teams that do not win the lottery& and negativeearnings are indicated with a minus sign in the record ta!le !elow. he

cumulative earnings column on the right !egins with ,00&000& reflecting your



initial financial ca#ital. *arnings should !e added to or su!tracted from this

amount. Are there any 'uestionsN

3ound ;um!er

Cards

8layed

Cost #er

Card

8layed

otal

Cost

1icense

Value

Mour

*arnings

Cumulative

*arnings

,00&000

, &000 ,%&000

he net lottery is for a second license. Mour team !egins again with thirteen

cards& !ut the cost of each card #layed is reduced to ,&000& due to a government

efficiency move that re'uires less #a#erwor" for each a##lication. his license is

worth ,%&000 as !efore& whether or not your team already ac'uired a license.

3ound ;um!er

Cards8layed

Cost #er

Card8layed

otal

Cost

1icense

Value

Mour

*arnings

Cumulative

*arnings,00&000

2 ,&000 ,%&000

n the net lottery& the value of the license may differ from team to team. Mour

team !egins again with thirteen cards& and the cost of each card #layed remains at

,&000. Mour instructor will inform you of the license value& which you should

write in the a##ro#riate #lace in the ta!le !elow.

3ound ;um!er

Cards

8layed

Cost #er

Card

8layed

otal

Cost

1icense

Value

Mour

*arnings

Cumulative

*arnings

,00&000

,&000

n the final round& the license will !e worth the same to you as it was in round &

!ut there is no lottery and no a##lication fee. nstead& will conduct an auction

!y starting with a low #rice of &000 and calling out successively higher #rices

until there is only one team actively !idding. he winning team will have to #aythe amount of its final !id. he losing teams do not have to #ay anything for the

license that they did not #urchaseB the winning team earns an amount that e'uals

its license value minus the #rice #aid. he revenue from the auction will !e

divided e'ually among the teams:



2 i*,t alls1 ar/ all

1 i*,t all2 ar/ alls

3ound Mour *arnings

=your license value minus your !id if you win& 0

otherwise>

Cumulative *arnings

,00&000

/

7ayes 3ule nstructions =Cha#ter 0>

;ote: he setu# re'uires a %+sided die& a ,0+sided die& light mar!les& dar"

mar!les& a #lastic cu# for ma"ing draws& two #a#er envelo#es mar"ed A and 7&

and a way to hide the cu# selection #rocess from view. he reading of the

instructions and the e#eriment itself can !e done in a!out 25 minutes.

n this e#eriment& you will o!serve Z!allsZ =colored mar!les> drawn from one of

two #ossi!le Zcu#s.Z Mou will see the !alls drawn& !ut you will not "now for sure

which cu# is !eing used. hen you will !e as"ed to indicate your !eliefs a!out

which cu# is !eing used. will !egin !y choosing one of you to serve as a

monitor who assists in setting things u# and drawing the mar!les.

he two cu#s will !e called cu# A and cu# 7. Cu# A contains 2 light !alls and ,

dar" !all& and Cu# 7 contain , light !all and 2 dar" !alls. he cu# will !e chosen

!y the throw of a % sided die: cu# A is used if the roll of the die yields a ,& 2& or &

and cu# 7 is used if the roll of the die yields a /& 5& or %. hus it is e'ually li"ely

that either cu# will !e selected.

Cu# A Cu# 7

=used if the die is ,& 2& or > =used if the die is /& 5& or %>

Ince a cu# is determined !y the roll of the die& we will em#ty the contents of that

cu# into a container. he container is always the same& regardless of which cu# is

!eing used& so you cannot guess the cu# !y loo"ing at the container. hen we will

draw one or more !alls from the container. =f more than one draw is to !e made&

then the first !all drawn will !e #ut !ac" into the container& which is then sha"en

!efore a second draw is made& etc.>



Recording Qo$r Be%iefs

After the draw has !een made& we will as" you to tell us your !eliefs a!out the

chances that cu# A is !eing used. Mou will indicate a num!er !etween 0 and ,00&

which we will call P & such that the chances that cu# A is !eing used are Z P out of

,00.Z f you could !e sure that cu# A is !eing used& you should choose 8 J ,00 to

indicate that the chances are ,00 out of ,00 that cu# A is !eing used. f you could

!e sure that cu# A is not !eing used& you should choose P J 0 to indicate that the

chances are 0 out of ,00 that cu# A is !eing used. hus the magnitude of P

corres#onds to the chances that cu# A is !eing used. ?or eam#le& if the manner

in which the cu# is selected and the !alls that you see drawn cause you to thin"

that cu# A is 9ust as li"ely as cu# 7& then you should choose P J 50& indicating

that the chances of A are 50 out of ,00.

Mou will use the attached (ecision 6heets to record your information and

decisions. At the start of each Z#eriod&Z the monitor will throw a %+sided die to

select the cu# =A if the throw is ,& 2& or & and 7 if the throw is /& 5& or %>. he

monitor will then #lace the !alls for that cu# in the #lastic container from which

we ma"e the draws. he #eriod num!er is shown in column =,> of the decisionsheet. he results of the draws are to !e recorded in column =2>. 8lease loo" at

the decision sheet for #eriods ,+-. n the first #eriod& there will !e no draws& as

indicated in column =2>& so the only information that you have is the information

a!out how the cu# is selected.

n su!se'uent #eriods& you will see one or more draws& and you can use column

=2> to record the draw=s>. $rite 1 =for 1ight> or ( =for (ar"> in this column at the

time the draw is made. n #eriods 2 and & there will !e only a single draw. n

#eriods / and 5& there will !e two draws from the same cu# =with re#lacement

after the first draw>. n #eriods % and -& there will !e draws =with each !all

drawn !eing #ut !ac" into the cu# !efore the net draw is made>.

After seeing the draw=s>& if any& for the #eriod& you will !e as"ed to indicate thechances that cu# A is !eing used. (o this !y writing a num!er& 8& !etween , and

,00 in column =>.

"arnings

;et& let me descri!e a #rocedure that will hel# you ma"e your decision. he

#rocedure is com#licated& !ut the underlying idea is sim#le. t s as if you send

your friend to a fruit stand for tomatoes& and they as" you what to do if there are

!oth red and yellow tomatoes. Mou should tell them the truth a!out your

#reference& since only then can they ma"e the !est choice for you. $e ll set u#

the incentives so that your re#ort a!out the chances of cu# A will ena!le us to

choose a lottery that gives you the highest chance of winning a , #ri<e.

6u##ose that& after seeing the draw or draws from the un"nown cu#& you

thin" the chances are 8 out of ,00 that cu# A is !eing used. f you were to receive



, in the event that the cu# actually used was A& then this determines a 8 lottery

which #ays , with chances of 8 out of ,00& nothing otherwise.

After you write down the num!er 8 that re#resents your !eliefs a!out li"elihood

of cu# A& we will use a ten+sided die to determine a second lottery& the dice

lottery. o do this& we throw the die twice& with the first throw giving the tens

digit =0& ,& > and the second throw giving the ones digit =0& ,& >. hus the

random num!er will !e !etween 0 and . f the num!er is ;& then the dice

lottery will give you an ; out of ,00 chance of earning ,. =o #lay the dice

lottery for a given value of ;& we would throw the ten+sided die two more times

to get a num!er that is e'ually li"ely to !e 0& ,& & and you would earn , if this

second throw is less than ;.>

U# to this #oint& you will have written down a num!er 8 for what you thin" are

the chances out of ,00 that cu# A is !eing used. he resulting 8 lottery will #ay

, if cu# A is used. hen we will have used a throw of dice to determine a dice

lottery that will #ay , with chance ; out of ,00. $hich would you rather haveN

t is !etter to have the dice lottery if ; is greater than 8& and it is !etter to have the

8 lottery otherwise. $e ll give you the one that is !est for you.

Case ': P 8

f you write down a num!er 8 and the num!er ; determined su!se'uently !y the

dice throws is greater& you will get to #lay the dice lottery =throwing the dice

again to see if you get the ,& i.e. when the num!er determined !y the second

throw is less than or e'ual to ;.>

Case : P ^ 8

f you write down a num!er 8 that is greater than or e'ual to the num!er ;determined su!se'uently !y the dice throws& then your earnings are determined

!y the 8 lottery& i.e. you will get , if the cu# used turns out to !e cu# A.



Cu# A Cu# 7

=used if the die is ,& 2& or > =used if the die is /& 5& or %>

2 1ight 7alls , 1ight 7all

, (ar" 7all 2 (ar" 7alls

=,> =2> => =/> =5> =%> =-> =>

8eriod

;um!er

Mour draws1 or (

Chancesof urn A 8

(icethrow ;

if 8 Q ;& use

(ice 1otteryif 8 ^ ;& use

the 8 1ottery

(icethrow(

*arnings J

, if ( Q ;

Cu# =A

or 7>

*arnings J

, if cu# A

, +++

2

/

5 &

% &

- &

& &

& &

,0 & &



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Cason& imothy ;. =,5> Chea# al" and 8rice 6ignaling in 1a!oratory 4ar"ets&

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Cason& imothy ;. =,-RR> he I##ortunity for Cons#iracy in Asset 4ar"ets

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Chen& M. =,>: ZAsynchronicity and 1earning in Cost 6haring 4echanisms&Z

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Chen& M.& and . 6oRRnme< =,>: ZAn *#erimental 6tudy of House Allocation

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SS =,>: Zm#roving the *fficiency of on+Cam#us Housing: An *#erimental

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Christie& $. . and 3. (. Huang =,5>: ?ollowing the 8ied 8i#er: (o ndividual

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Coller& 4.& and 4. $illiams =,>: Z*liciting ndividual (iscount 3ates&Z

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4conomic Kourna# & ,0-& 55+5-5.

Coo#er& 3.& (. V. (eFong& 3. ?orsythe& and . $. 3oss =,>: ZCommunication

in the 7attle of the 6ees ame: 6ome *#erimental 3esults&Z 'and

Kourna# of 4conomics& 20& 5%+5-.

=,2>: ZCommunication in Coordination ames&Z <uarter# Kourna# of 4conomics& ,0-& -+--,

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4conomic 'evie;& & ,0+,,%.

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Market 'esearch& ed. !y 6. 4oriarity. ;orman& I"lahoma: University of

I"lahoma& Center for *conometrics and 4anagement 3esearch& ,5+,%.

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$illingness to Acce#t and $illingness to 8ay 4easures of Value&Z

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Co& Fames =,>: rust and 3eci#rocity: m#lications of ame riads and

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Hy#othetical 3eferenda ncentive Com#ati!leN&Z Kourna# of Po#itica#

4conom& ,05& %0+%2,.

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*m#loyment& Kourna# of 4conomic Behavior and /rgani)ation& & -+

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=,>: 45perimenta# 4conomics. 8rinceton& ;.F.: 8rinceton University 8ress.

=,/>: Z4ar"et 8ower and 4ergers in 4ar"ets with 8osted 8rices&Z '!F(

Kourna# of 4conomics& 25& /%-+/-.

=,5>: ZAn *amination of the (iamond 8arado: nitial 1a!oratory 3esults&Z

!ctas de #as Kornadas de 4conomia Industria# & %-+-0.

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=200!>: Zhe *ffects of Collusion in 1a!oratory *#eriments&Z forthcoming in

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*#erimental Auctions: nstitutional *ffects and 4ar"et 8ower&Z

4conomic InHuir& 2& 2%,+2-/.

(avis& (ouglas (. and 7art $ilson =,RRR> Zhe *ffects of 6ynergies on the

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(avis& (ouglas (. and 7art $ilson =,RRR> Z(etecting Collusion from 7idding

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(eroot& 4. H. =,-0>: /ptima# Statistica# (ecisions& ;ew Mor": 4craw Hill.

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3e#utation ?ormation in Agency 3elationshi#s: A 1a!oratory 4ar"et

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4uropean 4conomic 'evie;& /0& %0+%,5.

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Behavior$ 4ssas in Aonor of !nato# 'apoport & ed. !y A. (ie"mann& and

8. 4itter. Heidel!erg: 8hysica+Verlag& ,-+,-.

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*avey& C. 1.& and . F. 4iller =,/>: Z*#erimental *vidence on the ?alla!ility

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AgendaControlled Committee *#eriments&Z !merican 4conomic 'evie;&

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3einforcement 1earning in *#erimental ames with Uni'ue& 4ied

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6im#le 7argaining ames&Z Games and 4conomic Behavior & %& /-+%.

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8reference 3eversal 8henomenon&Z !merican 4conomic 'evie;& %& %2+

%.



=,/> he *ffects of 4ar"et 8ractices in Iligo#olistic 4ar"ets: An

*#erimental *amination of the 4th# Case& 4conomic InHuir& && /-+

50-.

rether& (avid 4.& Alan 6chwart<& and 1ouis 1. $ilde =,> ZUncertainty and

6ho##ing 7ehavior: An *#erimental Analysis&Z 'evie; of 4conomic

Studies& ""& 2+/2.

th& $.& 3. 6chmitt!erger& and 7. 6chwar<e =,2>: ZAn *#erimental Analysis

of Ultimatum 7argaining&Z Kourna# of 4conomic Behavior and

/rgani)ation& & %-+.

uyer& 4.& and A. 3a#o#ort =,-2>: Z22 ames 8layed Ince&Z Kourna# of

Conf#ict 'eso#ution& ,%& /0+/,.

u<i"& Victor 6. =200/> Contetual ?raming *ffects in a Common 8ool 3esource

*#eriment& *conomics Honors hesis& 4iddle!ury College.

Har!augh& $. .& and D. Drause =,>: ZChildren)s Contri!utions in 8u!lic ood

*#eriments: he (evelo#ment of Altruistic and ?ree+3iding 7ehaviors&Z

University of Iregon.

Har!augh& $. .& D. Drause& and . 7erry =2000>: Zar# for Dids: In the(evelo#ment of 3ational Choice 7ehavior&Z University of Iregon.

Har!augh& $. .& D. Drause& and 1. Vesterlund =,>: Zhe *ndowment *ffect in

Children&Z University of Iregon.

=,>: ZAre Adults 7etter 7ehaved han ChildrenN Age& *#erience& and the

*ndowment *ffect&Z University of Iregon.

=2002RRR>: 8ro!a!ility $eighting in Children and Adults& RRR.

Hardin& arrett =,%>: he ragedy of the Commons& Science& ,%2& ,2/,2/.

Harless& (. $. =,>: Z4ore 1a!oratory *vidence on the (is#arity !etween

$illingness to 8ay and Com#ensation (emanded&Z Kourna# of 4conomic

Behavior and /rgani)ation& ,,& 5+-. Harrison& . $. =,>: Zheory and

4is!ehavior of ?irst+8rice Auctions&Z !merican 4conomic 'evie;& -& -/+-%2.

Harrison& . $.& and F. Hirshleifer =,>: ZAn *#erimental *valuation of

$ea"est 1in"7est 6hot 4odels of 8u!lic oods&Z Kourna# of Po#itica#

4conom& -& 20,+225.

Harrison& . $.& 4. 1au& and 4. $illiams =,>: Z*stimating ndividual

(iscount 3ates in (enmar"&Z University of 6outh Carolina.

Harrison& . $.& and 4. 4cDee =,5>: Z*#erimental *valuation of the Coase

heorem&Z Kourna# of -a; and 4conomics& 2& %5+%-0.

Harrison& . $.& RRRRRRR =2005>: Z&Z !merican 4conomic 'evie;& & RRRR.

Haruvy& *.& . *rev& and (. 6onsino =2000>: Zhe 4edium 8ri<e 8arado:

*#erimental *vidence&Z echnion+sreal.Haw"es& Dristen =,> $hy Hunter+atherers $or"& Current !nthropo#og&

/=/>& /,+%, =with commentaries and author s re#ly>.

Ha<lett& (. =,>: Znflation Uncertainty and nvestment in an *#erimental

Credit 4ar"et&Z $hitman College.



Ha<lett& . $. and 3. F. 4ichaels =,>: Zhe Cost of 3ent+6ee"ing: *vidence

from Cellular ele#hone 1icense 1otteries&Z Southern 4conomic Kourna# &

"%=>& /25+/5.

Henrich& F. et al. =200,>: n 6earch of Homo *conomicus: 7ehavioral

*#eriments in ,5 6mall+6ale 6ocieties& !merican 4conomic 'evie;&

%1=2>& -+/.

Herrnstein& 3. F.& and (. 8relec =,,>: Z4elioration: A heory of (istri!uted

Choice&Z Kourna# of 4conomic Perspectives& 5& ,-+,5%.

Hertwig& 3. and A. Irtmann =200,>: *#erimental 8ractices in *conomics: A

4ethodological Challenge to 8sychologistsN Behaviora# and Brain Sciences&

2/=>& +/0.

Hewett& 3.& C. A. Holt& . Dosmo#oulou& C. Dymn& C. X. 1ong& 6. 4ousavi& 6.

6arangi =2002>: A Classroom *ercise: Voting !y 7allots and ?eet&

(iscussion 8a#er& University of Virginia.

Hay& eorge A. and (aniel Delley =,-/> An *m#irical 6urvey of 8rice ?iing

Cons#iracies& Kourna# of -a; and 4conomics& 17 & ,+.

Hey& F. (. =,,>: Z6earch for 3ules of 6earch&Z Kourna# of 4conomic Behavior and /rgani)ation& & %5+,.

=,->: Z6till 6earching&Z Kourna# of 4conomic Behavior and /rgani)ation& &

,-+,//.

=,/>: 45perimenta# 4conomics& Heidel!erg: 6#ringer+Verlag.

=,5>: Z*#erimental nvestigations of *rrors in (ecision 4a"ing under 3is"&Z

4uropean 4conomic 'evie;& & %+%/0.

Hillman& A. 1. and (. 6amet =,->: Z(issi#ation of Contesta!le 3ents !y 6mall

;um!ers of Contenders&Z Pu+#ic Choice& "&=,>& %+2.

Hoffman& *.& D. 4cCa!e& D. 6hachat& and V. 1. 6mith =,/>: Z8references&

8ro#erty 3ights& and Anonymity in 7argaining ames&Z Games and

4conomic Behavior & -& /%+0.Hoffman& *.& D. 4cCa!e& and V. 1. 6mith =,%>: ZIn *#ectations and

4onetary 6ta"es in Ultimatum ames&Z Internationa# Kourna# of Game

8heor& 25=>& 2+0,.

Hoffman& *.& and 4. 6#it<er =,0>: Zhe (ivergence !etween $illingness+to8ay

and $illingness+to+Acce#t 4easures of Values&Z California nstitute of

echnology.

Hoffman& *.& and 4. 1. 6#it<er =,2>: Zhe Coase heorem: 6ome *#erimental

ests&Z Kourna# of -a; and 4conomics& 25& -+.

=,5>: Z*ntitlements& 3ights and ?airness: An *#erimental *amination of

6u!9ects) Conce#ts of (istri!utive Fustice&Z Kourna# of -ega# Studies& ,/&25+2-.

=,5>: Z*#erimental 1aw and *conomics: An ntroduction&Z Co#um+ia -a;

'evie;& 5& ,+,0%.



Hoggatt& A. C. =,5>: ZAn *#erimental 7usiness ame&Z Behaviora# Science& /&

,2+20.

Holt& C. A. =,5>: ZAn *#erimental est of the Consistent+Con9ectures

Hy#othesis&Z !merican 4conomic 'evie;& -5& ,/+25.

=,%>: Z8reference 3eversals and the nde#endence Aiom&Z !merican

4conomic 'evie;& -%& 50+5,5.

=,>: Zhe *ercise of 4ar"et 8ower in 1a!oratory *#eriments&Z Kourna# of

-a; and 4conomics& 2& 6,0-+6,,.

=,2>: Z6I 8ro!a!ility 4atching&Z University of Virginia.

=,5>: Zndustrial Irgani<ation: A 6urvey of 1a!oratory 3esults&Z in Aand+ook

of 45perimenta# 4conomics& ed. !y F. Dagel& and A. 3oth. 8rinceton& ;.F.:

8rinceton University 8ress& /+//.

=,%>: ZClassroom ames: rading in a 8it 4ar"et&Z Kourna# of 4conomic

Perspectives& ,0& ,+20.

=,>: Zeaching *conomics with Classroom *#eriments: A 6ym#osium&Z

Southern 4conomics Kourna# & %5& %0+%,0.

=200>: Z*conomic 6cience: An *#erimental A##roach for eaching and3esearch =2002 8residential Address>& Southern 4conomics Kourna# &

%=/>& -55+--,.

Holt& C. A.& and 1. 3. Anderson =,>: ZAgendas and 6trategic Voting&Z

Southern 4conomic Kourna# & %5& %22+%2.

Holt& Charles A. and (avis& (ouglas (. =,0> Zhe *ffects of ;on+7inding 8rice

Announcements on 8osted Iffer 4ar"ets&Z 4conomics -etters& 3&& ##.

0-+,0.

Holt& C. A.& 1. 1angan& and A. Villamil =,%>: Z4ar"et 8ower in Iral (ou!le

Auctions&Z 4conomic InHuir& 2/& ,0-+,2.

Holt& C. A.& and 6. D. 1aury =,->: ZClassroom ames: Voluntary 8rovision of a

8u!lic ood&Z Kourna# of 4conomic Perspectives& ,,& 20+2,5.

SS =,>: heoretical *#lanations of reatment *ffects in Voluntary

Contri!utions *#eriments& in C. 3. 8lott and V. 1. 6mith& eds. Aand+ook

of 45perimenta# 4conomics 'esu#ts. ;ew Mor": *lsevier& forthcoming.

SS =200,>: ZVarying the 6cale of ?inancial ncentives under 3eal and

Hy#othetical Conditions&Z Behaviora# and Brain Sciences& 2/=>& #. 2,-+

2,.

SS =2002>: Z3is" Aversion and ncentive *ffects&Z !merican 4conomic 'evie;&

2=5> (ecem!er& ,%//+,%55.

SS =2002> ?urther 3eflections on 8ros#ect heory& (iscussion 8a#er&

University of Virginia. SS =2005> 3is" Aversion and ncentive *ffects: ;ew (ata without Irder *ffects&

!merican 4conomic 'evie;& RRRRforthcoming.

Holt& Charles A. and Angela 4oore =200/a> he 7ullwhi# *ffect and Vertical

8rice Com#etition& un#u!lished wor"ing #a#er& University of Virginia.



SS =200/!> A 6urvey of *#erimental 6tudies in 8olitical 6cience& un#u!lished

wor"ing #a#er& University of Virginia& to !e #resented at the ?all 200/

*6A 4eetings in ucson.

Holt& C. A.& and . 4c(aniel =,>: Z*#erimental *conomics in the

Classroom&Z in 8eaching >ndergraduate 4conomics$ ! Aand+ook for

Instructors& ed. !y $. 7. $alstad& and 8. 6aunders. ;ew Mor": 4craw

Hill& 25-+2%.

Holt& C. A.& and A *. 3oth =200/> Zhe ;ash *'uili!rium: A 8ers#ective&

Proceedings of the Fationa# !cadem of Sciences, >.S.!& ,0,=,2>& +

/002.

Holt& C. A.& and (. 6cheffman =,->: Z?acilitating 8ractices: he *ffects of

Advance ;otice and 7est+8rice 8olicies&Z '!F( Kourna# of 4conomics&

,& ,-+,-.

Holt& Charles A. and 3oger 6herman =,0> ZAdvertising and 8roduct Luality on

8osted+Iffer *#eriments&Z 4conomic InHuir& =>& +5%.

Holt& C. A.& and 3. 6herman =,/>: Zhe 1oser)s Curse&Z !merican 4conomic

'evie;& /& %/2+%52. SS =,>: ZClassroom ames: A. 4ar"et for 1emons&Z Kourna# of 4conomic

Perspectives& ,& 205+2,/.

SS =2000>: 3is" Aversion and the $inner s Curse& (iscussion 8a#er& University of

Virgina.

Holt& C. A.& and ?. 6olis+6o!eron =,2>: Zhe Calculation of *'uili!rium 4ied

6trategies in 8osted+Iffer Auctions&Z in 'esearch in 45perimenta#

4conomics, :o#. "& ed. !y 3. 4. saac. reenwich& Conn.: FA 8ress& ,+

22.

Hong& Fames . and Charles 3. 8lott =,2> Z3ate ?iling 8olicies for nland $ater

rans#ortation: An *#erimental A##roach&Z Be## Kourna# of 4conomics&

13&,+,.

Hung& A. A. and C. 3. 8lott =200,> nformation Cascades: 3e#lication and an

*tension to 4a9ority 3ule and Conformity 3ewarding nstitutions&

!merican 4conomic 'evie;& RR& RRRR.

saac& 3. 4.& and C. A. Holt =,>: Z*missions 8ermit *#eriments&Z 6tamford&

Conn.: FA 8ress.

saac& 3. 4.& D. 4cCue& ?.& and C. 3. 8lott =,5>: Z8u!lic oods 8rovision in an

*#erimental *nvironment&Z Kourna# of Pu+#ic 4conomics& 2%& 5,+-/.

saac& 3. 4.& and C. 3. 8lott =,,>: Zhe I##ortunity for Cons#iracy in 3estraint

of rade&Z Kourna# of 4conomic Behavior and /rgani)ation& 2& ,+0.saac& 3. 4ar"& Valerie 3amey& and Arlington $illiams =,/> he *ffects of

4ar"et Irgani<ation on Cons#iracies in 3estraint of rade& Kourna# of

4conomic Behavior and /rgani)ation, ", ,, 222.



saac& 3. 4.& and F. 4. $al"er =,5>: Znformation and Cons#iracy in 6ealed 7id

Auctions&Z Kourna# of 4conomic Behavior and /rgani)ation& %& ,+

,5.

=,a>: ZCommunication and ?ree+3iding 7ehavior: he Voluntary

Contri!utions 4echanism&Z 4conomic InHuir& 2%& 55+%0.

=,!>: Zrou# 6i<e Hy#otheses of 8u!lic oods 8rovision: he Voluntary

Contri!utions 4echanism&Z <uarter# Kourna# of 4conomics& ,0& ,-+

200R.

saac& 3. 4.& F. 4. $al"er& and 6. H. homas =,/>: Z(ivergent *vidence on

?ree 3iding: An *#erimental *amination of 8ossi!le *#lanations&Z

Pu+#ic Choice& /& ,,+,/.

saac& 3. 4.& F. 4. $al"er& and A. $. $illiams =,/>: Zrou# 6i<e and the

Voluntary 8rovision of 8u!lic oods: *#erimental *vidence Utili<ing

1arge rou#s&Z Kourna# of Pu+#ic 4conomics& 5/& ,+%.

Fose#h& 4. =,%5>: Z3ole 8laying in eaching *conomics&Z !merican 4conomic

'evie;& 55& 55%+5%5.

Dachelmeier& 6. F.& and 4. 6hehata =,2>: Z*amining 3is" 8references under High 4onetary ncentives: *#erimental *vidence from the 8eo#le)s

3e#u!lic of China&Z !merican 4conomic 'evie;& 2& ,,20+,,/,.

Dagel& F. H. =,->: Z*conomics According to the 3ats =and 8igeons oo>B $hat

Have $e 1earned and $hat Can $e Ho#e to 1earnN&Z in -a+orator

45perimentation in 4conomics$ Si5 Points of :ie;& ed. !y A. *. 3oth.

Cam!ridge: Cam!ridge University 8ress& ,55+,2.

=,5>: ZAuctions: A 6urvey of *#erimental 3esearch&Z in 8he Aand+ook of

45perimenta# 4conomics& ed. !y F. H. Dagel& and A. *. 3oth. 8rinceton:

8rinceton University 8ress& 50,+55.

Dagel& F. H.& 3. C. 7attalio& and 1. reen =,>: Z4atching Versus 4aimi<ing:

Comments on 8relec)s 8a#er&Z Pscho#og 'evie;& 0& 0+,.

Dagel& F. H.& 3. C. 7attalio& H. 3achlin& and 1. reen =,,>: Z(emand Curves

for Animal Consumers&Z <uarter# Kourna# of 4conomics& %& ,+,%.

Dagel& F. H.& 3. C. 7attalio& and F. 4. $al"er =,->: ZVolunteer Artifacts in

*#eriments in *conomics: 6#ecification of the 8ro!lem and 6ome nitial

(ata from a 6mall+6cale ?ield *#eriment&Z in 'esearch in 45perimenta#

4conomics, 1& ed. !y V. 1. 6mith. reenwich& Conn.: FA 8ress& ,%+,-.

Dagel& F. H.& and (. 1evin =,%>: Zhe $inner)s Curse and 8u!lic nformation in

Common Value Auctions&Z !merican 4conomic 'evie;& -%& /+20.

=,>: Znde#endent 8rivate Value 4ulti+Unit (emand Auctions: An

*#eriment Com#aring Uniform 8rice and Vic"rey Auctions&Z Ihio 6tateUniversity.

Dagel& F. H.& (. 1evin& 3. C. 7attalio& and (. F. 4eyer =,>: Z?irst+8rice

Common Value Auctions: 7idder 7ehavior and the Z$inner)s CurseZ&Z

4conomic InHuir& 2-& 2/,+25.



Dagel& F. H.& and A. *. 3oth =,5>: 8he Aand+ook of 45perimenta# 4conomics&

8rinceton: 8rinceton University 8ress& -2,.

Dagel& F. H.& and $. Vogt =,>: he 7uyers 7id (ou!le Auction: 8reliminary

*#erimental 3esults& in (. ?riedman and F. 3ust& eds.& 3edwood City&

CA: Addison+$esley& 25+05.

Dahneman& (.& F. 1. Dnetsch& and 3. H. haler =,,>: Zhe *ndowment *ffect&

1oss Aversion& and 6tatus Luo 7ias: Anomalies&Z Kourna# of 4conomic

Perspectives& 5& ,+20%.

Dahneman& (.& 8. 6lovic& and A. vers"y =,2>: ZFudgement under Uncertainty:

Heuristics and 7iases&Z Cam!ridge: Cam!ridge University 8ress.

Dahneman& (.& and A. vers"y =,->: In the 8sychology of 8rediction&

Pscho#ogica# 'evie;& 0& 2-+25,.

Dahneman& (.& and A. vers"y =,->: Z8ros#ect heory: An Analysis of

(ecision under 3is"&Z 4conometrica& /-& 2%+2,.

Dachelmeier& 6. F. and 4. 6hehata =,2>: Z*amining 3is" 8references Under

High 4onetary ncentives: *#erimental *vidence from the 8eo#le)s

3e#u!lic of China& !merican 4conomic 'evie;& 2& ,,20+,,/,.Deser& C. =,%>: ZVoluntary Contri!utions to a 8u!lic ood $hen 8artial

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Detcham& F.& V. 1. 6mith& and A. $. $illiams =,/>: ZA Com#arison of

8ostedIffer and (ou!le+Auction 8ricing nstitutions&Z 'evie; of

4conomic Studies& 5,& 55+%,/.

Deynes& F. 4. =,%> 8he Genera# 8heor of 4mp#oment, Interest, and Mone&

1ondon: 4acmillan.

Dlem#erer& 8aul =2002>: $hat 3eally 4atters in Auction (esign& Kourna# of

4conomic Perspectives& ,%=,>& $inter& ,%+,.

Dnetsch& F. 1. =,>: Zhe *ndowment *ffect and *vidence of ;on+3eversi!le

ndifference Curves&Z !merican 4conomic 'evie;& -& ,2--+,2/RR.nedC arcC an! +olin +amerer =1994>_ `+reatin* :pectational

@ssets in t,e aboratory_ +oor!ination in ea/estin/\amesC` Strategic anagement JournalC 1!C 101119

Dogut& C. A. =,0>: ZConsumer 6earch 7ehavior and 6un" Costs&Z Kourna# of

4conomic Behavior and /rgani)ation& ,/& ,+2.

=,2>: Z3ecall in Consumer 6earch&Z Kourna# of 4conomic Behavior and

/rgani)ation& ,-& ,/,+,5,.

Drause& D.& and $. . Har!augh =,>: Z(o Children 7ehave According to

8ros#ect heory&Z University of ;ew 4eico.

Dre#s& (avid =,,> ! Course in Microeconomic 8heor& 8rinceton: 8rincetonUniversity 8ress.

Dre#s& (avid and Fose 6chein"man =,> ZLuantity 8recommitment and

7ertrand

Com#etition Mield Cournot Iutcomes&Z Be## Kourna# of 4conomics& 1&& 2%+



-.

Drueger& A. I. =,-/>: Zhe 8olitical *conomy of the 3ent+see"ing 6ociety&Z

!merican 4conomic 'evie;& %/& 2,+0.

Druse& F. 7.& 6. 3assenti& 6. 3eynolds& and V. 6mith =,/>: Z7ertrand+*dgeworth

Com#etition in *#erimental 4ar"ets&Z 4conometrica& %2& /+-2.

1ai!son& (. =,->: Zolden *ggs and Hy#er!olic (iscounting&Z <uarter#

Kourna# of 4conomics& ,,2& //+/--.

1aury& 6. D.& and C. A. Holt =,>: Z4a"ing 4oney&Z Kourna# of 4conomic

Perspectives& ,/& 205+2,.

=,>: Z4ulti+4ar"et *'uili!rium and the 1aw of Ine 8rice&Z Southern

4conomic Kourna# & %5& %,,+%2,.

SS =2000>: Voluntary 8rovision of 8u!lic oods: *#erimental 3esults with

nterior ;ash *'uili!ria& in C. 3. 8lott and V. 1. 6mith& eds. Aand+ook of

45perimenta# 4conomics 'esu#ts. ;ew Mor": *lsevier& forthcoming.

1ave& 1. 7. =,%2>: ZAn *m#irical A##roach to the 8risoner)s (ilemma&Z

<uarter# Kourna# of 4conomics& -%& /2/+/%.

=,%5>: Z?actors Affecting Coo#eration in the 8risoner)s (ilemma&Z Behaviora# Science& ,0& 2%+.

1edyard& F. I. =,5>: Z8u!lic oods: A 6urvey of *#erimental 3esearch&Z in !

Aand+ook of 45perimenta# 4conomics& ed. !y A. 3oth& and F. Dagel.

8rinceton: 8rinceton University 8ress& ,,,+,/.

1edyard& F.I.& (. 8orter& and 3. $essen =2000>: ZA 4ar"et+7ased 4echanism for

Allocating 6#ace 6huttle 6econdary 8ayload 8riority&Z 45perimenta#

4conomics& 2& ,-+,5.

1ee& H.1.& V. 8admana!han& and 6. $hang =,-a>: nformation (istortion in a

6u##ly Chain: he 7ullwhi# *ffect& Management Science& /=/>& 5/%+

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S#oan Management 'evie; RRR& RRR. 1evine& 4. *.& and C. 3. 8lott =,-->:

ZAgenda nfluence and ts m#lications&Z :irginia -a; 'evie;& %& 5%,+%0/.

1ian& 8.& and C. 3. 8lott =,>: Zeneral *'uili!rium& 4ar"ets& 4acroeconomics

and 4oney in a 1a!oratory *#erimental *nvironment&Z 4conomic

8heor& ,2& 2,+-%.

1ist& F.A.& and . 1. Cherry =2000>: Z1earning to Acce#t in Ultimatum ames:

*vidence from an *#erimental (esign hat enerates 1ow Iffers&Z

45perimenta# 4conomics& & ,,+2.

1ist& F.A.& and (. 1uc"ing+3eiley =2000>: Z(emand 3eduction in 4ulti+Unit

Auctions: *vidence from a 6#ortscard ?ield *#eriment&Z !merican 4conomic 'evie;& 0=/>& %,+-2. SS =200,>: he *ffects of 6eed 4oney and 3efunds on

Charita!le iving: *#erimental *vidence from a University Ca#ital Cam#aign&

Kourna# of Po#itica# 4conom& forthcomingRRRRR.



1oewenstein& .& and 3. H. haler =,>: ZAnomalies: ntertem#oral Choice&Z

Kourna# of 4conomic Perspectives& & ,,+,.

1uce& 3. (. =,5>: Individua# Choice Behavior & ;ew Mor": Fohn $iley Y 6ons.

1uce& 3. (.& and H. 3aiffa =,5->: Games and (ecisions. ;ew Mor": Fohn $iley

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1uc"ing+3eiley& (avid *#erimental *vidence on the *ndogenous *ntry of

7idders in nternet Auctions& (iscussion 8a#er& University of Ari<ona&

,.

1uc"ing+3eiley& (avid =2000>: Vic"rey Auctions in 8ractive: ?rom

;ineteenthCentury 8hilately to wenty+?irst Century *+Commerce&

Kourna# of 4conomic Perspectives& ,/=>& 6ummer& ,+,2.

1uc"ing+3eiley& (.& and F. A. 1ist =2000>: Z(o 6eed 4oney and 3efunds

nfluence Charita!le ivingN *#erimental *vidence from a University

Ca#ital Cam#aign&Z Vander!ilt University.

1ynch& 4.& 3. 4. 4iller& C. 3. 8lott& and 3. 8orter =,%>: Z8roduct Luality&

Consumer nformation and )1emons) in *#erimental 4ar"ets&Z in

4mpirica# !pproaches to Consumer Protection 4conomics& ed. !y 8. 4.##olito& and (. . 6cheffman. $ashington& (.C.: ?ederal rade

Commission& 7ureau of *conomics& 25,+0%.

4ac"ay& C. =,5>: 45traordinar Popu#ar (e#usions and the Madness of

Cro;ds& Hertfordshire& *ngland: $ordsworth *ditions 1td. =originally

,/,>.

4alouf& 4. $. D.& and A. *. 3oth =,,>: Z(isagreement in 7argaining: An

*#erimental 6tudy&Z Kourna# of Conf#ict 'eso#ution& 25& 2+/. 4arwell& .&

and 3. *. Ames =,,>: Z*conomists ?ree 3ide& (oes Anyone *lseN *#eriments

on the 8rovision of 8u!lic oods& V&Z Kourna# of Pu+#ic 4conomics& ,5& 25+,0.

4cCa!e& D. A.& 6. F. 3assenti& and V. 1. 6mith =,>: Z(esigning )6mart)

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4conomic Kourna# & ,02& +2.

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*valuation&Z in (. ?riedman and F. 3ust& eds& 8he (ou+#e !uction Market$

Institutions, 8heor, and 4vidence, S6I Studies in the Sciences of

Comp#e5it& 8roceedings& ,5& 3eading& 4ass.: Addison+$esley.

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the games people play, extended

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