09. decision analysis

8/21/2019 09. Decision Analysis

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CHAPTER 9 DECISION ANALYSIS

9.1 Introduction to Decision Analysis

Eleents!

"1#Decision maker "$%e'

"(# Choices"decisions' strate)ies#'

"*#Payoffs

Selection problems. De+ine a ,inary -aria,leyj+or eac

decision %it yj / 1' i+ %e a0e decision j' and

oter%ise. A selection 2ro,le %it n 2ossi,le

decisions %ill ten +eature te constrainty13y(3 43

yn/ 1.

5ulti2le criteria "MCDM/ multiple criteria decision

making-s ulti2le states o+ nature 6 decision analysis

/games against nature.

Example:Consider a 2ro,le %it tree decisions d1'

d(' and d*' and +our states o+ nature s1' s(' s*' and s7. Te

2ayo++s are so%n ,elo%.


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s1 s( s* s7

d1 * ( 7 8

d( ( 7 1

d* ( *

Tis )ae is

"1# asyetric! te decision a0er is rational"loo0s at

te 2ayo++s#' %ile nature is a rando 2layer

"(# a siultaneous )ae "%e do not 0no% in ad-ance

%at state o+ nature %ill ,e cosen#.

Consider te continuu ,et%een

certainty: 4 : risk: 4: uncertainty

Certainty! %e 0no% e;actly %ic strate)y nature %ill

2lay.

Risk! %e 0no% te 2ro,a,ility distri,ution nature uses

to 2lay er strate)ies "e.).' ,y %ay o+ 2ast

o,ser-ations#

ncertainty: %e do not 0no% e-en te 2ro,a,ility

distri,ution o+ nature


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Certainty is tri-ial! Since %e 0no% %at colun

nature 2lays' all %e a-e to do is coose te ro% tat

leads to te i)est 2ayo++.

9.( =isuali>ations o+ Decision Pro,les

5acro -ie%!!nfluence diagrams.

5icro -ie%!Decision trees.

!nfluence diagrams: Decision nodes' rando nodes'

conse?uence nodes. So%s )eneral interrelations

,et%een decisions' cance e-ents' @ resultsoutcoes.

Example:

Decision Rando e-ent Conse?uenceAdd electronics

de2artent

Beneral econoic

conditions

Pro+it

Relocate

de2artent

into a se2arate

,uildin)

Local acce2tance

o+ ser-ices


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Te ,ro0en arcs! 2ossi,le in+luences "local acce2tanceo+ an electronics de2artent or store ay ,e

in+luenced ,y te e;istence o+ an electronics

de2artent in our de2artent store and our

co2etitors< reaction to our introduction o+ te

de2artent#.

Decision tree +or te sae 2ro,le!


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9.* Decision Rules nder ncertainty and

Ris0

Start %it uncertainty. "Little in2ut on our 2art' onlycrude in+oration %ill coe out#.


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E;a2le!

s1 s( s*

d1 ( ( d( 1

d* ( 1 1

d7 ( * 7

e+ore coencin)' cec0 +or dominances. One

decision "ro%# doinates anoter' i+ its 2ayo++s are all)reater or e?ual tan tose o+ a sin)le oter ro%.

Here' d1 doinates d7. Doinated decisions can ,e

deleted. Colun doinances do not e;ist.

Cec0in) +or doinances re?uires a total o+ Fm"m1#

2airs o+ co2arisons. Beneral 2rocedure +or all

decision rules! Deterine anticipated outcomes +or all

decisions.

Decision rules under uncertainty!

"1# "ald#s rule "2essiist


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In our e;a2le' te antici2ated outcoes are (' 1' 1'

@ * "in case %e did not eliinate decision d7#' te

a;iu is 1' %ic ,elon)s to d*. Tis is te cosen

decision.

"(# O2tiist


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te. Tis a0es Jald


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is M' %ile te o22ortunity cost +or ne%s2a2ers as

,een estiated to ,e 1M +or eac ne%s2a2er tat

could a-e ,een sold ,ut %as not due to te lac0 o+

su22ly. Su22ose tat te 2urcasin) strate)ies are 1'(' *' @ 7.

Payo++ atri;

s1 s( s* s7

A /

4

3

2

1

d

d

d

d

00.2850.1900.1150.2

50.1900.2150.1200.400.1150.1200.1450.5

50.200.450.500.7$

Bi-en 2ro,a,ilities o+ 2 / K.8' .(' .1' .1' te decisions

a-e e;2ected 2ayo++s o+ .9' .7' .9' and .7'

,uy * ne%s2a2ers @ e;2ect a daily 2ayo++ o+ .9.

"# sin) target $alues *. Idea! Plot 2ayo++ -alues

a)ainst te 2ro,a,ility tat te -alue can ,e acie-ed.


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Te ori)inal 2ayo++ atri; %as

s1 s( s*

d1 ( ( d( 1

d* ( 1 1

d7 ( * 7

2 / K.' .*' .(.

d1! solid line

d(! ,ro0en line

d*! dotted line


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Decision rules ",y %ay o+ u22er en-elo2e#!

I+ * (' any decision %ill acie-e te tar)et.

I+ *K(' 1' d(and d*are ,est. ot %ill reacte tar)et %it a 2ro,a,ility o+ 1.

I+ *K1' 1' d*is ,est. It reaces te tar)et %it

a 2ro,a,ility o+ 1.

I+ *K1' (' d1is ,est. It reaces te tar)et %it

a 2ro,a,ility o+ ..

I+ *K(' ' d1and d(are ,est. ot acie-e tetar)et %it a 2ro,a,ility o+ .(.

I+ *K' ' d(is ,est. It acie-es te tar)et %it

a 2ro,a,ility o+ .(.

I+ *Q ' none o+ te decisions %ill ,e a,le to reac

te tar)et.

9.7 Sensiti-ity Analyses

$Jat i+& indi-idual 2ayo++s aijcan)e

s1 s( s*

d1 ( ( d( 1

d* ( 1 1

d7 ( * 7


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2 . .* .(

Su22ose tat %e are uncertain a,out a(*. Re%rite te

2ayo++ as a(* / 3 %it an un0no%n K(' *'

eanin) tat %e e;2ect te 2ayo++ to ,e ,et%een @1.

E;2ected 2ayo++s!

EM(/

+

9.

5.1

2.1.1

4.1

.

Clearly' d1 @ d7 are doinated. Te 2ayo++s +or te

reainin) strate)ies are so%n in te +i)ure ,elo%.


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Tis leads to

I+ Q ( "i.e.' a(*Q 9#' ten decision d(is ,est' @

i+ ( "i.e.' a(*9#' decision d*is ,est.

Anoter source o+ uncertainty relates to te a)nitude

o+ te 2ro,a,ilities.

Su22ose tat %e are unsure a,out p1. Siilar to te

a,o-e' %e can use p1 3 %it soe un0no%n .Ho%e-er! Te su o+ 2ro,a,ilities ust e?ual 1' so i+

p1increases' te oter 2ro,a,ilities ust decrease ,y

. Assue tat te oter t%o 2ro,a,ilities decrease ,y

te sae aounts' i.e.

2 / K. 3 ' .* F' .( F.

Bi-en te sae 2ayo++ atri;

s1 s( s*

d1 ( (

d( 1

d* ( 1 1d7 ( * 7

2 . .* .(

%e can co2ute te e;2ected -alues as


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EM("# /

+

+

+

5.19.

15.1

31.1

5.4.1

.

Su22ose %e estiate tat p1%ill ,e ,et%een .* @ .8.

"Alternati-ely' startin) %it p1 /.' te can)e

K .(' 3.1.

Jitin tis ran)e' decision d*doinates d1@ d7' %ic

can ,e deleted.

Te e;2ected onetary -alues +or d(@ d*are so%n in

te +i)ure ,elo%!


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Decision rule!

I+ .1 "i.e.'p1.7#' ten decision d(is ,est' @

i+ Q .1 "i.e.'p1Q .7#' ten decision d*is ,est.

Di++erent e;a2le! Sae 2ayo++ atri;' ,ut asp1 '

p( @p* . Te e;2ected 2ayo++s are ten

+

+

+

3

8

3

5

3

5

9.

5.1

1.1

4.1

.

Decision rule!

I+

.1 "i.e.'p1 .*#' ten decision d(is o2tial'i+ .1 "i.e.'p1.*#' ten decision d*is o2tial.


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9. Decision Trees and te =alue o+

In+oration

=alue o+ in+oration. E;tree case +irst!

Expected $alue of perfect information "E(P!#!

Di++erence ,et%een te e;2ected 2ayo++ %it 2er+ect

in+oration inus te e;2ected 2ayo++ %itout

in+oration ",eyond te 2rior 2ro,a,ilitiesp#.

Pre-ious e;a2le!

s1 s( s*

d1 (U (

d( 1 U

d* (U 1U 1

d7 (U * 72 . .* .(

2 / K.' .*' .(.

Te ,est res2onses to nature


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As te e;2ected 2ayo++ %itout 2er+ect in+oration

"i.e.' te e;2ected onetary -alue o+ te ,est strate)y

EM(,# %as 1. "acie-ed ,y usin) d*#' te expected

$alue of perfect informationis ten

E(P!/EPP! EM(U / (. 1. / 1.(.

No% i2er+ect in+oration.

Ia)e a +orecastin) institute tat uses indicators !1'!('

4 to +orecast te states o+ nature s1' s(' 4 . Clearly' te

indicators @ te states o+ nature sould ,e related.

"ty2ical e;a2les are deand @ %olesaler


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Decision tree!

ro le+t to ri)t' te tree de2icts te se?uence o+

e-ents.Decision modes"%e decide# are s?uares' e$ent

nodes"nature a0es a rando coice# are circles' @

terminal nodes "tere are no +urter u-es# as

trian)les.


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Note! te lo%er 2art o+ te tree e?uals %at %e a-e

already done in te atri; %en %e deterined te

EM(U strate)y.

Nu,ers tat are needed!

"1#Payoffs"at te terinal nodes#

"(# !ndicator probabilitiesP"!# at nature


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Here!

or!1' %e co2uteP"!1# andP"sV!1#

s P"s# P"!1Vs# P"!1Vs#P"s# P"sV!1#

s1 . .8 .* .791

s( .* .9 .( .77(8

s* .( .( .7 .88

P"!1# / .81

or!(' %e co2uteP"!(# andP"sV!(#

s P"s# P"!(Vs# P"!(Vs#P"s# P"sV!(#

s1 . .7 .( .1(

s( .* .1 .* .89

s* .( . .18 .71*P"!(# /.*9

Te co2lete decision tree is ten as +ollo%s!


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Te nu,ers ne;t to te nodes are co2uted ,yback'ard recursion. Te recursion starts at te

trian)ular terinal nodes @ %or0s ,ac0%ards to te

root o+ te tree.


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T%o rules a22ly!

"1# ac0 into a decision node! coose te ,est strate)y'

i.e.' te one %it te i)est "e;2ected# 2ayo++.

"(# ac0 into an e-ent node! co2ute te e;2ected

-alue o+ all successor nodes.

Here' te result is tepayoff 'ith imperfect information

EP!!/ (..

No% co2are! %itout in+oration ",eyond 2rior

2ro,a,ilities#' %e can )etEM(U / 1.. Jit additional

in+oration' %e can )et EPII / (..

Hence te expected $alue of perfect information

E(S!/ (. 1. / ..

A standardi>ed easure is e++iciencyE' %ic is

E/ .1.( / .7*.


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Te sae e;a2le %it di++erent nu,ers!

s1 s( s*

P"!-s#! !1 .9 .8 .(!( .1 .7 .

Note! Tis atri; is te sae as ,e+ore %it soe

coluns e;can)es. Ho%e-er' no% s1is stron)ly lin0ed

to te indicators' @P"s1# / .' so te result sould ,e

"at least soe%at# ,etter.

Co2utation o+P"!1# andP"sV!1#!

s P"s# P"!1Vs# P"!1Vs#P"s# P"sV!1#

s1 . .9 .7 .818

s( .* .8 .1 .(8

s* .( .( .7 .9P"!1# / .8

Co2utation o+P"!(# andP"sV!(#!

s P"s# P"!(Vs# P"!(Vs#P"s# P"sV!(#

s1 . .1 . .11

s( .* .7 .1( .*8*8s* .( . .18 .77

P"!(# /.**

Je ten o,tainEP!!/ (.1(' so tat


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E(S!/ (.1( 1. / .8( @E/ .8(1.( / .18.

E;tree e;a2les! Rando indicators result in te2rior 2ro,a,ilities e?ualin) te 2osterior 2ro,a,ilities

@ E=SI / .

On te oter and' %ile a +orecast tat is al%ays

correct %ill a-eE(S!/E(P!@E/ 1' a +orecast tat

is al%ays %ron) as te sae +eatures it is te

consistency o+ te indicators tat is i2ortant' not

teir actual eanin).

9.8. tility Teory

E;2ected -alues are eanin)+ul' only i+ decisiona0ers are risk neutral. Tis eans' tey sould ,e

indi++erent to eiter

"1# recei-in) 1' in cas' no ?uestions as0ed' or

"(# 2layin) te lottery %it a W o+ %innin) ('

@ a W cance o+ %innin) notin).

5ost 2eo2le %ould 2re+er "1# i.e.' tey are not ris0

neutral.


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Once te utilities a-e ,een deterined' tey can ,e

used instead o+ 2ayo++s. All 2rocedures reain

uncan)ed.

09. decision analysis

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