1

Raising Revenue With Raffles:Evidence from a Laboratory Experiment

Wooyoung Lim, University of Pittsburgh

Alexander Matros, University of Pittsburgh

Theodore Turocy, Texas A&M University

2

Lotteries

As of 2008, 43 States have State Lotteries

33% - 50% of USA population participates

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Lotteries

A lottery is a salutary instrument and a tax...laid on the willing only, that is to say, on those who can risk the price of a ticket without sensible injury, for the possibility of a higher prize.

Thomas Jefferson

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Lotteries

Too many players buy too many tickets

Why?

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Literature (A) Buy Hope?

Clotfelter and Cook (1989, 1990, 1993)

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Literature (A) Buy Hope?

Clotfelter and Cook (1989, 1990, 1993)

(B) Charity/Fund raising?

Morgan (2000), Morgan and Sefton (2000)

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Literature (A) Buy Hope?

Clotfelter and Cook (1989, 1990, 1993)

(B) Charity/Fund raising?

Morgan (2000), Morgan and Sefton (2000)

What if no (A) and no (B)?

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Plan

Theory Experiments Data Behavioral Models Results Conclusion

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Theory

n risk neutral players

V – prize value

W – endowment

xi 0 player i’s expenditure

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Players’ maximization problem

Player i solves the following problem

.0,

,0,,...,,...,

)1(,...,,...,max

1

1

1

i

in

jj

ii

nii

niix

xifw

xifVx

xxw

xxxu

xxxui

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Rationalizable choices

12

Rationalizable choices

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Nash equilibrium

Absolute performance

Unique Nash equilibrium!

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Evolutionary Stable Strategies

Relative performance (spiteful behavior)

n

VxESS

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Experimental Design

V = 1,000 tokens (= $10)W = 1,200 tokens (= $12)QuizzesExpected payoff tables

N = 2, 3, 4, 5, 93 sessions for each NPittsburgh Experimental Economics LaboratoryOctober 2007 – March 2008

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Experimental Design

Quiz 1

Assume that your contribution is 100 tokens and your opponent’s contribution is 900 tokens. What is your chance to win the lottery?

100 / 900 100 / 1,000 100 / 800 800 / 900 900 / 1,000

Assume that your contribution is 900 tokens and your opponent’s contribution is 100 tokens. What is your chance to win the lottery?

100 / 900 100 / 1,000 800 / 900 900 / 1,000 900 / 900

17

Experimental Design

Quiz 2

Assume that your contribution is 100 tokens and your opponent’s contribution is 900 tokens. What is your expected payoff?

-100 0 100 900 1,000

Assume that your contribution is 900 tokens and your opponent’s contribution is 100 tokens. What is your expected payoff?

- 900 - 100 0 100 900

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Experimental Design

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Summary 1Session # of Participants N # of Groups

2/1 18 2 9

2/2 20 2 10

2/3 12 2 6

3/1 12 3 4

3/2 15 3 5

3/3 12 3 4

4/1 20 4 5

4/2 16 4 4

4/3 16 4 4

5/1 20 5 4

5/2 15 5 3

5/3 15 5 3

9/1 18 9 2

9/2 18 9 2

9/3 18 9 2

Total 245 - -

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N = 2

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N = 3

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N = 2, 3: Nash and ESS

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N = 4

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N = 5, 9

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26

27

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Data

Integer multiples of 100 in 78.1% Integer multiples of 50 in 87.7% (+9.6%)

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Behavioral Predictions Quantal Response Equilibrium Level – k reasoning Learning Direction Theory

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Quantal Response Equilibrium McKelvey and Palfrey (1995) Noisy optimization process - the best parameter (from the data) = 0 – all choices are random = – no noise (QRE Nash)

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QRE

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QRE

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Level – k reasoning Stahl and Wilson (1994, 1995) Level – 0: random Level – 1: best reply to Level – 0 Level – 2: best reply to Level – 1

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N = 2

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N = 3

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N = 4

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N = 5

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N = 9

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Level – k reasoning Ho, Camerer and Weigelt (1998) Level – 0: uniform on [0, V] – density B0

Level – 1:

simulate N-1 draws from B0

compute best reply Level – 2:

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100 200 300 400 500 600 700

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0LB1LB

2LB

250228.6

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Level - k

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Level – k reasoning Level – 0 in N Level – 1 in N

Costa-Gomes and Crawford (2004)

classify subjects: at least 6 out of 10

96% can be classified!

Iterated elimination of dominated strategies: No

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Learning Direction Theory Selten and Buchta (1994) “Subjects are more likely to change their past actions

in the directions of a best response to the others’ previous period actions.”

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Learning Direction Theory

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Learning Direction Theory If you lose, you change“Small lotteries” YesOther lotteries No

If you win: you overpaid; if you lose: you underpaid“Small lotteries” YesOther lotteries No

Adjust in the best reply direction“Small lotteries” YesOther lotteries No

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Conclusion

Subjects’ behavior in lotteries w/t (A) and (B)

a) Nash equilibriumb) ESSc) QREd) Level – k reasoninge) Leaning direction theory

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Conclusion

Data

a) “Almost” do not change to change in Nb) Overspending even for N = 4, 5, 9

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Conclusion

Data: N = 2

a) Nashc) QRE (the least noise)d) Level – k reasoning (Level – 1)e) Leaning direction theory (BR changes)

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Conclusion

Data: N = 3

b) SSE

c) QRE (noise)

d) Level – k reasoning (Level – 1)

e) Leaning direction theory (some BR changes)

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Conclusion

Data: N = 4, 5, 9

c) QRE (noise)

d) Level – k reasoning (Level – 0)

e) Leaning direction theory (random changes)

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MP: Millner and Pratt (1989) SP: Shogren and Baik (1991) PdVvW: Potters, de Vries, van Winden (1998)

DR: Davis and Reilly (1998) AS: Anderson and Stafford(2003) Fonseca: Fonseca (2006)

LMT: Our result

Previous Literature

0

100

200

300

400

500

600

0 2 4 6 8 10 12

Number of Players

Indi

vidu

al S

pend

ing Nash

SBPdVvWDRASFonsecaLMT

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Conclusion

Lotteries: N > 4

Boundedly rational subjects “Random” choices Overspending!