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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
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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
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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
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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!