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

3 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

4 Lotteries Too many players buy too many tickets Why?

5 Literature (A) Buy Hope? Clotfelter and Cook (1989, 1990, 1993)

6 Literature (A) Buy Hope? Clotfelter and Cook (1989, 1990, 1993) (B) Charity/Fund raising? Morgan (2000), Morgan and Sefton (2000)

7 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)?

8 Plan Theory Experiments Data Behavioral Models Results Conclusion

9 Theory n risk neutral players V – prize value W – endowment x i  0 player i’s expenditure

10 Players’ maximization problem Player i solves the following problem

11 Rationalizable choices

12 Rationalizable choices

13 Nash equilibrium Absolute performance Unique Nash equilibrium!

14 Evolutionary Stable Strategies Relative performance (spiteful behavior)

15 Experimental Design V = 1,000 tokens (= $10) W = 1,200 tokens (= $12) Quizzes Expected payoff tables N = 2, 3, 4, 5, 9 3 sessions for each N Pittsburgh Experimental Economics Laboratory October 2007 – March 2008

16 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 / / 1, / / / 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 / / 1, / / 1, / 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? ,000 Assume that your contribution is 900 tokens and your opponent’s contribution is 100 tokens. What is your expected payoff?

18 Experimental Design

19 Summary 1 Session# of ParticipantsN# of Groups 2/ / / / / / / / / / / / / / /31892 Total245--

20 N = 2

21 N = 3

22 N = 2, 3: Nash and ESS

23 N = 4

24 N = 5, 9

25

26

27

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

29

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

31 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)

32 QRE

33 QRE

34 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

35 N = 2

36 N = 3

37 N = 4

38 N = 5

39 N = 9

40 Level – k reasoning Ho, Camerer and Weigelt (1998) Level – 0: uniform on [0, V] – density B 0 Level – 1: simulate N-1 draws from B 0 compute best reply Level – 2:

41

42 Level - k

43 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

44 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.”

45 Learning Direction Theory

46 Learning Direction Theory If you lose, you change “Small lotteries”Yes Other lotteriesNo If you win: you overpaid; if you lose: you underpaid “Small lotteries”Yes Other lotteriesNo Adjust in the best reply direction “Small lotteries”Yes Other lotteriesNo

47 Conclusion Subjects’ behavior in lotteries w/t (A) and (B) a) Nash equilibrium b) ESS c) QRE d) Level – k reasoning e) Leaning direction theory

48 Conclusion Data a) “Almost” do not change to change in N b) Overspending even for N = 4, 5, 9

49 Conclusion Data: N = 2 a) Nash c) QRE (the least noise) d) Level – k reasoning(Level – 1) e) Leaning direction theory(BR changes)

50 Conclusion Data: N = 3 b) SSE c) QRE (noise) d) Level – k reasoning(Level – 1) e) Leaning direction theory(some BR changes)

51 Conclusion Data: N = 4, 5, 9 c) QRE (noise) d) Level – k reasoning(Level – 0) e) Leaning direction theory(random changes)

52

53

54 Conclusion Lotteries: N > 4 Boundedly rational subjects “Random” choices Overspending!