Jonathan W. Leland The National Science Foundation* June 2007

1. (Non) Equilibrium Selection, Similarity Judgments and the “Nothing to Gain / Nothing to Lose” Effect Jonathan W. Leland The National Science Foundation* June 2007 *The research discussed here was funded by an Italian Ministry of Education “Rientro dei Cervelli” fellowship. Views expressed do not necessarily represent the views the National Science Foundation nor the United States government. Not for quote without permission

2. Motivation “Many interesting games have more than one Nash equilibrium. Predicting which of these equilibria will be selected is perhaps the most important problem in behavioral game theory.” (Camerer, 2003)

3. Games with Multiple Equilibria • Two pure strategy equilibria • One pareto superior, one pareto inferior • Incentives are compatible – problem is coordination • Stag-hunt is harder – achieving pareto outcome is riskier

4. Equilibrium Selection Criteria • Payoff Dominance • choose the equilibrium offering all players their highest payoff – predicts UL • Security-mindedness • choose the strategy that minimizes the worst possible payoff – predicts DR • Risk Dominance • choose the strategy that minimizes loses incurred by players as a consequence of unilaterally deviating from their eq. strategy – predicts DR

5. An Alternative Approach Similarity Judgments in Choice R:{$X , p ; $Y , 1-p } S:{$M , q ; $N , 1-q } Choose R (S) if it is favored in some comparisons and not disfavored in any, otherwise choose at random. X similar / dissimilar M p similar / dissimilar q Y similar / dissimilar N 1-p similar / dissimilar 1-q Favor R, Favor S, Inconclusive, Inconsequential Favor R, Favor S, Inconclusive, Inconsequential

6. Similarity Judgments The “Nothing to Gain/Nothing to Lose” Effect R:{$10 , .90 ; $0 , .10 } S:{$9 , .90 ; $9 , .10 } R’:{$10 , .10 ; $0 , .90 } S’:{$1 , .10 ; $1 , .90 } antg 10 ~x 9 .90 ~ p .90 9 >x 0 .10 ~ p .10 Inconsequential Favors S antl 10 >x 1 .10 ~ p .10 1 ~x 0 .90 ~p .90 Inconsequential Favors R’

7. Similarity Judgments and the Allais Paradox a S:{$3000 , .90 ; $0 , .10 } R:{$6000, .45 ; $0 , .55 } S’:{$3000, .02 ; $0 , .98 } R’:{$6000, .01 ; $0 , .99 } 6000 >x 3000 .9 >p .45 0 ~x 0 .55 >p .10 Inconclusive Favors S antl 6000 >x 3000 .02 ~p .01 0 ~x 0 .99~p .98 Favors R’ Inconsequential

8. Similarity Judgments and Intertemporal Choice T1:{$20, 1 month } T2:{$25 , 2 months } T11:{$20, 11 month } T12:{$25 , 12 months } S or D? $25 >x $20 S or D? 2 >t 1 Inconclusive Choose either S or D? $25 >x $20 S or D? 12 ~t 11 Nothing to lose Choose T12

9. Sources of prediction in Similarity based models • Intransitivity of the similarity relation • E.g., 20 ~x 17, 17 ~x 15 but 20 >x 15 • Theoretically inconsequential manipulation of prizes, probabilities, dates of receipt may have consequences if they influence perceived similarity or dissimilarity. • Framing of the choice • Framing determines what is compared with what – theoretically inconsequential changes in the description of the choice may influence what is compared with what.

10. Application to Games - Preliminaries • Assume • Players all have the same Bernoullian utility function • Let: • >x mean “is dissimilar and greater than” • A strict partial order (asymmetric and transitive) • ~x mean “is similar to” • Symmetric but not necessarily transitive (e.g., 20 ~x 15, 15 ~x 10 but 20 >x10.

11. Similarity Judgments in Games • Payoffs to Player 1: h(igh) > m(edium) > l(ow) • Payoffs to Player 2: t(op) > c(enter) > b(ottom) • Decision process • Do I have a dominating strategy, if so, choose it. • Do I have dominating strategy in similarity – if so, choose it. • Does Other have dominating strategy, if so, best respond • Does Other have dominating strategy in similarity, if so, best respond • ?

12. Similarity Judgments in the Stag-Hunt • Player 1 (2): • Checks for dominance • Checks for dominance in similarity • Checks for dominance for P2 (1), and best responds • Checks for dominance in similarity for P2(1), best responds • Chooses at random

13. An Example • If Other L and You U Y=$8, O=$8 • D $5 $2 • If Other R and You U Y=$2 O=$2 • D $5 $5 If U(D) favored in some and not disfavored in any, Choose U(D), otherwise If L(R ) favored in some and not disfavored in any, best response to L(R ), otherwise random. Favors U,D,I Favors L,R,I Favors U,D,I Favors L,R,I

14. The “Nothing to Lose” Effect and the Payoff Dominant Eq. • Decrease m and c. • For Player 1: h >x m ~x l, Choose U – ntl • For Player 2: t >x c ~x b, Choose L – ntl • Outcome is payoff dominant UL

15. The “Nothing to Gain” Effect and the Security-minded Eq. • Increase m and c • For Player 1: h ~x m >x l, Choose D – ntg • For Player 2: t ~x c >x b, Choose R – ntg • Outcome is security- minded DR

16. The “Nothing to Gain/Nothing to Lose” Effect and Non-eq. Outcomes • Increase m, decrease c. • For Player 1: h ~x m >x l, Choose D – ntg • For Player 2: t >x c ~x b, Choose L – ntl • Outcome is non-equilibrium DL

17. Predictions in the Stag Hunt

18. Testing the Ntg/Ntl Effect – Experiment Details • 76 students at the University of Trento • Experiment consisted of 3 parts, 1st of which involved games. • 9 games – 5 stag hunts, 3 matching pennies games, 1 additional stag hunt (always last) • Order otherwise randomized • Subjects played 1 of games at end of session – payouts between 1.20 and 8.00 euro.

19. Games and Individual Results

20. Results Regarding Game Outcomes

21. Performance Relative to Proposed Selection Criteria

22. Games of Pure Conflict • Player’s interests are diametrically opposed • No equilibrium in pure strategies, only a mixed strategy

23. Games of Pure Conflict and Ntg/Ntl Effects • Player 1 compares: • high and low and • high and middle • Increasing m produces • “Nothing to Gain” effect • - choose U • Player 2 compares top and bottom and bottom and center. Decreasing c produces “Nothing to Lose” effect – choose R

24. Predictions in Conflict Games

25. Results

26. Additional Results in Conflict Games

27. Across Game Results

28. Across Game Results cont.

29. Other Implications - The Relativity of Similarity Judgments

30. Similarity Judgments and Framing Effects In Choice Under Uncertainty

31. Framing Effects in Games - Own First vs Other First and non-Equilibrium Outcomes

32. Framing and Question Format in Games

33. Results and Implications for Quantal Response Models

34. What Would You Choose?

35. Level-1 Bounded Rationality vs. Similarity

36. A Speculation - the social benefit of individual irrationality?

37. A Speculation - the social benefit of individual irrationality? (cont.)

38. A Speculation - the social benefit of non-strategic thinking and limits to learning

39. Some Other Speculations and Conjectures • Things will matter that shouldn’t • Time, recalibration and regret • Differences in similarity perceptions and acrimony in negotiations

40. Conclusions • Many choice anomalies can be explained if people employ “nothing to gain/nothing to lose reasoning” • The same reasoning process applied to games predicts: • play in coordination and conflict games and • the successes and failures of equilibrium selection criteria and mixed strategy choice • systematic differences in play as a consequence of theoretically inconsequential changes in the way strategy choices are elicited.

41. References • Camerer, C. Behavioral Game Theory: Experiments on Strategic Interaction, • Princeton, 2003. • Camerer, C., Teck-Hua Ho and Juin Kuan Chong. Behavioral Game Theory: Thinking, Learning and Teaching," with Teck-Hua Ho and Juin Kuan Chong. Forthcoming in a book edited by Steffen Huck, Essays in Honor of Werner Guth." • Goerree, J. and C. Holt. “Ten Little Treasures of Game Theory and Ten Intuitive • Contradictions.” American Economic Review. 2001. Vl. 91(5), pp 1402-1422. • Haruvy, E. and D. Stahl. “Deductive versus Inductive equilibrium selection” • experimental results.” Journal of Economic Behavior and Organization. 2004, 53, 319-331. • Keser, C. and B. Vogt. “Why do experimental subjects choose an equilibrium which • is neither risk nor payoff dominant?” Cirano Working Paper. 2000. http://www.cirano.qc.ca/pdf/publication/2000s-34.pdf • Leland, J. "Generalized Similarity Judgments: An Alternative Explanation for Choice Anomalies." Journal of Risk and Uncertainty, 9, 1994, 151-172. • Leland, J. “Similarity Judgments in Choice Under Uncertainty: A Reinterpretation of Regret Theory.” Management Science, 44(5), 1998, 1-14. • Leland, J. “Similarity Judgments and Anomalies in Intertemporal Choice.” Economic Inquiry Vol. 40, No. 4, October 2002, 574-581. • Lowenstein, G. and D. Prelec. "Anomalies in Intertemporal Choice: Evidence and Interpretation." The Quarterly Journal of Economics, May 1992, 573-597. • Rubinstein, A. "Similarity and Decision-making Under Risk (Is There a Utility Theory Resolution to the Allais Paradox?)." Journal of Economic Theory, 46, 1988, 145-153. • Rubinstein, A. “Economics and Psychology"? The Case of Hyperbolic Discounting,  International Economic Review 44, 2003, 1207-1216.  • Standord Encycolpedia of Philosophy. http://plato.stanford.edu/entries/game-theory/

42. Testing the Ntg/Ntl Effect – Question Format

43. The Problem • “existing deductive selection rules have been shown to do poorly in experiments” (Haruvy & Stahl, 2004) • Should we be surprised? • “…Game theory is the study strategic interactions among rational players..” • We know people behave irrationally in risky and intertemporal choice situations – why would we expect them to do better in complex strategic settings?

44. A Speculation - the problem with being strategic in a non-strategic world