(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

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)

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 The Matching GameThe Stag Hunt Game Player 2 LR LR Player 1 U$80, $80$10, $10 Player 1 U$80, $80$10, $50 D $10, $10$50,$50D$50, $10$50,$50

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 The Stag Hunt Game Player 2 LR Player 1 U $8, $8 $1, $5 D $5, $1 $5,$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 Favor R, Favor S, Inconclusive, Inconsequential Favor R, Favor S, Inconclusive, Inconsequential 1-p similar / dissimilar 1-q

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} 10 ~ x 9.90 ~ p.90 9 > x 0.90 ~ p.90 InconsequentialFavors S  ntg 10 > x 1.10 ~ p.10 Favors R’ 1 ~ x 0.10 ~ p.10 Inconsequential  ntl

Similarity Judgments and the Allais Paradox S:{$3000,.90;$0,.10} R:{$6000,.45;$0,.55} S’:{$3000,.02;$0,.98} R’:{$6000,.01;$0,.99} 6000 > x > p > x > p.45 0 ~ x 0.55 > p.10 InconclusiveFavors S  Favors R’ Inconsequential  ntl 6000 > x ~ p.01 0 ~ x 0.99~ p.98

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 Nothing to lose Choose T12 S or D? $25 > x $20 S or D? 12 ~ t 11

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

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

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 ?

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 LR Player 1 U 8, 82, 5 D5, 25, 5

An Example If Other L and You U Y=$8, O=$8 If Other L and You U Y=$8, O=$8 D $5 $2 D $5 $2 If Other R and You U Y=$2 O=$2 If Other R and You U Y=$2 O=$2 D $5 $5 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

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 LR Player 1 U 8, 82, 2.1 D2.1, 22.1, 2.1

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 LR Player 1 U 8, 82, 7.9 D7.9, 27.9, 7.9

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 LR Player 1 U 8, 82, 2.1 D7.9, 27.9, 2.1

Predictions in the Stag Hunt

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

Games and Individual Results

Results Regarding Game Outcomes

Performance Relative to Proposed Selection Criteria

Games of Pure Conflict Player’s interests are diametrically opposed No equilibrium in pure strategies, only a mixed strategy LR Player 1 U h, bm, t Dl, ch, b

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 LR Player 1 U h, bm, t Dl, ch, b

Predictions in Conflict Games

Results

Additional Results in Conflict Games

Across Game Results

Across Game Results cont.

Other Implications - The Relativity of Similarity Judgments

Similarity Judgments and Framing Effects In Choice Under Uncertainty

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

Framing and Question Format in Games

Results and Implications for Quantal Response Models

What Would You Choose?

Level-1 Bounded Rationality vs. Similarity

A Speculation - the social benefit of individual irrationality?

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

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

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

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

References Camerer, C. Behavioral Game Theory: Experiments on Strategic Interaction, Camerer, C. Behavioral Game Theory: Experiments on Strategic Interaction, Princeton, Princeton, 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." 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."Behavioral Game Theory: Thinking, Learning and Teaching,"Behavioral Game Theory: Thinking, Learning and Teaching," Goerree, J. and C. Holt. “Ten Little Treasures of Game Theory and Ten Intuitive Goerree, J. and C. Holt. “Ten Little Treasures of Game Theory and Ten Intuitive Contradictions.” American Economic Review Vl. 91(5), pp Contradictions.” American Economic Review Vl. 91(5), pp Haruvy, E. and D. Stahl. “Deductive versus Inductive equilibrium selection” Haruvy, E. and D. Stahl. “Deductive versus Inductive equilibrium selection” experimental results.” Journal of Economic Behavior and Organization. 2004, 53, experimental results.” Journal of Economic Behavior and Organization. 2004, 53, Keser, C. and B. Vogt. “Why do experimental subjects choose an equilibrium which Keser, C. and B. Vogt. “Why do experimental subjects choose an equilibrium which is neither risk nor payoff dominant?” Cirano Working Paper is neither risk nor payoff dominant?” Cirano Working Paper Leland, J. "Generalized Similarity Judgments: An Alternative Explanation for Choice Anomalies." Journal of Risk and Uncertainty, 9, 1994, Leland, J. "Generalized Similarity Judgments: An Alternative Explanation for Choice Anomalies." Journal of Risk and Uncertainty, 9, 1994, Leland, J. “Similarity Judgments in Choice Under Uncertainty: A Reinterpretation of Regret Theory.” Management Science, 44(5), 1998, Leland, J. “Similarity Judgments in Choice Under Uncertainty: A Reinterpretation of Regret Theory.” Management Science, 44(5), 1998, Leland, J. “Similarity Judgments and Anomalies in Intertemporal Choice.” Economic Inquiry Vol. 40, No. 4, October 2002, Leland, J. “Similarity Judgments and Anomalies in Intertemporal Choice.” Economic Inquiry Vol. 40, No. 4, October 2002, Lowenstein, G. and D. Prelec. "Anomalies in Intertemporal Choice: Evidence and Interpretation." The Quarterly Journal of Economics, May 1992, Lowenstein, G. and D. Prelec. "Anomalies in Intertemporal Choice: Evidence and Interpretation." The Quarterly Journal of Economics, May 1992, Rubinstein, A. "Similarity and Decision-making Under Risk (Is There a Utility Theory Resolution to the Allais Paradox?)." Journal of Economic Theory, 46, 1988, Rubinstein, A. "Similarity and Decision-making Under Risk (Is There a Utility Theory Resolution to the Allais Paradox?)." Journal of Economic Theory, 46, 1988, Rubinstein, A. “Economics and Psychology"? The Case of Hyperbolic Discounting, International Economic Review 44, 2003, Rubinstein, A. “Economics and Psychology"? The Case of Hyperbolic Discounting, International Economic Review 44, 2003, Standord Encycolpedia of Philosophy. Standord Encycolpedia of Philosophy.

Testing the Ntg/Ntl Effect – Question Format

The Problem “, 2004) “existing deductive selection rules have been shown to do poorly in experiments” (Haruvy & Stahl, 2004) Should we be surprised? 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?

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