Learning aspiration in repeated games

Abstract We study infinitely repeated symmetric 2 × 2 games played by bounded rational agents who follow a simple rule of thumb: each agent continues to play the same action if the current payoff exceeds the average of the past payoffs, and switches to the other action with a positive probability otherwise. By applying the stochastic approximation technique, we characterize the asymptotic outcomes for all 2×2 games. In the prisoners’ dilemma game, for example, the players cooperate in the limit if and only if the gain from defecting against cooperation is “modest.”

[1]  Jörg Oechssler,et al.  Cooperation as a Result of Learning with Aspiration Levels , 2002 .

[2]  I. Gilboa,et al.  Case-Based Decision Theory , 1995 .

[3]  Tamer Basar,et al.  Analysis of Recursive Stochastic Algorithms , 2001 .

[4]  R. Rob,et al.  Learning, Mutation, and Long Run Equilibria in Games , 1993 .

[5]  Harold J. Kushner,et al.  wchastic. approximation methods for constrained and unconstrained systems , 1978 .

[6]  A. Rubinstein,et al.  The Structure of Nash Equilibrium in Repeated Games with Finite Automata (Now published in Econometrica, 56 (1988), pp.1259-1282.) , 1986 .

[7]  Stefan Napel,et al.  Aspiration adaptation in the ultimatum minigame , 2003, Games Econ. Behav..

[8]  Carlos S. Kubrusly,et al.  Stochastic approximation algorithms and applications , 1973, CDC 1973.

[9]  Sidney G. Winter,et al.  Satisficing, Selection, and the Innovating Remnant , 1971 .

[10]  Akihiko Matsui,et al.  Expected utility and case-based reasoning , 2000, Math. Soc. Sci..

[11]  T. Sargent,et al.  Convergence of Least Squares Learning Mechanisms in Self- Referential Linear Stochastic Models* , 1989 .

[12]  Seppo Honkapohja,et al.  Expectations and the Stability Problem for Optimal Monetary Policies , 2000, SSRN Electronic Journal.

[13]  H. Simon,et al.  Models of Man. , 1957 .

[14]  R. Sarin,et al.  Payoff Assessments without Probabilities: A Simple Dynamic Model of Choice , 1999 .

[15]  Jonathan Bendor,et al.  Reinforcement Learning in Repeated Interaction Games , 2001 .

[16]  G. Evans,et al.  Learning and expectations in macroeconomics , 2001 .

[17]  W. Brown.  Animal Intelligence: Experimental Studies , 1912, Nature.

[18]  Itzhak Gilboa,et al.  Reaction to price changes and aspiration level adjustments , 2001 .

[19]  M. Woodford.  Learning to Believe in Sunspots , 1990 .

[20]  Dov Samet,et al.  Learning to play games in extensive form by valuation , 2001, J. Econ. Theory.

[21]  K. Sigmund,et al.  The efficiency of adapting aspiration levels , 1998, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[22]  William H. Sandholm,et al.  ON THE GLOBAL CONVERGENCE OF STOCHASTIC FICTITIOUS PLAY , 2002 .

[23]  Debraj Ray,et al.  Evolving Aspirations and Cooperation , 1998 .

[24]  Win Stay, Lose Shift Or Imitation – Only The Choice Of Peers Counts , 2001 .

[25]  P. Kline.  Models of man , 1986, Nature.

[26]  P. Dupuis,et al.  Stochastic approximation and large deviations: upper bounds and w.p.1 convergence , 1989 .

[27]  Itzhak Gilboa,et al.  A theory of case-based decisions , 2001 .

[28]  W. MacLeod,et al.  Gift Giving and the Evolution of Cooperation , 1997 .

[29]  M. Benaïm,et al.  Deterministic Approximation of Stochastic Evolution in Games , 2003 .

[30]  Peter Secretan.  Learning , 1965, Mental Health.

[31]  Robert J. Aumann,et al.  16. Acceptable Points in General Cooperative n-Person Games , 1959 .