Evolutionary minority games with memory

We propose a simple dynamic adjustment mechanism, equivalent to the standard replicator dynamics in discrete time, to study the time evolution of a population of players facing a binary choice game, and apply this mechanism to minority games in order to investigate the effects of memory on the stability of the unique Nash equilibrium. Two different kinds of memory are considered, one where the players take into account the current and the previous payoffs in order to decide the strategy chosen in the next period, and the other one where the players consider the whole series of payoffs observed in the past through a discounted sum with exponentially fading weights. Both the memory representations proposed lead to an analytically tractable two-dimensional dynamical system, so that analytical results can be given for the stability of the Nash equilibrium. However, a global analysis of the models – performed by numerical methods and guided by the analytical results – shows that some complexities arise for intermediate values of the memory parameter, even if the stabilization effect of uniform memory is stated in both cases.

[1]  J. Sobel,et al.  On the limit points of discrete selection dynamics , 1992 .

[2]  S. Elaydi.  An introduction to difference equations , 1995 .

[3]  Alfredo Medio,et al.  Nonlinear Dynamics: Contents , 2001 .

[4]  R. Devaney.  An Introduction to Chaotic Dynamical Systems , 1990 .

[5]  Francesca Aicardi,et al.  MEMORY EFFECTS IN DISCRETE DYNAMICAL SYSTEMS , 1992 .

[6]  T. Schelling.  Hockey Helmets, Concealed Weapons, and Daylight Saving , 1973 .

[7]  Antje Sommer.  Nonlinear Dynamical Economics And Chaotic Motion , 2016 .

[8]  Yi-Cheng Zhang,et al.  Minority Games , 2008, Encyclopedia of Complexity and Systems Science.

[9]  P. Whitney,et al.  Impulsive decision making and working memory. , 2003, Journal of experimental psychology. Learning, memory, and cognition.

[10]  Axel Leijonhufvud,et al.  Effective Demand Failures , 1973 .

[11]  B. Jones.  BOUNDED RATIONALITY , 1999 .

[12]  Yi Tao,et al.  The replicator equation and other game dynamics , 2014, Proceedings of the National Academy of Sciences.

[13]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[14]  J. Hovi.  Binary Games as Models of Public Goods Provision , 1986 .

[15]  Alfredo Medio,et al.  Nonlinear Dynamics: Subject index , 2001 .

[16]  T. Schelling.  Micromotives and Macrobehavior , 1978 .

[17]  Vijay Mahajan,et al.  New Product Diffusion Models in Marketing: A Review and Directions for Research: , 1990 .

[18]  James M. Holmes,et al.  Memory and market stability: The case of the cobweb , 1988 .

[19]  W. Arthur.  Inductive Reasoning and Bounded Rationality , 1994 .

[20]  Laura Gardini,et al.  Impulsivity in Binary Choices and the Emergence of Periodicity , 2009 .

[21]  Mark S. Granovetter.  Threshold Models of Collective Behavior , 1978, American Journal of Sociology.

[22]  Ali Jadbabaie,et al.  Duopoly pricing game in networks with local coordination effects , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[23]  D. Stahl,et al.  Level-n bounded rationality and dominated strategies in normal-form games , 2008 .

[24]  Jaap van Ginneken.  Collective Behavior and Public Opinion : Rapid Shifts in Opinion and Communication , 2003 .

[25]  Alfredo Medio,et al.  Nonlinear Dynamics: Bibliography , 2001 .

[26]  Reiner Franke,et al.  Reinforcement learning in the El Farol model , 2003 .

[27]  Ugo Merlone,et al.  Binary choices in small and large groups: A unified model , 2010 .

[28]  S. Fiske,et al.  Social Psychology , 2019, Encyclopedia of Personality and Individual Differences.

[29]  A tractable evolutionary model for the Minority Game with asymmetric payoffs , 2005 .

[30]  A. Cavagna.  Irrelevance of memory in the minority game , 1998, cond-mat/9812215.

[31]  Ugo Merlone,et al.  Global Dynamics in Binary Choice Models with Social Influence , 2009 .

[32]  J. Hofbauer,et al.  Evolutionary game dynamics , 2011 .

[33]  Yi-Cheng Zhang,et al.  Emergence of cooperation and organization in an evolutionary game , 1997 .

[34]  Mark S. Granovetter,et al.  Threshold models of diffusion and collective behavior , 1983 .

[35]  Challet,et al.  Relevance of memory in minority games , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.