Deterministic Equations for Stochastic Spatial Evolutionary Games

Spatial evolutionary games model individuals who are distributed in a spa- tial domain and update their strategies upon playing a normal form game with their neighbors. We derive integro-differential equations as deterministic ap- proximations of the microscopic updating stochastic processes. This generalizes the known mean-field ordinary differential equations and provide a powerful tool to investigate the spatial effects in populations evolution. The determin- istic equations allow to identify many interesting features of the evolution of strategy profiles in a population, such as standing and traveling waves, and pattern formation, especially in replicator-type evolutions.

[1]  G. T. Vickers,et al.  Travelling waves and dominance of ESS's , 1992 .

[2]  C. Budd,et al.  Spatial patterns in population conflicts , 1993 .

[3]  T. Liggett.  Interacting Particle Systems , 1985 .

[4]  R. Durrett.  Mutual invadability implies coexistence in spatial models , 2002 .

[5]  Panagiotis E. Souganidis,et al.  Stochastic Ising models and anisotropic front propagation , 1997 .

[6]  Glenn Ellison.  Learning, Local Interaction, and Coordination , 1993 .

[7]  H. Young,et al.  Competition and Custom in Economic Contracts: A Case Study of Illinois Agriculture , 2001 .

[8]  J M Smith,et al.  Evolution and the theory of games , 1976 .

[9]  Elias M. Stein,et al.  Fourier Analysis: An Introduction , 2003 .

[10]  H. Young,et al.  The Evolution of Conventions , 1993 .

[11]  Sebastian Grauwin,et al.  Dynamic Models of Residential Ségrégation: An Analytical Solution , 2010 .

[12]  L. Blume,et al.  POPULATION GAMES , 1995 .

[13]  G. Szabó,et al.  Evolutionary games on graphs , 2006, cond-mat/0607344.

[14]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[15]  Errico Presutti,et al.  Travelling fronts in non-local evolution equations , 1995 .

[16]  Markos A. Katsoulakis,et al.  Mesoscopic Modeling for Continuous Spin Lattice Systems: Model Problems and Micromagnetics Applications , 2005 .

[17]  Xinfu Chen,et al.  Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations , 1997, Advances in Differential Equations.

[18]  J. Marsden,et al.  Linearization stability of nonlinear partial differential equations , 1975 .

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

[20]  Enza Orlandi,et al.  Glauber evolution with Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics , 1994 .

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

[22]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[23]  L. Blume.  The Statistical Mechanics of Best-Response Strategy Revision , 1995 .

[24]  Thomas C. Schelling,et al.  Dynamic models of segregation , 1971 .

[25]  David Aldous,et al.  Stopping Times and Tightness. II , 1978 .

[26]  Enza Orlandi,et al.  Travelling fronts in nonlocal models for phase separation in an external field , 1997, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[27]  T. Kurtz.  Solutions of ordinary differential equations as limits of pure jump markov processes , 1970, Journal of Applied Probability.

[28]  Michael J. Smith,et al.  The Stability of a Dynamic Model of Traffic Assignment - An Application of a Method of Lyapunov , 1984, Transp. Sci..

[29]  H. Young,et al.  Individual Strategy and Social Structure: An Evolutionary Theory of Institutions , 1999 .

[30]  Steven N. Durlauf,et al.  A formal model of theory choice in science , 1999 .

[31]  H. Peyton Young,et al.  Individual Strategy and Social Structure , 2020 .

[32]  Alexander Matros,et al.  Stochastic Imitation in Finite Games , 2002, Games Econ. Behav..

[33]  Rick Durrett,et al.  Stochastic Spatial Models , 1999, SIAM Rev..

[34]  R. D. Passo,et al.  The heat equation with a nonlocal density dependent advection term , 1991 .

[35]  Pierre Collet,et al.  Instabilities and fronts in extended systems , 1990 .

[36]  Samuel Bowles,et al.  Microeconomics: Behavior, Institutions, and Evolution , 2003 .

[37]  Josef Hofbauer,et al.  The spatially dominant equilibrium of a game , 1999, Ann. Oper. Res..

[38]  L. Blume.  The Statistical Mechanics of Strategic Interaction , 1993 .

[39]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[40]  J. Carr,et al.  Metastable patterns in solutions of ut = ϵ2uxx − f(u) , 1989 .

[41]  J. Hofbauer.  Equilibrium Selection Via Travelling Waves , 1998 .

[42]  William H. Sandholm,et al.  Population Games And Evolutionary Dynamics , 2010, Economic learning and social evolution.

[43]  Glenn Ellison.  Basins of Attraction, Long-Run Stochastic Stability, and the Speed of Step-by-Step Evolution , 2000 .

[44]  Josef Hofbauer,et al.  Evolutionary Games and Population Dynamics , 1998 .

[45]  P. Bates,et al.  Traveling Waves in a Convolution Model for Phase Transitions , 1997 .

[46]  William A. Brock,et al.  Discrete Choice with Social Interactions , 2001 .

[47]  D. Fudenberg,et al.  The Theory of Learning in Games , 1998 .

[48]  David K. Levine,et al.  The evolution of cooperation through imitation , 2002, Games Econ. Behav..

[49]  F. Comets.  Nucleation for a long range magnetic model , 1987 .

[50]  Alberto Bisin,et al.  The Economics of Cultural Transmission and Socialization , 2010 .

[51]  C. Landim,et al.  Scaling Limits of Interacting Particle Systems , 1998 .

[52]  Jörg Oechssler,et al.  Imitation - Theory and Experimental Evidence , 2003, J. Econ. Theory.

[53]  Jörg Oechssler,et al.  Imitation and the Evolution of Walrasian Behavior: Theoretically Fragile but Behaviorally Robust , 2008, J. Econ. Theory.

[54]  John C. Harsanyi,et al.  Общая теория выбора равновесия в играх / A General Theory of Equilibrium Selection in Games , 1989 .

[55]  Stefan Grosskinsky Warwick,et al.  Interacting Particle Systems , 2016 .

[56]  Enza Orlandi,et al.  Glauber evolution with Kac potentials: III. Spinodal decomposition , 1996 .

[57]  N. Shigesada,et al.  Spatial distribution of dispersing animals , 1980, Journal of mathematical biology.

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

[59]  Josef Hofbauer,et al.  Travelling waves for games in economics and biology , 1997 .

[60]  Errico Presutti,et al.  Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics , 2008 .

[61]  Dan Bernhardt,et al.  Cooperation through imitation , 2009, Games Econ. Behav..

[62]  William H. Sandholm,et al.  Pairwise Comparison Dynamics and Evolutionary Foundations for Nash Equilibrium , 2009, Games.

[63]  J. Norris,et al.  Differential equation approximations for Markov chains , 2007, 0710.3269.

[64]  E. Hopkins.  A Note on Best Response Dynamics , 1999 .

[65]  Paul C. Fife,et al.  Mathematical Aspects of Reacting and Diffusing Systems , 1979 .

[66]  R. Ellis,et al.  Symmetry breaking and random waves for magnetic systems on a circle , 1983 .

[67]  O. Penrose,et al.  Rigorous Treatment of the Van Der Waals-Maxwell Theory of the Liquid-Vapor Transition , 1966 .

[68]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .

[69]  Vitaly Volpert,et al.  Traveling Wave Solutions of Parabolic Systems , 1994 .

[70]  F. Vega-Redondo,et al.  Efficient Equilibrium Selection in Evolutionary Games with Random Matching , 1996 .