Large-Scale Dynamics of Mean-Field Games Driven by Local Nash Equilibria

We introduce a new mean field kinetic model for systems of rational agents interacting in a game-theoretical framework. This model is inspired from non-cooperative anonymous games with a continuum of players and Mean-Field Games. The large time behavior of the system is given by a macroscopic closure with a Nash equilibrium serving as the local thermodynamic equilibrium. An application of the presented theory to a social model (herding behavior) is discussed.

[1]  M J Lighthill,et al.  On kinematic waves II. A theory of traffic flow on long crowded roads , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[2]  Shlomo Weber,et al.  Pure Strategy Nash Equilibrium in a Group Formation Game with Positive Externalities , 1997 .

[3]  Robert J. Aumann,et al.  EXISTENCE OF COMPETITIVE EQUILIBRIA IN MARKETS WITH A CONTINUUM OF TRADERS , 2020, Classics in Game Theory.

[4]  B. Kiker.  Investment in human capital , 1971 .

[5]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[6]  C. Villani.  Topics in Optimal Transportation , 2003 .

[7]  Kerson Huang.  Statistical Mechanics, 2nd Edition , 1963 .

[8]  E. Lazear,et al.  Rank-Order Tournaments as Optimum Labor Contracts , 1979, Journal of Political Economy.

[9]  Sara van de Geer,et al.  Ecole d'été de probabilités de Saint-Flour XLV , 2016 .

[10]  P. Lions,et al.  Mean field games , 2007 .

[11]  M. A. Cayless.  Statistical Mechanics (2nd edn) , 1977 .

[12]  Vicsek,et al.  Novel type of phase transition in a system of self-driven particles. , 1995, Physical review letters.

[13]  Nicholas Kevlahan,et al.  Principles of Multiscale Modeling , 2012 .

[14]  R. Aumann.  Markets with a continuum of traders , 1964 .

[15]  Filippo Santambrogio,et al.  Existence and Uniqueness of Equilibrium for a Spatial Model of Social Interactions , 2016 .

[16]  L. Shapley,et al.  Values of Large Games, I , 1977 .

[17]  A. Sznitman.  Topics in propagation of chaos , 1991 .

[18]  Guillaume Carlier,et al.  Optimal Transport and Cournot-Nash Equilibria , 2012, Math. Oper. Res..

[19]  Pierre Degond,et al.  HYDRODYNAMIC MODELS OF SELF-ORGANIZED DYNAMICS: DERIVATION AND EXISTENCE THEORY ∗ , 2011, 1108.3160.

[20]  E. Weinan.  Principles of Multiscale Modeling , 2011 .

[21]  Jian-Guo Liu,et al.  Dynamics in a Kinetic Model of Oriented Particles with Phase Transition , 2011, SIAM J. Math. Anal..

[22]  G. Theraulaz,et al.  Vision-based macroscopic pedestrian models , 2013, 1307.1953.

[23]  Jian‐Guo Liu,et al.  Evolution of the Distribution of Wealth in an Economic Environment Driven by Local Nash Equilibria , 2013, 1307.1685.

[24]  M J Lighthill,et al.  ON KINEMATIC WAVES.. , 1955 .

[25]  Luc Mieussens,et al.  Macroscopic Fluid Models with Localized Kinetic Upscaling Effects , 2006, Multiscale Model. Simul..

[26]  Pierre Degond,et al.  Continuum limit of self-driven particles with orientation interaction , 2007, 0710.0293.

[27]  R. Rosenthal.  A class of games possessing pure-strategy Nash equilibria , 1973 .

[28]  Edward P. Lazear,et al.  Rank-Order Tournaments as Optimum Labor Contracts , 1979 .

[29]  Elton P. Hsu.  Stochastic analysis on manifolds , 2002 .

[30]  D. Kinderlehrer,et al.  Free energy and the Fokker-Planck equation , 1997 .

[31]  J. Nash.  Equilibrium Points in N-Person Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[32]  G. Psacharopoulos,et al.  Investment in Human Capital. , 1972 .

[33]  Pierre-Louis Lions,et al.  Long time average of mean field games , 2012, Networks Heterog. Media.

[34]  L. Shapley,et al.  Potential Games , 1994 .

[35]  R. Porter,et al.  NONCOOPERATIVE COLLUSION UNDER IMPERFECT PRICE INFORMATION , 1984 .

[36]  P. Degond,et al.  A Hierarchy of Heuristic-Based Models of Crowd Dynamics , 2013, 1304.1927.

[37]  Jian-Guo Liu,et al.  Macroscopic Limits and Phase Transition in a System of Self-propelled Particles , 2011, Journal of Nonlinear Science.

[38]  L. Shapley,et al.  REGULAR ARTICLEPotential Games , 1996 .

[39]  M. Lighthill,et al.  On kinematic waves I. Flood movement in long rivers , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[40]  N. Z. Shapiro,et al.  Values of Large Games, I: A Limit Theorem , 1978, Math. Oper. Res..

[41]  A. Mas-Colell.  On a theorem of Schmeidler , 1984 .

[42]  O. A. B. Space,et al.  EQUILIBRIUM POINTS OF NONATOMIC GAMES , 2010 .