BEST RESPONSE DYNAMICS FOR CONTINUOUS ZERO{SUM GAMES

We study best response dynamics in continuous time for continuous concave-convex zero-sum games and prove convergence of its trajectories to the set of saddle points, thus providing a dynamical proof of the minmax theorem. Consequences for the corresponding discrete time process with small or diminishing step-sizes are established, including convergence of the fictitious play procedure.

[1]  J. Neumann.  Zur Theorie der Gesellschaftsspiele , 1928 .

[2]  J. Neumann,et al.  SOLUTIONS OF GAMES BY DIFFERENTIAL EQUATIONS , 1950 .

[3]  J. Robinson.  AN ITERATIVE METHOD OF SOLVING A GAME , 1951, Classics in Game Theory.

[4]  K Fan,et al.  Minimax Theorems. , 1953, Proceedings of the National Academy of Sciences of the United States of America.

[5]  M. Sion.  On general minimax theorems , 1958 .

[6]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[7]  I. Gilboa,et al.  Social Stability and Equilibrium , 1991 .

[8]  Akihiko Matsui,et al.  Best response dynamics and socially stable strategies , 1992 .

[9]  D. Monderer,et al.  Belief Affirming in Learning Processes , 1997 .

[10]  A. R. Humphries,et al.  Dynamical Systems And Numerical Analysis , 1996 .

[11]  P. Rivière.  Quelques modeles de jeux d'evolution , 1997 .

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

[13]  C. Harris.  On the Rate of Convergence of Continuous-Time Fictitious Play , 1998 .

[14]  Yu. S. Ledyaev,et al.  Nonsmooth analysis and control theory , 1998 .

[15]  M. Benaïm.  Dynamics of stochastic approximation algorithms , 1999 .

[16]  J. Frédéric Bonnans,et al.  Perturbation Analysis of Optimization Problems , 2000, Springer Series in Operations Research.

[17]  J. Hofbauer.  From Nash and Brown to Maynard Smith: Equilibria, Dynamics and ESS , 2001 .

[18]  Josef Hofbauer,et al.  Stochastic Approximations and Differential Inclusions , 2005, SIAM J. Control. Optim..