Local stability under evolutionary game dynamics

We prove that any regular evolutionarily stable strategy (ESS) is asymptotically stable under any impartial pairwise comparison dynamic, including the Smith dynamic; under any separable excess payoff dynamic, including the BNN dynamic; and under the best response dynamic. Combined with existing results for imitative dynamics, our analysis validates the use of regular ESS as a blanket sufficient condition for local stability under evolutionary game dynamics.

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

[2]  T. Koopmans,et al.  Studies in the Economics of Transportation. , 1956 .

[3]  J. M. Smith,et al.  The Logic of Animal Conflict , 1973, Nature.

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

[5]  P. Taylor,et al.  Evolutionarily Stable Strategies and Game Dynamics , 1978 .

[6]  K Sigmund,et al.  A note on evolutionary stable strategies and game dynamics. , 1979, Journal of theoretical biology.

[7]  W. Hines Three characterizations of population strategy stability , 1980 .

[8]  E. C. Zeeman,et al.  Population dynamics from game theory , 1980 .

[9]  R. Selten A note on evolutionarily stable strategies in asymmetric animal conflicts. , 1980, Journal of theoretical biology.

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

[11]  B. Thomas Evolutionary stability: States and strategies , 1984 .

[12]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[13]  Josef Hofbauer,et al.  The theory of evolution and dynamical systems , 1988 .

[14]  Brian Skyrms,et al.  The Dynamics Of Rational Deliberation , 1990 .

[15]  R. Cressman,et al.  Strong stability and evolutionarily stable strategies with two types of players , 1991 .

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

[17]  D. Friedman EVOLUTIONARY GAMES IN ECONOMICS , 1991 .

[18]  L. Samuelson,et al.  Evolutionary Stability in Asymmetric Games , 1992 .

[19]  Ross Cressman,et al.  The Stability Concept of Evolutionary Game Theory , 1992 .

[20]  Ross Cressman,et al.  The Stability Concept of Evolutionary Game Theory: A Dynamic Approach , 1992 .

[21]  Jeroen M. Swinkels Evolutionary stability with equilibrium entrants , 1992 .

[22]  Jeroen M. Swinkels Adjustment Dynamics and Rational Play in Games , 1993 .

[23]  J. Weibull,et al.  Nash Equilibrium and Evolution by Imitation , 1994 .

[24]  Jörgen W. Weibull,et al.  Evolutionary Game Theory , 1996 .

[25]  J. Weibull,et al.  Does Neutral Stability Imply Lyapunov Stability , 1995 .

[26]  Ross Cressman,et al.  Local stability of smooth selection dynamics for normal form games , 1997 .

[27]  K. Schlag Why Imitate, and If So, How?, : A Boundedly Rational Approach to Multi-armed Bandits , 1998 .

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

[29]  M. Nowak,et al.  Evolutionary game theory , 1995, Current Biology.

[30]  J Hofbauer,et al.  Evolutionary stability concepts for N-species frequency-dependent interactions. , 2001, Journal of theoretical biology.

[31]  Karl H. Schlag,et al.  Evolutionarily stable sets , 2000, Int. J. Game Theory.

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

[33]  William H. Sandholm,et al.  Potential Games with Continuous Player Sets , 2001, J. Econ. Theory.

[34]  R. Cressman Evolutionary Dynamics and Extensive Form Games , 2003 .

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

[36]  Josef Hofbauer,et al.  Learning in perturbed asymmetric games , 2005, Games Econ. Behav..

[37]  William H. Sandholm,et al.  Excess payoff dynamics and other well-behaved evolutionary dynamics , 2005, J. Econ. Theory.

[38]  R. Cressman Uninvadability in N-species frequency models for resident-mutant systems with discrete or continuous time. , 2006, Theoretical population biology.

[39]  Karl H. Schlag,et al.  On the evolutionary selection of sets of Nash equilibria , 2007, J. Econ. Theory.

[40]  William H. Sandholm,et al.  Evolution in Bayesian games II: Stability of purified equilibria , 2007, J. Econ. Theory.

[41]  William H. Sandholm,et al.  The projection dynamic and the replicator dynamic , 2008, Games Econ. Behav..

[42]  Ziv Gorodeisky Stochastic Approximation of Discontinuous Dynamics , 2008 .

[43]  Josef Hofbauer,et al.  Stable games and their dynamics , 2009, J. Econ. Theory.

[44]  William H. Sandholm,et al.  Evolutionary Game Theory , 2009, Encyclopedia of Complexity and Systems Science.

[45]  Ziv Gorodeisky Deterministic approximation of best-response dynamics for the Matching Pennies game , 2009, Games Econ. Behav..

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

[47]  William H. Sandholm,et al.  Survival of dominated strategies under evolutionary dynamics , 2011 .