Stability and trembles in extensive-form games

Heller, Yuval (2013): Stability and trembles in extensive-form games. Forthcoming in: Games and Economic Behaivor No. forthcoming

Abstract

A leading solution concept in the evolutionary study of extensive-form games is Selten's (1983) (selten1983evolutionary) notion of limit ESS. We demonstrate that a limit ESS does not imply neutral stability, and that it may be dynamically unstable (almost any small perturbation takes the population away). These problems arise due to an implicit assumption that “mutants” are arbitrarily rare relative to “trembling” incumbents. Finally, we present a novel definition that solves this issue and has appealing properties.

• Item Type: MPRA Paper
• Language: English
• Keywords: Limit ESS, evolutionary stability, extensive-form games.
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C73 - Stochastic and Dynamic Games ; Evolutionary Games ; Repeated Games

References

Bhaskar, V. 1995. On the neutral stability of mixed strategies in asymmetric contests. Mathematical Social Sciences, 30(3), 273-284.

Bhaskar, V. 1998. Noisy communication and the evolution of cooperation. Journal of Economic Theory, 82(1), 110-131.

Bolton, Gary E. 1997. The rationality of splitting equally. Journal of Economic Behavior and Organization, 32(3), 365-381.

Bomze, Immanuel M, & Weibull, Jörgen W. 1995. Does neutral stability imply Lyapunov stability? Games and Economic Behavior, 11(2), 173-192.

Cressman, Ross. 1997. Local stability of smooth selection dynamics for normal form games. Mathematical Social Sciences, 34(1), 1-19.

Heller, Y. 2013. Three steps ahead. mimeo.

Kim, Yong-Gwan. 1993. Evolutionary stability in the asymmetric war of attrition. Journal of theoretical biology, 161(1), 13.

Kim, Yong-Gwan. 1994. Evolutionarily stable strategies in the repeated prisoner's dilemma. Mathematical Social Sciences, 28(3), 167-197.

Kuhn, Harold W. 1953. Extensive games and the problem of information. Contributions to the Theory of Games, 2(28), 193-216.

Leimar, Olof. 1997. Repeated games: a state space approach. Journal of Theoretical Biology, 184(4), 471-498.

Maynard Smith, J. 1982. Evolution and the theory of games. Cambridge University Press.

Maynard-Smith, J., & Price, G.R. 1973. The Logic of Animal Conflict. Nature, 246, 15.

Samuelson, Larry, & Zhang, Jianbo. 1992. Evolutionary stability in asymmetric games. Journal of economic theory, 57(2), 363-391.

Sandholm, William H. 2010. Local stability under evolutionary game dynamics. Theoretical Economics, 5(1), 27-50.

Selten, Reinhard. 1983. Evolutionary stability in extensive two-person games. Mathematical Social Sciences, 5(3), 269-363.

Taylor, P.D., & Jonker, L.B. 1978. Evolutionary stable strategies and game dynamics. Mathematical Biosciences, 40(1), 145-156.

van Damme, E. 1987. Stability and Perfection of Nash Equilibria. Springer, Berlin.

Weibull, Jörgen W. 1995. Evolutionary game theory. The MIT press.