Published on Fri Aug 06 2021

Monomorphic ESS does not imply the stability of the corresponding polymorphic state in the replicator dynamics in matrix games under time constraints

Varga, T., Garay, J.

Monomorphic ESS condition implies the stability of the corresponding state of the polymorphic replicator dynamics. In this paper we demonstrate that this relationship generally does not hold in three or higher dimensions.

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Abstract

One of the main result in the theory of classical evolutionary matrix games (Maynard Smith and Price 1973, Maynard Smith 1982) claims that monomorphic ESS condition implies the stability of the corresponding state of the polymorphic replicator dynamics (Hofbauer et al. 1979, Zeeman 1980). The picture was then refined by Cressman (1990) introducing the strong stability concept which says that if there is a monomorphic ESS then stable polymorphism is established in polymorphic populations. In this paper we demonstrate with examples that this relationship generally does not hold in three or higher dimensions if times related to the interactions vary with the strategies of the participants.