The ESS and replicator equation in matrix games under time constraints.

József Garay,Ross Cressman,Tamás Varga,Tamás F Móri

doi:10.1007/s00285-018-1207-0

Abstract

Recently, we introduced the class of matrix games under time constraints and characterized the concept of (monomorphic) evolutionarily stable strategy (ESS) in them. We are now interested in how the ESS is related to the existence and stability of equilibria for polymorphic populations. We point out that, although the ESS may no longer be a polymorphic equilibrium, there is a connection between them. Specifically, the polymorphic state at which the average strategy of the active individuals in the population is equal to the ESS is an equilibrium of the polymorphic model. Moreover, in the case when there are only two pure strategies, a polymorphic equilibrium is locally asymptotically stable under the replicator equation for the pure-strategy polymorphic model if and only if it corresponds to an ESS. Finally, we prove that a strict Nash equilibrium is a pure-strategy ESS that is a locally asymptotically stable equilibrium of the replicator equation in n-strategy time-constrained matrix games.

Highlights

In ecology, the number of individuals ready to interact with the conspecifics they meet is less than the total number of individuals in the species
Given the prominence of two-strategy games in applications of evolutionary game theory, these results mean that evolutionary outcomes of ecological models that incorporate the dynamic effects of different individual behaviors may continue to be predicted through static game-theoretic reasoning
We emphasize that the replicator equation is an ecological model since the question it addresses is whether the long time existence of different strategies is possible

Summary

Introduction

The number of individuals ready to interact with the conspecifics they meet (we call these active individuals) is less than the total number of individuals in the species. In optimal foraging theory (Charnov 1976; Garay et al 2012) and in ecological games (e.g. Broom et al 2008, Broom and Rychtar 2013; Garay et al 2015a), activity dependent time constraints have an essential effect on the expected evolutionary outcome Motivated by these facts, we recently developed the theory of singlespecies matrix games under time constraints and characterized the concept of a (monomorphic) evolutionarily stable strategy (ESS) in them (Garay et al 2017). Under the replicator equation (Taylor and Jonker 1978) of the standard polymorphic population model (i.e. each phenotype existing in the population is a pure strategy), the evolutionary outcome is characterized as a locally asymptotically stable rest point of this dynamical system. An ESS still corresponds to a polymorphic rest point of the replicator equation (see Lemma 3.2), we conjecture that counterexamples to stability of this polymorphic distribution already appear in three-strategy games

Matrix games under time constraints and the monomorphic model

Monomorphic versus polymorphic populations

Evolutionary and Dynamic Stability in the Pure-Strategy Model

Discussion

Preliminaries