The ESS and replicator equation in matrix games under time constraints

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.

The ESS for evolutionary matrix games under time constraints and its relationship with the asymptotically stable rest point of the replicator dynamics

“…For classical matrix games the corresponding result makes it easier to verify the asymptotic stability of the corresponding rest point of the replicator dynamics. Using these characterizations we extend the main result of Garay et al (2018) showing that, for two dimensional strategies, a strategy is a UESS if and only if the set of the corresponding states is asymptotically stable with respect to the replicator dynamics independently of how many two dimensional phenotypes are considered in the replicator dynamics. The proof, however, makes essential use of the fact that the strategy space is a one dimensional manifold.…”

“…In this article we continue the investigation begun in Garay et al (2018) and seek an answer to what the relationship is between an ESS and the corresponding rest point of the replicator dynamics in matrix games under time constraints. More precisely, we use the notion of uniformly evolutionarily stable strategy (UESS) instead of ESS.…”

“…If ε 0 is allowed to be chosen depending on q we come to the definition of evolutionarily stable strategy of the matrix game under time constraints which was used in our previous work [Garay et al 2018] and we get back Taylor and Jonker's wording. It is not too difficult to see that for N = 2 the two notions are equivalent, whereas in higher dimensions, the equivalence is not known in our setup.…”

“…Proposition 2.8 (Garay et al 2018, p. 9-10) Let p * be the convex combination of the strategies p 1 , . .…”

“…This is natural since other activities such as handling the prey [Holling 1959;Garay and Móri 2010], recovering from an injury [Garay et al 2015;Sirot 2000] decrease the number of individuals able to interact in the consumer/predator population. Moreover, the time necessary for the previous activities can depend on the strategy of the given individual resulting in other evolutionary outcomes [Křivan and Cressman 2017, Sections 3.1.1-2 and 3.2.1-2; Garay et al 2017, Section 4;Garay et al 2018, Example 1] 1 to those in the classical setup of Maynard Smith [Maynard Smith 1982] in which the effect of time constraints can be considered as a constant and so a negligible factor in the payoff function (see the remark after (2.2)). Also, other theories such as optimal foraging theory [Charnov 1976;Garay et al 2012] or ecological games [Broom and Ruxton 1998;Broom et al 2008;Broom et al 2009;Broom et al 2010, Broom andRychtář 2013;Garay et al 2015] consider the substantial effect of time constraints on the expected evolutionary outcomes.…”