Evolutionary stability for matrix games under time constraints

Abstract

Game theory focuses on payoffs and typically ignores time constraints that play an important role in evolutionary processes where the repetition of games can depend on the strategies, too. We introduce a matrix game under time constraints, where each pairwise interaction has two consequences: both players receive a payoff and they cannot play the next game for a specified time duration. Thus our model is defined by two matrices: a payoff matrix and an average time duration matrix. Maynard Smith's concept of evolutionary stability is extended to this class of games. We illustrate the effect of time constraints by the well-known prisoner's dilemma game, where additional time constraints can ensure the existence of unique evolutionary stable strategies (ESS), both pure and mixed, or the coexistence of two pure ESS. Our general results may be useful in several fields of biology where evolutionary game theory is applied, principally in ecological games, where time constraints play an inevitable role.

The ESS and replicator equation in matrix games under time constraints

“…Moreover, as shown by Garay et al (2017), the stationary distribution of the Markov process for fixed ε satisfies the system of equations…”

“…Hence our time-constrained matrix game is characterized by two matrices, the intake matrix A = (a ij ) n×n , and the time constraint matrix T = (τ ij ) n×n . For individual fitness in this game, we follow Garay et al (2017) who assume a continuous time Markov model is used where a focal individual's time between encounters when active and the amount of time it is inactive are independent and exponentially distributed with prescribed mean. 2 For a large population in this situation, they show that the fitness of the focal individual is given by the quotient…”

“…From the monomorphic approach of Maynard Smith (1982), we have the following definition (see also Garay et al (2017)). …”

“…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). This theory is briefly summarized in Section 2 below by following the static ESS approach of Maynard Smith (1982).…”

“…An ESS is then a resident phenotype whose fitness is higher than that of any possible mutant (cf. Maynard Smith 1982, Garay et al 2017). Although the static ESS concept relies implicitly upon an underlying dynamic (see Maynard Smith (1982) as well as the discussion following Definition 2.1 below), its basic intuition concentrates on the evolutionary question: What phenotype is uninvadable by an arbitrary rare mutant?…”