Random Time Delays in Evolutionary Game Dynamics

In this paper, we study evolutionary games and we examine the stability of the Evolutionarily Stable Strategy (ESS) in the continuous-time replicator dynamics with random time delays. In many examples the interactions between individuals take place instantaneously but their impacts are not immediate and take a certain amount of time, which is usually random. In this paper we study the effect of randomly distributed time delays in the replicator dynamics. We show that, under the exponential delay distribution, the ESS is asymptotically stable for any value of the rate parameter. For the uniform and Erlang distributions, we derive necessary and sufficient conditions for the asymptotic stability of the ESS. We also study the case of one delayed strategy.