Deterministic Approximation of a Stochastic Imitation Dynamics with Memory

Deterministic Approximation of a Stochastic Imitation Dynamics with Memory 
Abstract: We provide deterministic approximation results for non-Markovian stochastic processes modeling finite populations of individuals who recurrently play symmetric finite games and imitate each other according to payoffs. The deterministic approximation is defined as a system of delay differential equations in continuous time. We show that if the initial states of two (deterministic and stochastic) processes are close enough, then the trajectory of the stochastic process stay close to that of the deterministic model up to any given finite time horizon with a probability exponentially approaching one as the population size increases. We use this result to obtain results regarding the absorption time of the non-Markovian process. Additionally, we obtain the replicator equations with a distributed delay term as an example and analyze how the memory of individuals can affect the evolution of cooperation in a two-player symmetric Snow-drift game. The stability of the evolutionarily stable state of the game will be investigated when agents have the memory of past population states and implications of these results will be given for the stochastic model.