Long-run analysis of the stochastic replicator dynamics in the presence of random jumps

Andrew Vlasic

doi:10.3934/jdg.2018018

Abstract

The effect of anomalous events on the replicator dynamics with aggregate shocks is considered. The anomalous events are described by a Poisson integral, where this stochastic forcing term is added to the fitness of each agent. Contrary to previous models, this noise is assumed to be correlated across the population. A formula to calculate a closed form solution of the long run behavior of a two strategy game will be derived. To assist with the analysis of a two strategy game, the stochastic Lyapunov method will be applied. For a population with a general number of strategies, the time averages of the dynamics will be shown to converge to the Nash equilibria of a relevant modified game. In the context of the modified game, the almost sure extinction of a dominated pure strategy will be derived. As the dynamic is quite complex, with respect to the original game a pure strict Nash equilibrium and an interior evolutionary stable strategy will be considered. Respectively, conditions for stochastically stability and the positive recurrent property will be derived. This work extends previous results on the replicator dynamics with aggregate shocks.

Highlights

Consider a two-player symmetric game, where aij is the payoff to a player using strategy Si against an opponent employing strategy Sj, and define A =, as the payoff matrix
Within a population we assume that every individual is programmed to play a pure strategy Si
Let ri(t) be the size of the subpopulation that plays strategy Si at time t, which we denote as the ith subpopulation

Summary

Introduction

We consider effects that impact each subpopulation’s fitness, and where the affects slowly decrease To capture this phenomenon, a compensated Poisson integral is added to the Fudenberg and Harris model, so that the expectation of this perturbation is zero, and a continuous integrand models the various intensities of each impact of the anomaly. The method follows the proof of the theorem that Fudenberg and Harris applied in their analysis [12] The proof of this theorem first considers a subinterval of [0, 1] where the initial condition lies, and a second-order differential equation determines the probabilities of the process first leaving from the right or left endpoint. The conditions for the stability of pure Nash equilibria and extinction of pure dominated strategies are more general and creates situations for either behavior to hold that would otherwise not be possible with just considering the stochastic replicator dynamic.

Deriving the Extended Stochastic Replicator Dynamic

Part II

Dominated Strategies

Conditions for Recurrence Near an Interior Evolutionary Stable Strategy