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

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.