Long-Run Analysis of the Stochastic Replicator Dynamics in the Presence of Random Jumps

TL;DR

This paper extends the stochastic replicator dynamic model by incorporating Poisson jumps to simulate anomalous events, analyzing long-term behavior and stability in multi-strategy populations.

Contribution

It introduces a Poissonian integral into the replicator dynamic and derives conditions for stability, extinction, and recurrence in complex multi-strategy settings.

Findings

Conditions for stability of pure Nash equilibria

Criteria for extinction of dominated strategies

Recurrence near internal stable strategies

Abstract

A further generalization of the stochastic replicator dynamic derived by Fudenberg and Harris \cite{FH92} is considered. In particular, a Poissonian integral is introduced to the fitness to simulate the affects of anomalous events. For the two strategy population, an estimation of the long run behavior of the dynamic is derived. For the population with many strategies, conditions for stability to pure strict Nash equilibria, extinction of dominated pure strategies, and recurrence in a neighborhood of an internal evolutionary stable strategy are derived. This extends the results given by Imhof \cite{I05}.