Evolutionary stochastic games

We extend the notion of Evolutionarily Stable Strategies introduced by Maynard Smith and Price (Nature 246: 15-18, 1973) for models ruled by a single fitness matrix A, to the framework of stochastic games developed by Lloyd Shapley (Proc. Natl. Acad. Sci. USA 39: 1095-1100, 1953) where, at discrete stages in time, players play one of finitely many matrix games, while the transitions from one matrix game to the next follow a jointly controlled Markov chain. We show that this extension from a single-state model to a multistate model can be done on the assumption of having an irreducible transition law. In a similar way, we extend the notion of Replicator Dynamics introduced by Taylor and Jonker (Math. Biosci. 40: 145-156, 1978) to the multistate model. These extensions facilitate the analysis of evolutionary interactions that are richer than the ones that can be handled by the original, single-state, evolutionary game model. Several examples are provided.