Evolutionary Stochastic Games

We extend the notion of Evolutionarily Stable Strategies introduced by Maynard Smith & Price [6] in 1973 for models ruled by a single fitness matrix A, to the framework of stochastic games developed by Lloyd Shapley [13] in 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 multi-state 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 & Jonker [16] in 1978 to the multi-state 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.