Poisson geometry in evolutionary game theory.

In this work we show that zero-sum replicator evolutionary games in presence of an interior fixpoint admit a Hamiltonian description with respect to a cubic Poisson structure on the simplex. In the first chapter Poisson manifolds are studied, with particular focus on the methods of Poisson reduction. Via a reduction procedure we derive in the second chapter a stratified Poisson structure for the simplex. In the third chapter we introduce normal games in a population dynamics setting and the notions of stable Nash strategy and evolutionarily stable strategy. This concept implies an underlying dynamics for the evolution of the average strategy of the population, modeled with the replicator equation. Simple examples are discussed, but the focus is mainly geometrical: the main result proves the Hamiltonian character of the replicator vector field with respect to the derived Poisson structure.