Evolutionary game theory in mixed strategies: from microscopic interactions to kinetic equations

TL;DR

This paper develops a kinetic framework for evolutionary game theory with mixed strategies, deriving a PDE from microscopic rules and linking it to classical replicator dynamics and Nash equilibria.

Contribution

It introduces a novel kinetic formulation for mixed strategy evolutionary games, connecting microscopic adaptive rules to macroscopic PDEs and classical game theory concepts.

Findings

Derived a Boltzmann equation for mixed strategies

Established the PDE as a limit of microscopic rules

Linked stationary solutions to Nash equilibria

Abstract

In this work we propose a kinetic formulation for evolutionary game theory for zero sum games when the agents use mixed strategies. We start with a simple adaptive rule, where after an encounter each agent increases the probability of play the successful pure strategy used in the match. We derive the Boltzmann equation which describes the macroscopic effects of this microscopical rule, and we obtain a first order, nonlocal, partial differential equation as the limit when the probability change goes to zero. We study the relationship between this equation and the well known replicator equations, showing the equivalence between the concepts of Nash equilibria, stationary solutions of the partial differential equation, and the equilibria of the replicator equations. Finally, we relate the long time behavior of solutions to the partial differential equation and the stability of the replicator equations.