The Evolution of Mixed Strategies in Population Games

We study the evolution of mixed strategies in population games. At any time, the distribution of mixed strategies over agents in a population is described by a density function. A pair of players is chosen randomly in each round of the game. After each round, players update their mixed strategies using certain reinforcement driven rules. The distribution over mixed strategies thus changes. In a continuous-time limit, this change is described by non-linear continuity equations. The updating rules we use generate the replicator continuity equations, and we provide the asymptotic solution for these equations for general 2 player asymmetric and symmetric normal form games. We use these results to study in greater detail mixed strategy evolution in 2× 2 symmetric and asymmetric games. A key finding is that, when agents carry mixed strategies, distributional considerations in general cannot be subsumed under a classical approach represented by the deterministic replicator dynamics.