Stochastic fictitious play with continuous action sets

Continuous action space games form a natural extension to normal form games with finite action sets. However, whilst learning dynamics in normal form games are now well studied, it is not until recently that their continuous action space counterparts have been examined. We extend stochastic fictitious play to the continuous action space framework. In normal form games the limiting behaviour of a discrete time learning process is often studied using its continuous time counterpart via stochastic approximation. In this paper we study stochastic fictitious play in games with continuous action spaces using the same method. This requires the asymptotic pseudo-trajectory approach to stochastic approximation to be extended to Banach spaces. In particular the limiting behaviour of stochastic fictitious play is studied using the associated smooth best response dynamics on the space of finite signed measures. Using this approach, stochastic fictitious play is shown to converge to an equilibrium point in single population negative definite games, two-player zero-sum games and N -player potential games, when they have Lipschitz continuous rewards.