Breadcrumb
Stochastic Fictitious Play with Continuous Action Sets
Fri 15 February 2013, 14:15
Steven Perkins
Bristol
Organisers: Nick Whiteley, Feng Yu
ABSTRACT
Stochastic approximation is a widely used tool which allows the limiting behaviour of a stochastic, discrete time, learning procedures on RK to be studied using an associated continuous time, deterministic, dynamical system. We extend the asymptotic pseudo-trajectory approach to stochastic approximation so that the processes can take place on any Banach space. This allows us to consider an iterative process of probability measures (or probability densities) on a compact subset of R as opposed to the regular stochastic approximation framework which is limited to probability mass functions on RK.
A common application of stochastic approximation in game theory is to study the limiting behaviour of a discrete time learning algorithm, such as stochastic fictitious play, in normal form games. 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. Our Banach space stochastic approximation framework shows that in a continuous action space game the limiting behaviour of stochastic fictitious play can be studied using the associated smooth best response dynamics on the space of finite signed measures. We show that stochastic fictitious play will 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 over a compact subset of R.