On the Global Convergence of Stochastic Fictitious Play

Econometrica · 2002 · Volume 70 · Issue 6

DOI: 10.1111/j.1468-0262.2002.00440.x

Abstract

We establish global convergence results for stochastic fictitious play for four classes of games: games with an interior ESS, zero sum games, potential games, and supermodular games. We do so by appealing to techniques from stochastic approximation theory, which relate the limit behavior of a stochastic process to the limit behavior of a differential equation defined by the expected motion of the process. The key result in our analysis of supermodular games is that the relevant differential equation defines a strongly monotone dynamical system. Our analyses of the other cases combine Lyapunov function arguments with a discrete choice theory result: that the choice probabilities generated by any additive random utility model can be derived from a deterministic model based on payoff perturbations that depend nonlinearly on the vector of choice probabilities.

Keywords

learning in games, stochastic fictitious play, supermodular games, discrete choice theory, chain recurrence, stochastic approximation theory

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