Fictitious play in networks

Abstract

This paper studies fictitious play in networks of noncooperative two-person games. We show that continuous-time fictitious play converges to the set of Nash equilibria if the overall n-person game is zero-sum. Moreover, the rate of convergence is 1/τ, regardless of the size of the network. In contrast, arbitrary n-person zero-sum games with bilinear payoff functions do not possess the continuous-time fictitious-play property. As extensions, we consider networks in which each bilateral game is either strategically zero-sum, a weighted potential game, or a two-by-two game. In those cases, convergence requires a condition on bilateral payoffs or, alternatively, that the network is acyclic. Our results hold also for the discrete-time variant of fictitious play, which implies, in particular, a generalization of Robinson's theorem to arbitrary zero-sum networks. Applications include security games, conflict networks, and decentralized wireless channel selection.