Decentralized Fictitious Play Converges Near a Nash Equilibrium in Near-Potential Games

Abstract

We investigate convergence of decentralized fictitious play (DFP) in near-potential games, wherein agents preferences can almost be captured by a potential function. In DFP agents keep local estimates of other agents' empirical frequencies, best-respond against these estimates, and receive information over a time-varying communication network. We prove that empirical frequencies of actions generated by DFP converge around a single Nash Equilibrium (NE) assuming that there are only finitely many Nash equilibria, and the difference in utility functions resulting from unilateral deviations is close enough to the difference in the potential function values. This result assures that DFP has the same convergence properties of standard Fictitious play (FP) in near-potential games. I. INTRODUCTIONGame theory deals with systems having multiple decisionmakers. In non-cooperative games, agents take actions to maximize their individual utility functions that depend on the actions of other agents. Potential games is a special class of games that capture scenarios where there exists a common function modeling the change in individual utilities, named as potential function. Applications of potential games appear in various large-scale networked systems including transportation systems [1], mobile robotic systems [2], and communication networks [3]. Decentralized decision-making protocols, e.g., best-response [4], [5], fictitious play (FP) [6], [7], are used to understanding emerging behavior or to design individual actions in such large-scale systems. A common assumption in the convergence of these protocols is that agents have full or common information about their utility functions or the potential function. Here, we lift this assumption by allowing the game agents are playing to deviate from an exact potential game.Near-potential games [8] extend potential games, by defining games as a deviation from a potential game. This deviation may stem from incomplete information about payoffrelevant environment parameters. Specifically, the deviation between two games is defined in terms of unilateral change of actions, where only one agent changes its and others stay in the same profile. If this deviation is bounded, traditional decision-making protocols, e.g., best-response, or FP, converge to a region around NE, i.e., an approximate-NE [8]. In this paper, we analyze convergence properties of a decentralized version of FP (DFP) where agents can only exchange information with a subset of their neighbors after S. Aydin and C. Eksin are with the