Dynamics in near-potential games

Abstract

Except for special classes of games, there is no systematic framework for analyzing the dynamical properties of multi-agent strategic interactions. Potential games are one such special but restrictive class of games that allow for tractable dynamic analysis. Intuitively, games that are "close" to a potential game should share similar properties. In this paper, we formalize and develop this idea by quantifying to what extent the dynamic features of potential games extend to "near-potential" games.We study convergence of three commonly studied classes of adaptive dynamics: discretetime better/best response, logit response, and discrete-time fictitious play dynamics. For better/best response dynamics, we focus on the evolution of the sequence of pure strategy profiles and show that this sequence converges to a (pure) approximate equilibrium set, whose size is a function of the "distance" from a close potential game. We then study logit response dynamics parametrized by a smoothing parameter that determines the frequency with which the best response strategy is played. Our analysis uses a Markov chain representation for the evolution of pure strategy profiles. We provide a characterization of the stationary distribution of this Markov chain in terms of the distance of the game from a close potential game and the corresponding potential function. We further show that the stochastically stable strategy profiles (defined as those that have positive probability under the stationary distribution in the limit as the smoothing parameter goes to 0) are pure approximate equilibria. Finally, we turn attention to fictitious play, and establish that in near-potential games, the sequence of empirical frequencies of player actions converges to a neighborhood of (mixed) equilibria of the game, where the size of the neighborhood increases with distance of the game to a potential game. Thus, our results suggest that games that are close to a potential game inherit the dynamical properties of potential games. Since a close potential game to a given game can be found by solving a convex optimization problem, our approach also provides a systematic framework for studying convergence behavior of adaptive learning dynamics in arbitrary finite strategic form games.

A Comprehensive Survey of Potential Game Approaches to Wireless Networks

“…2, i.e., I = {1, 2, 3}, C i = {1, 2} for every i, and G 13 > G 12 , G 21 > G 23 , and G 32 > G 31 † † . The game does not have a Nash equilibrium, i.e., for every channel allocation, at least one pair has an incentive to change his/her channel.…”

“…† There are a variety of generalized concepts of potential games, e.g., generalized ordinal potential games [126], bestresponse potential games [177], pseudo-potential games [56], nearpotential games [30], [31], and state-based potential games [117]. Applications of these games are beyond the scope of this paper.…”

Robust Stochastic Stability

“…Marden and Shamma (2012) describe different ways to restore the selection of potential maximizers for more general revision processes, including updating probabilities that depend on the level of noise and restrictions on actions sets or information available to the players. Marden and Shamma (2012) and Candogan, Ozdaglar, and Parrilo (2013) also investigate games that are approximately potential games.…”

Ordinal Potentials in Smooth Games

“…Potentials of di¤erent types have been identi…ed for a large variety of games. Moreover, the underlying methods have been found useful for the analysis of oligopolistic markets (Slade, 1994), learning processes (Monderer and Shapley, 1996b;Fudenberg and Levine, 1998;Young, 2004), population dynamics (Sandholm, 2001(Sandholm, , 2009Cheung, 2014), the robustness of equilibria (Frankel et al, 2003;Morris and Ui, 2005;Okada and Tercieux, 2012), the decomposition of games (Candogan et al, 2011), imitation strategies (Duersch et al, 2012), dynamics in near-potential games (Candogan et al, 2013a(Candogan et al, , 2013b, the existence of Nash equilibrium (Voorneveld, 1997;Kukushkin, 1994Kukushkin, , 2011, solution concepts (Peleg et al, 1996;Tercieux and Voorneveld, 2010), games with monotone best-response selections (Huang, 2002;Dubey et al, 2006;Jensen, 2010), supermodular and zero-sum games (Brânzei et al, 2003), and even issues in mechanism design (Jehiel et al, 2008).…”