Distributed Methods for Computing Approximate Equilibria

Abstract

We present a new, distributed method to compute approximate Nash equilibria in bimatrix games. In contrast to previous approaches that analyze the two payoff matrices at the same time (for example, by solving a single LP that combines the two players payoffs), our algorithm first solves two independent LPs, each of which is derived from one of the two payoff matrices, and then compute approximate Nash equilibria using only limited communication between the players. Our method has several applications for improved bounds for efficient computations of approximate Nash equilibria in bimatrix games. First, it yields a best polynomial-time algorithm for computing approximate wellsupported Nash equilibria (WSNE), which guarantees to find a 0.6528-WSNE in polynomial time. Furthermore, since our algorithm solves the two LPs separately, it can be used to improve upon the best known algorithms in the limited communication setting: the algorithm can be implemented to obtain a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication and finds a 0.6528-WSNE. The algorithm can also be carried out to beat the best known bound in the query complexity setting, requiring O(n log n) payoff queries to compute a 0.6528-WSNE. Finally, our approach can also be adapted to provide the best known communication efficient algorithm for computing approximate Nash equilibria: it uses poly-logarithmic communication to find a 0.382-approximate Nash equilibrium.Communication complexity of equilibria in games has been studied in previous works [3,13]. The recent paper of Goldberg and Pastink [11] initiated the study of communication complexity in the bimatrix game setting. There they showed Θ(n 2 ) communication is required to find an exact Nash equilibrium of an n × n bimatrix game. Since these games have Θ(n 2 ) payoffs in total, this implies that there is no communication efficient protocol for finding exact Nash equilibria in bimatrix games. For approximate equilibria, they showed that one can find a 3 4 -Nash equilibrium without any communication, and that in the nocommunication setting, finding an 1 2 -Nash equilibrium is impossible. Motivated by these positive and negative results, they focused on the most interesting setting, which allows only a polylogarithmic (in n) amount of communication (number of bits) between the players. They demonstrated that one can compute 0.438-Nash equilibria and 0.732-well-supported Nash equilibria in this setting.

Lipschitz Continuity and Approximate Equilibria

“…A similar analysis might be applied to games with L 1 penalties, which would lead to a constant approximation guarantee similar to the bound of 0.5 that was established in that paper. The other known techniques that compute approximate Nash equilibria [5] and approximate well supported Nash equilibria [9,19,25] solve a zero sum bimatrix game in order to derive the approximate equilibrium, and there is no obvious way to generalise this approach in penalty games.…”

Lipschitz Continuity and Approximate Equilibria

“…Under the L 1 penalties the analysis of the steepest descent algorithm may be similar to and therefore we may be able to obtain a constant approximation guarantee similar to the bound of 0.5 that was established in that paper. The other known techniques that compute approximate Nash equilibria [Bosse et al 2010] and approximate well supported Nash equilibria [Czumaj et al 2015;Fearnley et al 2012; Kontogiannis and Spirakis 2010] solve a zero sum bimatrix game in order to derive the approximate equilibrium, and there is no obvious way to generalise this approach in penalty games.…”

Combining local and global evolutionary trajectories of brain–behaviour relationships through game theory

“…In the classical example, called the prisoner's dilemma, the years in jail (payoff) of two prisoners (players) depend on their choice (action) to confess their crime or not (Figure 3a). Among the many variants of game theory, games can involve more than two players (n‐players game; Nash, 1950; Czumaj et al, 2018), can be sequential or simultaneous depending on the temporal delay between choices (Fudenberg, Yuhta, & Kominers, 2014), can entail stochastic elements if the strategies or the payoff follow probabilistic laws (Shapley, 1953; Solan & Vieille, 2015), and can be cooperative or non‐cooperative depending on whether or not players are allowed to form temporarily stable coalitions (Nash, 1951).…”