A Direct Reduction from k-Player to 2-Player Approximate Nash Equilibrium

Abstract

We present a direct reduction from k-player games to 2-player games that preserves approximate Nash equilibrium. Previously, the computational equivalence of computing approximate Nash equilibrium in k-player and 2-player games was established via an indirect reduction. This included a sequence of works defining the complexity class PPAD, identifying complete problems for this class, showing that computing approximate Nash equilibrium for k-player games is in PPAD, and reducing a PPAD-complete problem to computing approximate Nash equilibrium for 2-player games. Our direct reduction makes no use of the concept of PPAD, thus eliminating some of the difficulties involved in following the known indirect reduction.In the k-player case, the payoff of each player depends on the combined behavior of the other players. We can thus view each player's set of expected payoffs as a set of multiplicative functions in the other players' strategies. In a 2-player game, however, each player interacts with a single other player, and so the expected payoffs are linear [vS07]. The described gap calls for "linearization" of k-player games, and indeed the first step of our reduction replaces the multilateral interactions among the k players with bilateral interactions among pairs of players. In the next step, two representative "super-players" replace the multiple players, resulting in a 2-player game.In terms of techniques, the first step of the reduction uses and extends the machinery of gadget games developed by Goldberg and Papadimitriou [GP06]. We introduce a new gadget for performing approximate multiplication using linear operations, in order to bridge the gap between multiplicative and linear games. The second step of the reduction uses similar methods to [GP06] and [MT09] in order to replace multiple players by 2 players. The resulting 2-player game is a combination of a generalized Matching Pennies game [GP06] and an imitation game [MT09].

Smoothed Efficient Algorithms and Reductions for Network Coordination Games

“…Worst case complexity of NE computation in general non-potential games has been studied extensively. The computation is typically PPAD-complete, even for various special cases (e.g., Abbott et al [2005], Chen et al [2006b, Mehta [2014], Feige and Talgam-Cohen [2010]) and approximation (e.g., Chen et al [2006a], Rubinstein [2018]). On the other hand efficient algorithms have been designed for interesting sub-classes (e.g., Kannan and Theobald [2007], Tsaknakis and Spirakis [2007], Immorlica et al [2011], Adsul et al [2011], Cai and Daskalakis [2011], Cai et al [2015, Ahmadinejad et al [2016], Balcan and Braverman [2017], Barman [2018]), exploiting the structure of NE for the class to either enumerate, or through other methods such as parameterized LP and binary search.…”

A survey of PPAD-completeness for computing Nash equilibria

“…The reduction of [8] can in addition be used to show that 3-player games are as computationally hard to solve as k-players, but it does not highlight the expressive power of games as arithmetic circuits. Recently, Feige and Talgam-Cohen [32] give a reduction from (approximate) k-player Nash to 2-player Nash, that does not explicitly proceed via End of the line.…”

An Empirical Study on Computing Equilibria in Polymatrix Games

“…While there has been a large amount of theoretical work on polymatrix games [14,15,17,24,26,28], the practical aspects of computing equilibria in polymatrix games have yet to be studied. In this paper, we provide an empirical study of two prominent methods for computing equilibria in these games.…”