Multiplicative weights, equalizers, and P=PPAD

TL;DR

This paper demonstrates that symmetric bimatrix games either have an interior symmetric equilibrium or a weakly dominated pure strategy, and shows that such equilibria and strategies can be computed efficiently, leading to the conclusion that P equals PPAD.

Contribution

It introduces a polynomial-time method to find symmetric equilibria and weakly dominated strategies in symmetric bimatrix games, connecting game theory with computational complexity.

Findings

Symmetric bimatrix games have either an interior symmetric equilibrium or a weakly dominated pure strategy.

Weakly dominated strategies can be detected and eliminated in polynomial time via linear programming.

Interior symmetric equilibria are a special case of equalizers, also computable efficiently.

Abstract

We show that, by using multiplicative weights in a game-theoretic thought experiment (and an important convexity result on the composition of multiplicative weights with the relative entropy function), a symmetric bimatrix game (that is, a bimatrix matrix wherein the payoff matrix of each player is the transpose of the payoff matrix of the other) either has an interior symmetric equilibrium or there is a pure strategy that is weakly dominated by some mixed strategy. Weakly dominated pure strategies can be detected and eliminated in polynomial time by solving a linear program. Furthermore, interior symmetric equilibria are a special case of a more general notion, namely, that of an "equalizer," which can also be computed efficiently in polynomial time by solving a linear program. An elegant "symmetrization method" of bimatrix games [Jurg et al., 1992] and the well-known PPAD-completeness results on equilibrium computation in bimatrix games [Daskalakis et al., 2009, Chen et al., 2009] imply then the compelling P = PPAD.