Polynomial Expectation Property for Max-Polymatrix Games
Howard Dai
TL;DR
The paper tackles the open question of computing correlated equilibria in a Max-Polymatrix variant where a player's payoff is the maximum over edge payoffs. It proves that Max-Polymatrix has the polynomial expectation property by providing a polynomial-time algorithm to compute a player's expected utility under any product distribution, achieving a runtime of $O\left(n^2 m^2 \log m\right)$ per action and enabling a polynomial correlated equilibrium scheme via Papadimitriou and Roughgarden's framework. It further explores extensions to other payoff structures, showing NP-hardness for certain nonlinear convex functions and discussing potential generalizations like sorted-linear functions. The paper also investigates whether polynomial expectation is necessary, introducing an equilibria-checking oracle perspective that connects CE verification to constructive polynomial-time CE computation.
Abstract
We address an open problem on the computability of correlated equilibria in a variant of polymatrix where each player's utility is the maximum of their edge payoffs. We demonstrate that this max-variant game has the polynomial expectation property, and the results of Papadimitriou and Roughgarden can thus be applied. We propose ideas for extending these findings to other variants of polymatrix games, as well as briefly address the broader question of necessity for the polynomial expectation property when computing correlated equilibria.