A General Framework for Computing Optimal Correlated Equilibria in Compact Games

TL;DR

This paper introduces a unified framework for computing optimal correlated equilibria in compact games, extending tractability results to broader classes including graphical polymatrix and singleton congestion games.

Contribution

It proposes a general algorithmic approach applicable to all compact game representations and generalizes previous tractability conditions for optimal CE computation.

Findings

Optimal CE computation reduces to deviation-adjusted social welfare optimization.

Graphical polymatrix games on trees have tractable optimal CE computation.

Optimal coarse correlated equilibrium is tractable in singleton congestion games.

Abstract

We analyze the problem of computing a correlated equilibrium that optimizes some objective (e.g., social welfare). Papadimitriou and Roughgarden [2008] gave a sufficient condition for the tractability of this problem; however, this condition only applies to a subset of existing representations. We propose a different algorithmic approach for the optimal CE problem that applies to all compact representations, and give a sufficient condition that generalizes that of Papadimitriou and Roughgarden. In particular, we reduce the optimal CE problem to the deviation-adjusted social welfare problem, a combinatorial optimization problem closely related to the optimal social welfare problem. This framework allows us to identify new classes of games for which the optimal CE problem is tractable; we show that graphical polymatrix games on tree graphs are one example. We also study the problem of computing the optimal coarse correlated equilibrium, a solution concept closely related to CE. Using a similar approach we derive a sufficient condition for this problem, and use it to prove that the problem is tractable for singleton congestion games.