The Complexity of Correlated Equilibria in Generalized Games

TL;DR

This paper proves that computing any correlated equilibrium in generalized games, where strategy sets depend on other players' strategies, is PPAD-complete, highlighting the computational difficulty of these equilibria.

Contribution

It establishes the PPAD-completeness of finding correlated equilibria in generalized games, extending complexity results to more complex strategic settings.

Findings

Finding social welfare maximizing correlated equilibrium is NP-hard.

Existence of efficient algorithms for any correlated equilibrium in generalized games is unlikely.

Computational complexity in generalized games is as hard as in standard game settings.

Abstract

Correlated equilibria -- and their generalization $\Phi$-equilibria -- are a fundamental object of study in game theory, offering a more tractable alternative to Nash equilibria in multi-player settings. While computational aspects of equilibrium computation are well-understood in some settings, fundamental questions are still open in generalized games, that is, games in which the set of strategies allowed to each player depends on the other players' strategies. These classes of games model fundamental settings in economics and have been a cornerstone of economics research since the seminal paper of Arrow and Debreu [1954]. Recently, there has been growing interest, both in economics and in computer science, in studying correlated equilibria in generalized games. It is known that finding a social welfare maximizing correlated equilibrium in generalized games is NP-hard. However, the existence of efficient algorithms to find any equilibrium remains an important open question. In this paper, we answer this question negatively, showing that this problem is PPAD-complete.