The Complexity of Correlated Equilibria in Generalized Games

2025-06-02

AI Summary

This paper addresses the long-standing open problem of computing correlated equilibria (and, more generally, Φ-equilibria) in generalized games—where players’ strategy sets are interdependent, invalidating standard equilibrium computation frameworks. The authors establish, for the first time, that computing *any* correlated equilibrium in a generalized game is PPAD-complete, thereby rigorously ruling out the existence of polynomial-time algorithms unless PPAD ⊆ P. This result is achieved via a carefully constructed PPAD-hardness reduction that intricately integrates the structural constraints of generalized games with foundational existence arguments for correlated equilibria. The work fills a critical theoretical gap at the intersection of game theory and computational economics, providing the first definitive characterization of the intrinsic computational hardness of correlated equilibria in generalized games. It establishes an unassailable complexity benchmark, guiding future research on approximation algorithms and tractability under restricted game structures.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Philosophy and Ethics of AI: Artificial General Intelligence

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• Study computational complexity of correlated equilibria in generalized games
• Address existence of efficient algorithms for finding equilibria
• Prove PPAD-completeness for equilibrium computation in generalized games

Innovation

• Studying correlated equilibria in generalized games
• Proving PPAD-completeness for equilibrium computation
• Addressing NP-hardness in social welfare optimization