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The Complexity of Correlated Equilibria in Generalized Games

Martino Bernasconi, Matteo Castiglioni, Andrea Celli, Gabriele Farina

TL;DR

The paper proves that computing an approximate Constrained-$\Phi$-Equilibrium in generalized games is $PPAD$-complete, even with limited actions and a fixed number of players in some settings. It achieves this via a detailed reduction from PolyMatrix that arranges two opposing teams and couples their marginals through costs, demonstrating that correlation cannot overcome the complexity introduced by joint constraints. Additionally, it establishes $PPAD$-membership by a reduction to a QuasiVI problem and derives quasi-polynomial lower bounds under the ETH for $PPAD$, clarifying limits on convergence guarantees for no-regret learners in constrained environments. The results have implications for economic models and automated bidding platforms, where constrained correlated behavior must be computed or approximated efficiently.

Abstract

Correlated equilibria -- and their generalization $Φ$-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.

The Complexity of Correlated Equilibria in Generalized Games

TL;DR

The paper proves that computing an approximate Constrained--Equilibrium in generalized games is -complete, even with limited actions and a fixed number of players in some settings. It achieves this via a detailed reduction from PolyMatrix that arranges two opposing teams and couples their marginals through costs, demonstrating that correlation cannot overcome the complexity introduced by joint constraints. Additionally, it establishes -membership by a reduction to a QuasiVI problem and derives quasi-polynomial lower bounds under the ETH for , clarifying limits on convergence guarantees for no-regret learners in constrained environments. The results have implications for economic models and automated bidding platforms, where constrained correlated behavior must be computed or approximated efficiently.

Abstract

Correlated equilibria -- and their generalization -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.

Paper Structure

This paper contains 17 sections, 10 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions and Overview of Techniques
  3. Preliminaries
  4. Multi-player Games and Phi-equilibrium
  5. Constrained $\Phi$-equilibria
  6. Polymatrix and team games
  7. PPAD-hardness and Total Problems
  8. Reduction from PolyMatrix to Constrained-$\Phi$-Equilibrium
  9. Proof of \ref{['th:hardness']}
  10. Implications of the PPAD-hardness
  11. PPAD-membership
  12. Examples of $\Phi$-equilibria
  13. Missing proof from \ref{['sec:hardness']} (hardness)
  14. Missing proofs and additional details from \ref{['sec:membership']} (membership)
  15. Explicit definition of $A(\tilde{z})$
  16. ...and 2 more sections

Key Result

Theorem 2.1

There exists a constant $\epsilon^*$ such that PolyMatrix is PPAD-complete, even when the graph has degree $3$ and each player has $2$ actions.

Figures (1)

  • Figure 1: On the edges, we reported the matrix corresponding to the utilities of the players. Colors indicate the row player of the bi-matrix game and the player associated with the utility that is associated with the matrix.

Theorems & Definitions (23)

  • Theorem 2.1: rubinstein2015inapproximability
  • Theorem 3.1
  • Lemma 3.1
  • Lemma 3.1
  • Theorem 4.1: bernasconi2024role
  • Proposition 4.1
  • proof
  • Claim 4.1
  • Claim 4.1
  • Claim 4.1
  • ...and 13 more