On the Complexity of Correlated Equilibria Beyond Normal-Form Games

TL;DR

This paper investigates the computational complexity of correlated equilibria in complex game settings, establishing hardness results and proposing efficient algorithms for certain classes of games.

Contribution

It provides the first strong evidence of intractability for correlated equilibria in concave quadratic games and introduces a polynomial-time approximation scheme for $\

Findings

Computing correlated equilibria in concave quadratic games is as hard as finding fixed points of contraction mappings.

Any online learning algorithm requires exponentially many iterations to achieve low swap regret in high dimensions.

A fully polynomial-time approximation scheme (FPTAS) is developed for $\

Abstract

Correlated equilibria are a fundamental solution concept in game theory. However, despite decades of research, the complexity beyond games of polynomial type -- such as extensive-form games, congestion or routing games, and more broadly concave games -- has remained a major open problem, first highlighted by Papadimitriou and Roughgarden (JACM '08). In this paper, we resolve several long-standing questions concerning the complexity of correlated equilibria and swap regret minimization. First, we show that computing a correlated equilibrium in concave quadratic games is as hard as computing the fixed point of a contraction mapping (Contr), providing the first strong evidence of intractability. Moreover, we establish an unconditional, information-theoretic lower bound ruling out the existence of a strongly sublinear swap regret minimizer: any online learning algorithm requires exponentially many iterations in the dimension $d$ to guarantee at most $1/\text{poly}(d)$ (average) swap regret. To circumvent these hardness results, we examine the complexity of $\Phi$-equilibria -- tractable relaxations of correlated equilibria. We obtain a fully polynomial-time approximation scheme (FPTAS) for computing poly-dimensional $\Phi$-equilibria in general concave games. We complement this by showing that Contr-hardness persists even under poly-dimensional swap deviations in the regime where the precision $\epsilon$ is exponentially small. Finally, we show that Contr-hardness can be bypassed in the canonical setting of concave \emph{quadratic games}, for which we provide a $\text{poly}(d, \log(1/\epsilon))$-time algorithm for computing poly-dimensional $\Phi$-equilibria. As a byproduct, we obtain an algorithm for computing fixed points of a mapping that is contracting with respect to an unknown Mahalanobis norm, which could be of independent interest.