{"id":1173696,"date":"2026-05-27T13:53:00","date_gmt":"2026-05-27T20:53:00","guid":{"rendered":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/publication\/on-the-complexity-of-correlated-equilibria-beyond-normal-form-games\/"},"modified":"2026-05-29T11:44:10","modified_gmt":"2026-05-29T18:44:10","slug":"on-the-complexity-of-correlated-equilibria-beyond-normal-form-games","status":"publish","type":"msr-research-item","link":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/publication\/on-the-complexity-of-correlated-equilibria-beyond-normal-form-games\/","title":{"rendered":"On the Complexity of Correlated Equilibria Beyond Normal-Form Games"},"content":{"rendered":"<p>Correlated equilibria are a fundamental solution concept in game theory, introduced in seminal work by Robert Aumann in the 1970s. A textbook fact is that correlated equilibria enjoy more desirable computational properties <em>vis-`a-vis<\/em> Nash equilibria: they can be computed in polynomial time in normal form games via linear programming, and are also amenable to learning via decentralized <em>no-swap-regret<\/em> dynamics. However, despite decades of research, the complexity beyond games of polynomial type\u2014such as extensive-form games, congestion or routing games, and more broadly concave games\u2014has remained a major open problem, first highlighted by Papadimitriou and Roughgarden (JACM \u201908).<\/p>\n<p>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\/poly(d)\\) (average) swap regret.<\/p>\n<p>To circumvent these hardness results, we examine the complexity of \u03a6-equilibria\u2014tractable relaxations of correlated equilibria. We obtain a fully polynomial-time approximation scheme (FPTAS) for computing poly-dimensional \u03a6-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 \u03f5 is exponentially small. Finally, we show that Contr-hardness can be bypassed in the canonical setting of concave &lt;em>quadratic games&lt;\/em>, for which we provide a \\(poly(d,log(1\/\u03f5))\\)-time algorithm for computing poly-dimensional \u03a6-equilibria. As a byproduct, we obtain an algorithm for computing fixed points of a mapping that is contracting with respect to an <em>unknown Mahalanobis norm<\/em>, which could be of independent interest<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Correlated equilibria are a fundamental solution concept in game theory, introduced in seminal work by Robert Aumann in the 1970s. A textbook fact is that correlated equilibria enjoy more desirable computational properties vis-`a-vis Nash equilibria: they can be computed in polynomial time in normal form games via linear programming, and are also amenable to learning 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