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/poly(d) (average) swap regret. To circumvent these hardness results, we examine the complexity of Φ-equilibria – tractable relaxations of correlated equilibria. We obtain a fully polynomial-time approximation scheme (FPTAS) for computing poly-dimensional Φ-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 ε is exponentially small. Finally, we show that Contr-hardness can be bypassed in the canonical setting of concave quadratic games, for which we provide a poly(d, log(1/ε))-time algorithm for computing poly-dimensional Φ-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.
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