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 $Φ$-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 \emph{quadratic games}, for which we provide a $\text{poly}(d, \log(1/ε))$-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.

翻译：相关均衡是博弈论中的基本解概念。然而，尽管经过数十年研究，多项式类型博弈（如扩展式博弈、拥塞或路由博弈，以及更广泛的凹博弈）之外的复杂性仍是一个重大开放问题，这一问题最初由Papadimitriou和Roughgarden（JACM '08）提出。本文解决了关于相关均衡和交换遗憾最小化复杂性的若干长期未决问题。首先，我们证明计算凹二次博弈中的相关均衡与计算压缩映射的不动点（Contr）同等困难，这首次有力表明其难解性。此外，我们还建立了无条件的、信息论下界，排除了存在强次线性交换遗憾最小化器的可能性：任何在线学习算法都需要维度$d$的指数级迭代次数，才能保证（平均）交换遗憾至多为$1/\text{poly}(d)$。为规避这些困难结果，我们研究了$\Phi$-均衡（相关均衡的易处理松弛）的复杂性。我们为一般凹博弈中计算多维$\Phi$-均衡设计了一个完全多项式时间近似方案（FPTAS）。我们进一步证明，即使将精度$ε$设为指数级小，在多维交换偏差下Contr困难性依然存在。最后，我们证明在凹\emph{二次博弈}典型设置中可规避Contr困难性，为此提供了一种时间复杂度为$\text{poly}(d, \log(1/ε))$的算法，用于计算多维$\Phi$-均衡。作为副产品，我们获得了一种计算关于未知马氏范数压缩映射不动点的算法，这可能具有独立的研究价值。