Optimal Correlated Equilibria in General-Sum Extensive-Form Games: Fixed-Parameter Algorithms, Hardness, and Two-Sided Column-Generation

We study the problem of finding optimal correlated equilibria of various sorts in extensive-form games: normal-form coarse correlated equilibrium (NFCCE), extensive-form coarse correlated equilibrium (EFCCE), and extensive-form correlated equilibrium (EFCE). We make two primary contributions. First, we introduce a new algorithm for computing optimal equilibria in all three notions. Its runtime depends exponentially only on a parameter related to the information structure of the game. We also prove a fundamental complexity gap between NFCCE and the other two concepts. Second, we propose a two-sided column generation approach for use when the runtime or memory usage of the previous algorithm is prohibitive. Our algorithm improves upon an earlier one-sided approach by means of a new decomposition of correlated strategies which allows players to reoptimize their sequence-form strategies with respect to correlation plans which were previously added to the support. Experiments show that our techniques outperform the prior state of the art for computing optimal general-sum correlated equilibria. Funding: This work was supported by National Institutes of Health [Grant A240108S001]; Vannevar Bush Faculty Fellowship [Grant ONR N00014-23-1-2876]; National Science Foundation [Grants CCF-173355, IIS-190140, RI-1901403, RI-2312342]; and Army Research Office [Grants W911NF2010081, W911NF2210266].