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# The complexity of approximate (coarse) correlated equilibrium for incomplete information games

*Proceedings of Thirty Seventh Conference on Learning Theory*, PMLR 247:4158-4184, 2024.

#### Abstract

We study the iteration complexity of decentralized learning of approximate correlated equilibria in incomplete information games. On the negative side, we prove that in extensive-form games, assuming $\mathsf{PPAD} \not\subset \mathsf{TIME}(n^{\polylog(n)})$, any polynomial-time learning algorithms must take at least $2^{\log_2^{1-o(1)}(|\mathcal{I}|)}$ iterations to converge to the set of $\epsilon$-approximate correlated equilibrium, where $|\mathcal{I}|$ is the number of nodes in the game and $\epsilon > 0$ is an absolute constant. This nearly matches, up to the $o(1)$ term, the algorithms of (Peng and Rubinstein STOC’2024, Dagan et al. STOC’2024) for learning $\epsilon$-approximate correlated equilibrium, and resolves an open question of Anagnostides, Kalavasis, Sandholm, and Zampetakis (Anagnostides et al. ITCS 2024). Our lower bound holds even for the easier solution concept of $\epsilon$-approximate coarse correlated equilibrium. On the positive side, we give uncoupled dynamics that reach $\epsilon$-approximate correlated equilibria of a Bayesian game in polylogarithmic iterations, without any dependence of the number of types. This demonstrates a separation between Bayesian games and extensive-form games.