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

Binghui Peng, Aviad Rubinstein

TL;DR

The paper analyzes iteration complexity of decentralized learning toward approximate CCE/CE in incomplete-information games, showing a near-optimal exponential lower bound for EF games under standard complexity assumptions and a polylogarithmic, type-independent algorithm for Bayesian games achieving ε-CE. It introduces a Kibitzer-based reduction to translate Bayesian-Nash-hardness into low-rank CCE hardness in EF games, establishing strong limits on poly-time uncoupled dynamics for EF settings. Conversely, it develops a per-type, multi-scale MWU approach that yields ε-CE for Bayesian games with iteration counts independent of the number of types, highlighting a clear separation between Bayesian and EF models. The results sharpen our understanding of when decentralized learning can efficiently converge to equilibrium in incomplete-information environments and provide practical, uncoupled dynamics for Bayesian settings. Overall, the work advances the theoretical landscape of equilibrium computation by tying together online learning, complexity theory, and game-theoretic design under incomplete information.

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 $\mathit{extensive}$-$\mathit{form}$ $\mathit{games}$, assuming $\mathsf{PPAD} \not\subset \mathsf{TIME}(n^{\mathsf{polylog}(n)})$, any polynomial-time learning algorithms must take at least $2^{\log_2^{1-o(1)}(|\mathcal{I}|)}$ iterations to converge to the set of $ε$-approximate correlated equilibrium, where $|\mathcal{I}|$ is the number of nodes in the game and $ε> 0$ is an absolute constant. This nearly matches, up to the $o(1)$ term, the algorithms of [PR'24, DDFG'24] for learning $ε$-approximate correlated equilibrium, and resolves an open question of Anagnostides, Kalavasis, Sandholm, and Zampetakis [AKSZ'24]. Our lower bound holds even for the easier solution concept of $ε$-approximate $\mathit{coarse}$ correlated equilibrium On the positive side, we give uncoupled dynamics that reach $ε$-approximate correlated equilibria of a $\mathit{Bayesian}$ $\mathit{game}$ in polylogarithmic iterations, without any dependence of the number of types. This demonstrates a separation between Bayesian games and extensive-form games.

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

TL;DR

The paper analyzes iteration complexity of decentralized learning toward approximate CCE/CE in incomplete-information games, showing a near-optimal exponential lower bound for EF games under standard complexity assumptions and a polylogarithmic, type-independent algorithm for Bayesian games achieving ε-CE. It introduces a Kibitzer-based reduction to translate Bayesian-Nash-hardness into low-rank CCE hardness in EF games, establishing strong limits on poly-time uncoupled dynamics for EF settings. Conversely, it develops a per-type, multi-scale MWU approach that yields ε-CE for Bayesian games with iteration counts independent of the number of types, highlighting a clear separation between Bayesian and EF models. The results sharpen our understanding of when decentralized learning can efficiently converge to equilibrium in incomplete-information environments and provide practical, uncoupled dynamics for Bayesian settings. Overall, the work advances the theoretical landscape of equilibrium computation by tying together online learning, complexity theory, and game-theoretic design under incomplete information.

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 - , assuming , any polynomial-time learning algorithms must take at least iterations to converge to the set of -approximate correlated equilibrium, where is the number of nodes in the game and is an absolute constant. This nearly matches, up to the term, the algorithms of [PR'24, DDFG'24] for learning -approximate correlated equilibrium, and resolves an open question of Anagnostides, Kalavasis, Sandholm, and Zampetakis [AKSZ'24]. Our lower bound holds even for the easier solution concept of -approximate correlated equilibrium On the positive side, we give uncoupled dynamics that reach -approximate correlated equilibria of a in polylogarithmic iterations, without any dependence of the number of types. This demonstrates a separation between Bayesian games and extensive-form games.
Paper Structure (29 sections, 14 theorems, 40 equations, 5 algorithms)

This paper contains 29 sections, 14 theorems, 40 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. A lower bound on (poly-time) external regret in extensive-form games
  3. An algorithm for swap regret minimization in Bayesian games
  4. A brief overview of our approach
  5. Lower bound for extensive-form game
  6. Previous approach
  7. Our approach
  8. Algorithm for Bayesian games
  9. Related work
  10. Decentralized learning in games
  11. Lower bounds for games
  12. Preliminary
  13. Notation
  14. Bayesian game
  15. Extensive-form game
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $\epsilon > 0$ be an absolute constant. Assuming $\PPAD \not\subset \TIME(n^{\polylog(n)})$, there is no polynomial time learning dynamics that guarantee to converge to the set of $\epsilon$-CCE of a three player extensive-form game in $2^{\log_2^{1-o(1)}(|\mathcal{I}|)}$ iterations.

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Normal-form correlated equilibrium
  • Definition 2.2: Notions of approximate equilibria
  • Definition 2.3: Coarse correlated equilibrium
  • Remark 2.4: Mixed strategy, behaviour strategy and rank
  • Definition 2.5: Rank
  • Lemma 2.6: arora2012multiplicative
  • Lemma 2.7: vovk1990aggregating
  • Theorem 2.8: rubinstein2018inapproximability
  • ...and 20 more