# Computing Optimal Coarse Correlated Equilibria in Sequential Games

## TL;DR

This paper introduces a new class of equilibria for extensive-form games, showing that optimal solutions can be computed efficiently in two-player cases but are NP-hard in more complex scenarios.

## Contribution

It extends the concept of coarse correlated equilibrium to sequential games and provides polynomial-time algorithms for two-player cases, addressing computational complexity.

## Key findings

- Optimal equilibria computed in polynomial time for two-player games without chance.
- NP-hardness results for multi-player and games with chance.
- Practical algorithms proposed for large two-player games.

## Abstract

We investigate the computation of equilibria in extensive-form games where ex ante correlation is possible, focusing on correlated equilibria requiring the least amount of communication between the players and the mediator. Motivated by the hardness results on the computation of normal-form correlated equilibria, we introduce the notion of normal-form coarse correlated equilibrium, extending the definition of coarse correlated equilibrium to sequential games. We show that, in two-player games without chance moves, an optimal (e.g., social welfare maximizing) normal-form coarse correlated equilibrium can be computed in polynomial time, and that in general multi-player games (including two-player games with Chance), the problem is NP-hard. For the former case, we provide a polynomial-time algorithm based on the ellipsoid method and also propose a more practical one, which can be efficiently applied to problems of considerable size. Then, we discuss how our algorithm can be extended to games with Chance and games with more than two players.