Polynomial-Time Computation of Exact $Φ$-Equilibria in Polyhedral Games
Gabriele Farina, Charilaos Pipis
TL;DR
This work addresses the problem of exactly computing $\Phi$-equilibria in polyhedral (including extensive-form) games by generalizing the Ellipsoid Against Hope paradigm. It introduces a Good-Enough-Response (GER) oracle and a three-part sufficient condition (polynomial utility gradient, linear-transformations forming a polytope $\Phi$ with a polynomial-time separation oracle) that together enable polynomial-time computation of exact $\{\Phi_p\}$-equilibria, for a broad class of $\Phi$ including EFCE, EFCCE, NFCCE, and LCE. The authors present a general algorithmic framework that recovers all known polynomial-time results for CEs and, crucially, provides the first polynomial-time algorithm for exact linear-deviation CEs (LCE) in extensive-form games, offering a simpler and more unified solution. This advances the practical and theoretical tractability of strong equilibrium notions in games with large or exponential action spaces, with potential impact on learning dynamics and strategic decision-making in sequential settings.
Abstract
It is a well-known fact that correlated equilibria can be computed in polynomial time in a large class of concisely represented games using the celebrated Ellipsoid Against Hope algorithm (Papadimitriou and Roughgarden, 2008; Jiang and Leyton-Brown, 2015). However, the landscape of efficiently computable equilibria in sequential (extensive-form) games remains unknown. The Ellipsoid Against Hope does not apply directly to these games, because they do not have the required "polynomial type" property. Despite this barrier, Huang and von Stengel (2008) altered the algorithm to compute exact extensive-form correlated equilibria. In this paper, we generalize the Ellipsoid Against Hope and develop a simple algorithmic framework for efficiently computing saddle-points in bilinear zero-sum games, even when one of the dimensions is exponentially large. Moreover, the framework only requires a "good-enough-response" oracle, which is a weakened notion of a best-response oracle. Using this machinery, we develop a general algorithmic framework for computing exact linear $Φ$-equilibria in any polyhedral game (under mild assumptions), including correlated equilibria in normal-form games, and extensive-form correlated equilibria in extensive-form games. This enables us to give the first polynomial-time algorithm for computing exact linear-deviation correlated equilibria in extensive-form games, thus resolving an open question by Farina and Pipis (2023). Furthermore, even for the cases for which a polynomial time algorithm for exact equilibria was already known, our framework provides a conceptually simpler solution.