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 $\Phi$-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.

翻译：众所周知，在大量简洁表示的博弈中，可以利用著名的“埃利普索德对抗希望”算法（Papadimitriou与Roughgarden，2008；Jiang与Leyton-Brown，2015）在多项式时间内计算相关均衡。然而，在序贯（扩展形式）博弈中可高效计算的均衡图景仍属未知。“埃利普索德对抗希望”算法无法直接适用于此类博弈，因其不具备所需的“多项式类型”性质。尽管存在这一障碍，Huang与von Stengel（2008）通过改进该算法成功计算出精确的扩展形式相关均衡。本文对“埃利普索德对抗希望”算法进行泛化，并发展出一个简单算法框架，用于高效计算双线性零和博弈中的鞍点，即使其中一维度规模呈指数级增长。该框架仅需“足够好响应”预言机——这是最优响应预言机的一种弱化概念。借助此工具，我们构建了一个通用算法框架，能在任意多面体博弈（在温和假设下）中计算精确线性Φ-均衡，涵盖正规形式博弈中的相关均衡以及扩展形式博弈中的扩展形式相关均衡。这使我们能够提出首个在扩展形式博弈中计算精确线性偏差相关均衡的多项式时间算法，从而解决了Farina与Pipis（2023）提出的开放性问题。此外，即使对于已知存在精确均衡多项式时间算法的情形，我们的框架也提供了概念上更简洁的解决方案。