This article explores the minimum approximation ratio for Nash equilibrium in bi-matrix games, focusing on the Tsaknakis and Spirakis (TS) methods. The previous SOTA, TS algorithm, achieved an approximation ratio of 0.3393, but efforts to improve the analysis of the TS algorithm have been unsuccessful. This work demonstrates that the bound of 0.3393 is tight for the TS algorithm and presents a theoretical worst-case analysis. A condition for identifying tight instances is provided, along with a generator. While most generated instances are unstable, indicating potential improvements, stable instances exist where perturbations cannot enhance the 0.3393 bound. Other approximate algorithms, such as regret-matching and fictitious play, achieve better ratios on these instances. The generated instances can serve as benchmarks for approximate Nash equilibrium algorithms. The article also mentions progress in the TS algorithm, achieving an approximation ratio of 1/3, which can be further studied using the presented techniques.

翻译：本文探讨了双矩阵博弈中纳什均衡的最小近似比，重点研究Tsaknakis与Spirakis（TS）方法。此前的最优算法TS算法已实现0.3393的近似比，但改进其分析的努力均未成功。本文证明0.3393这一界对TS算法是紧致的，并给出理论最坏情况分析。我们提供了识别紧致实例的条件及生成器。虽然生成的大多数实例不稳定（表明存在改进空间），但存在扰动无法突破0.3393界的稳定实例。其他近似算法（如遗憾匹配和虚构博弈）在这些实例上能取得更优比率。生成的实例可作为近似纳什均衡算法的基准测试集。此外，本文提及TS算法的进展——已实现1/3的近似比，可借助本文技术进一步研究。