  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.
