On Tightness of Tsaknakis-Spirakis Descent Methods for Approximate Nash Equilibria

Finding the minimum approximation ratio for Nash equilibrium of bi-matrix games has derived a series of studies, starting with 3/4, followed by 1/2, 0.38, 0.36, and the previously best-known approximation ratio of 0.3393 by Tsaknakis and Spirakis (TS algorithm for short). The TS algorithm applies a descent method to locally minimize a loss function and then makes a further adjustment. Efforts to improve the analysis of the TS algorithm remain unsuccessful in the past 15 years. This work makes the first progress in showing that the bound of 0.3393 is indeed tight for the TS algorithm. We also present a thorough theoretical worst-case analysis and give a computable equivalent condition of tight instances. With this condition, we provide a tight instance generator for the TS algorithm. Empirically, most generated instances are unstable, that is, a small perturbation near a 0.3393 solution may help the TS algorithm find another faraway solution with a much better approximation ratio. Meanwhile, the existence of stable tight instances indicates the perturbation cannot improve 0.3393 bound in worst cases for the TS algorithm. Furthermore, we test approximate algorithms other than the TS algorithm on these generated instances. Two approximate algorithms, the regret-matching algorithm and the fictitious play algorithm, can find solutions with approximation ratios far better than 0.3393. Interestingly, the distributed approximate algorithm by Czumaj et al. finds solutions with the same approximation ratio of 0.3393 on these generated instances. Such results demonstrate that our generated instances against the TS algorithm serve as a necessary benchmark in design and analysis of approximate Nash equilibrium algorithms. Finally, we show that our techniques can be further generalized to prove the tightness of recently developed 1/3-approximation DFM algorithm.