Abstract
Finding the minimum approximate ratio for Nash equilibrium of bi-matrix games has derived a series of studies, started with 3/4, followed by 1/2, 0.38 and 0.36, finally the best approximate ratio of 0.3393 by Tsaknakis and Spirakis (TS algorithm for short). Efforts to improve the results remain not successful in the past 14 years.
This work makes the first progress to show that the bound of 0.3393 is indeed tight for the TS algorithm. Next, we characterize all possible tight game instances for the TS algorithm. It allows us to conduct extensive experiments to study the nature of the TS algorithm and to compare it with other algorithms. We find that this lower bound is not smoothed for the TS algorithm in that any perturbation on the initial point may deviate away from this tight bound approximate solution. Other approximate algorithms such as Fictitious Play and Regret Matching also find better approximate solutions. However, the new distributed algorithm for approximate Nash equilibrium by Czumaj et al. performs consistently at the same bound of 0.3393. This proves our lower bound instances generated against the TS algorithm can serve as a benchmark in design and analysis of approximate Nash equilibrium algorithms.
H. Li—Main technical contributor.
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Notes
- 1.
- 2.
We will see in Remark 1 that finding a stationary point is not enough to reach a good approximation ratio; therefore the adjustment step is necessary.
- 3.
Throughout the paper, we suppose that \((x, y), (x',y')\in \varDelta _m\times \varDelta _n\), and \(\rho \in [0, 1]\), \(w\in \varDelta _m\), \(\mathrm {supp}(w)\subseteq S_R(y)\), \(z\in \varDelta _n\), \(\mathrm {supp}(z)\subseteq S_C(x)\). These restrictions are omitted afterward for fluency.
- 4.
The denominator of \(p^*\) or \(q^*\) may be zero. In this case, we simply define \(p^*\) or \(q^*\) to be 0.
- 5.
One can verify that the value of \(\rho ^*\) in the dual solution of any tight stationary point has to be \(\mu _0/(\lambda _0+\mu _0)\), by the second part of Lemma 5.
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Acknowledgement
This work is supported by Science and Technology Innovation 2030 –“The Next Generation of Artificial Intelligence” Major Project No. (2018AAA0100901). We are grateful to Dongge Wang, Xiang Yan and Yurong Chen for their inspiring suggestions and comments. We thank a number of readers for their revision opinions. We are also grateful to anonymous reviewers for their kindness.
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Chen, Z., Deng, X., Huang, W., Li, H., Li, Y. (2021). On Tightness of the Tsaknakis-Spirakis Algorithm for Approximate Nash Equilibrium. In: Caragiannis, I., Hansen, K.A. (eds) Algorithmic Game Theory. SAGT 2021. Lecture Notes in Computer Science(), vol 12885. Springer, Cham. https://doi.org/10.1007/978-3-030-85947-3_7
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