Inapproximability Results for Approximate Nash Equilibria

Deligkas, Argyrios, Fearnley, John and Savani, Rahul ORCID: 0000-0003-1262-7831 (2016) Inapproximability Results for Approximate Nash Equilibria In: WINE 2016: The 12th Conference on Web and Internet Economics.

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Abstract

We study the problem of finding approximate Nash equilibria that satisfy certain conditions, such as providing good social welfare. In particular, we study the problem ϵ$$\epsilon $$-NE δ$$\delta $$-SW: find an ϵ$$\epsilon $$-approximate Nash equilibrium (ϵ$$\epsilon $$-NE) that is within δ$$\delta $$ of the best social welfare achievable by an ϵ$$\epsilon $$-NE. Our main result is that, if the randomized exponential-time hypothesis (RETH) is true, then solving 18-O(δ)$$\left( \frac{1}{8} - \mathrm {O}(\delta )\right) $$-NE O(δ)$$\mathrm {O}(\delta )$$-SW for an n×n$$n\times n$$ bimatrix game requires nΩ~(δΛlogn)$$n^{\mathrm {\widetilde{\Omega }}(\delta ^{\varLambda } \log n)}$$ time, where Λ$$\varLambda $$ is a constant.Building on this result, we show similar conditional running time lower bounds on a number of decision problems for approximate Nash equilibria that do not involve social welfare, including maximizing or minimizing a certain player’s payoff, or finding approximate equilibria contained in a given pair of supports. We show quasi-polynomial lower bounds for these problems assuming that RETH holds, and these lower bounds apply to ϵ$$\epsilon $$-Nash equilibria for all ϵ<18$$\epsilon < \frac{1}{8}$$. The hardness of these other decision problems has so far only been studied in the context of exact equilibria.

Item Type:	Conference Item (Unspecified)
Uncontrolled Keywords:	46 Information and Computing Sciences, Liver Disease, Hepatitis, Digestive Diseases
Depositing User:	Symplectic Admin
Date Deposited:	18 Oct 2016 13:51
Last Modified:	02 Jun 2026 06:57
DOI:	10.1007/978-3-662-54110-4_3
Related Websites:	Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3003819
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