Deligkas, Argyrios, Fearnley, John and Savani, Rahul 
ORCID: 0000-0003-1262-7831
(2016)
Inapproximability Results for Approximate Nash Equilibria.
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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 $\epsilon$-NE $\delta$-SW: find an $\epsilon$-approximate Nash equilibrium ($\epsilon$-NE) that is within $\delta$ of the best social welfare achievable by an $\epsilon$-NE. Our main result is that, if the exponential-time hypothesis (ETH) is true, then solving $\left(\frac{1}{8} - \mathrm{O}(\delta)\right)$-NE $\mathrm{O}(\delta)$-SW for an $n\times n$ bimatrix game requires $n^{\mathrm{\widetilde \Omega}(\log n)}$ time. 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 ETH holds, where these lower bounds apply to $\epsilon$-Nash equilibria for all $\epsilon < \frac{1}{8}$. The hardness of these other decision problems has so far only been studied in the context of exact equilibria.
| Item Type: | Conference or Workshop Item (Unspecified) |
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| Additional Information: | A short (14-page) version of this paper appeared at WINE 2016. Compared to that conference version, this new version improves the conditional lower bounds, which now rely on ETH rather than RETH (Randomized ETH) |
| Uncontrolled Keywords: | cs.GT, cs.GT |
| Divisions: | Faculty of Science and Engineering > School of Electrical Engineering, Electronics and Computer Science Faculty of Science and Engineering > School of Physical Sciences |
| Depositing User: | Symplectic Admin |
| Date Deposited: | 12 Dec 2022 09:52 |
| Last Modified: | 18 Jan 2023 19:40 |
| DOI: | 10.1007/978-3-662-54110-4_3 |
| Related URLs: | |
| URI: | https://livrepository.liverpool.ac.uk/id/eprint/3166597 |
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