# Inapproximability Results for Approximate Nash Equilibria

arXiv: https://arxiv.org/abs/1608.03574 · HTML: https://mlsanity.com/paper/1608.03574/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Aug 2016.

## Built on by (most cited first, 1 of 1)

- [Computing Equilibrium beyond Unilateral Deviation](https://mlsanity.com/paper/2604.28186/) (2026)

## Abstract (arXiv)

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-NE δ-SW: find an ε-approximate Nash equilibrium (ε-NE) that is within δ of the best social welfare achievable by an ε-NE. Our main result is that, if the exponential-time hypothesis (ETH) is true, then solving left(frac18 - O(δ)right)-NE O(δ)-SW for an n× n bimatrix game requires nwidetilde 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 ε-Nash equilibria for all ε < frac18. The hardness of these other decision problems has so far only been studied in the context of exact equilibria.