Smoothed Complexity of 2-player Nash Equilibria

TL;DR

This paper proves that computing Nash equilibria in two-player games remains PPAD-hard even after smoothing with constant noise, challenging previous conjectures and separating smoothed complexity from approximation hardness.

Contribution

It establishes PPAD-hardness of Nash equilibrium computation under smoothed analysis with constant noise, using a novel reduction involving random zero-sum games.

Findings

PPAD-hardness persists after smoothing with constant noise

All Nash equilibria of random zero-sum games are far from pure

Smoothed complexity and approximation hardness are separated for Nash equilibria

Abstract

We prove that computing a Nash equilibrium of a two-player ($n \times n$) game with payoffs in $[-1,1]$ is PPAD-hard (under randomized reductions) even in the smoothed analysis setting, smoothing with noise of constant magnitude. This gives a strong negative answer to conjectures of Spielman and Teng [ST06] and Cheng, Deng, and Teng [CDT09]. In contrast to prior work proving PPAD-hardness after smoothing by noise of magnitude $1/\operatorname{poly}(n)$ [CDT09], our smoothed complexity result is not proved via hardness of approximation for Nash equilibria. This is by necessity, since Nash equilibria can be approximated to constant error in quasi-polynomial time [LMM03]. Our results therefore separate smoothed complexity and hardness of approximation for Nash equilibria in two-player games. The key ingredient in our reduction is the use of a random zero-sum game as a gadget to produce two-player games which remain hard even after smoothing. Our analysis crucially shows that all Nash equilibria of random zero-sum games are far from pure (with high probability), and that this remains true even after smoothing.