The complexity of approximate Nash equilibrium in congestion games with negative delays

TL;DR

This paper investigates the computational complexity of finding approximate Nash equilibria in congestion games with arbitrary delay functions, revealing polynomial-time convergence for negative delays and PLS-completeness for general delays.

Contribution

It extends the complexity analysis of congestion games to include negative delays, showing both polynomial convergence and computational hardness results.

Findings

Polynomial-time convergence for negative delay functions

PLS-completeness for general delay functions

Extension of complexity results to arbitrary sign delays

Abstract

We extend the study of the complexity of finding an $\eps$-approximate Nash equilibrium in congestion games from the case of positive delay functions to delays of arbitrary sign. We first prove that in symmetric games with $\alpha$-bounded jump the $\eps$-Nash dynamic converges in polynomial time when all delay functions are negative, similarly to the case of positive delays. We then establish a hardness result for symmetric games with $\alpha$-bounded jump and with arbitrary delay functions: in that case finding an $\eps$-Nash equilibrium becomes $\PLS$-complete.