The Weihrauch degree of finding Nash equilibria in multiplayer games

TL;DR

This paper investigates the computational complexity of finding Nash equilibria in multiplayer games with real payoffs by classifying the problem's Weihrauch degree, extending previous two-player results to multiplayer scenarios.

Contribution

It advances the understanding of the non-computability of Nash equilibria in multiplayer games by classifying their Weihrauch degree using polynomial root-finding and cylindrical algebraic decomposition.

Findings

Classified the Weihrauch degree for multiplayer Nash equilibria.

Extended the two-player classification to multiplayer cases.

Linked the problem to polynomial root-finding complexity.

Abstract

Is there an algorithm that takes a game in normal form as input, and outputs a Nash equilibrium? If the payoffs are integers, the answer is yes, and lot of work has been done in its computational complexity. If the payoffs are permitted to be real numbers, the answer is no, for continuity reasons. It is worthwhile to investigate the precise degree of non-computability (the Weihrauch degree), since knowing the degree entails what other approaches are available (eg, is there a randomized algorithm with positive success change?). The two player case has already been fully classified, but the multiplayer case remains open and is addressed here. Our approach involves classifying the degree of finding roots of polynomials, and lifting this to systems of polynomial inequalities via cylindrical algebraic decomposition.