Tropical complementarity problems and Nash equilibria

TL;DR

This paper demonstrates that the tropical analogue of the complementarity problem related to Nash equilibria can be solved efficiently in polynomial time, contrasting with the classical PPAD-complete case, and extends the Lemke--Howson algorithm to this setting.

Contribution

It introduces a polynomial-time solution for the tropical complementarity problem linked to Nash equilibria and adapts the Lemke--Howson algorithm to this tropical context.

Findings

Tropical complementarity problems are solvable in polynomial time.

The Lemke--Howson algorithm extends to the tropical setting with linear pivot steps.

A new class of bimatrix games allows polynomial-time Nash equilibrium computation.

Abstract

Linear complementarity programming is a generalization of linear programming which encompasses the computation of Nash equilibria for bimatrix games. While the latter problem is PPAD-complete, we show that the tropical analogue of the complementarity problem associated with Nash equilibria can be solved in polynomial time. Moreover, we prove that the Lemke--Howson algorithm carries over the tropical setting and performs a linear number of pivots in the worst case. A consequence of this result is a new class of (classical) bimatrix games for which Nash equilibria computation can be done in polynomial time.