Unit Vector Games

TL;DR

This paper introduces 'unit vector games,' a generalization of imitation games, to simplify the construction of bimatrix games where key algorithms for finding Nash equilibria require exponential time.

Contribution

It extends the concept of imitation games to unit vector games, providing a simplified framework for analyzing complex equilibrium-finding algorithms.

Findings

Simplifies the construction of hard bimatrix games

Shows exponential complexity for Lemke-Howson algorithm

Demonstrates support enumeration's exponential steps

Abstract

McLennan and Tourky (2010) showed that "imitation games" provide a new view of the computation of Nash equilibria of bimatrix games with the Lemke-Howson algorithm. In an imitation game, the payoff matrix of one of the players is the identity matrix. We study the more general "unit vector games", which are already known, where the payoff matrix of one player is composed of unit vectors. Our main application is a simplification of the construction by Savani and von Stengel (2006) of bimatrix games where two basic equilibrium-finding algorithms take exponentially many steps: the Lemke-Howson algorithm, and support enumeration.