Colorful linear programming, Nash equilibrium, and pivots

TL;DR

This paper explores colorful linear programming, connecting it to game theory and combinatorics, and introduces algorithms and complexity results for finding solutions like Nash equilibria and sets guaranteed by the colorful Carathéodory theorem.

Contribution

It establishes that computing Nash equilibria can be formulated as colorful linear programming and introduces a variant of Bárány's algorithm linked to the simplex method.

Findings

Computing Nash equilibrium is a colorful linear programming problem.

Deciding and optimizing in colorful linear programming are computationally equivalent.

A new algorithm variant connects colorful Carathéodory sets with the simplex algorithm.

Abstract

The colorful Carath\'eodory theorem, proved by B\'ar\'any in 1982, states that given d+1 sets of points S_1,...,S_{d+1} in R^d, with each S_i containing 0 in its convex hull, there exists a subset T of the union of the S_i's containing 0 in its convex hull and such that T contains at most one point from each S_i. An intriguing question -- still open -- is whether such a set T, whose existence is ensured, can be found in polynomial time. In 1997, B\'ar\'any and Onn defined colorful linear programming as algorithmic questions related to the colorful Carath\'eodory theorem. The question we just mentioned comes under colorful linear programming. The traditional applications of colorful linear programming lie in discrete geometry. In this paper, we study its relations with other areas, such as game theory, operations research, and combinatorics. Regarding game theory, we prove that computing a Nash equilibrium in a bimatrix game is a colorful linear programming problem. We also formulate an optimization problem for colorful linear programming and show that as for usual linear programming, deciding and optimizing are computationally equivalent. We discuss then a colorful version of Dantzig's diet problem. We also propose a variant of the B\'ar\'any algorithm, which is an algorithm computing a set T whose existence is ensured by the colorful Carath\'eodory theorem. Our algorithm makes a clear connection with the simplex algorithm and we discuss its computational efficiency. Related complexity and combinatorial results are also provided.