Patience of Matrix Games

TL;DR

This paper investigates the minimal nonzero probabilities in optimal strategies for matrix games, establishing bounds and constructing explicit examples that require extremely small probabilities, highlighting the complexity of optimal strategies.

Contribution

It provides tight bounds on the smallest nonzero probabilities needed in optimal strategies for nxn win-lose-draw matrix games and constructs explicit examples demonstrating these bounds.

Findings

Nonzero probabilities smaller than n^{-O(n)} are never needed in optimal strategies.

An explicit nxn win-lose game requires nonzero probabilities as small as n^{-Omega(n)}.

Constructed matrices have inverses with only nonnegative entries and some entries of size n^{Omega(n)}.

Abstract

For matrix games we study how small nonzero probability must be used in optimal strategies. We show that for nxn win-lose-draw games (i.e. (-1,0,1) matrix games) nonzero probabilities smaller than n^{-O(n)} are never needed. We also construct an explicit nxn win-lose game such that the unique optimal strategy uses a nonzero probability as small as n^{-Omega(n)}. This is done by constructing an explicit (-1,1) nonsingular nxn matrix, for which the inverse has only nonnegative entries and where some of the entries are of value n^{Omega(n)}.