Patience of Matrix Games
Kristoffer Arnsfelt Hansen, Rasmus Ibsen-Jensen, Vladimir V., Podolskii, Elias Tsigaridas

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)}.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsMatrix Theory and Algorithms · Game Theory and Applications · Computability, Logic, AI Algorithms
