Tropical polyhedra are equivalent to mean payoff games

TL;DR

This paper establishes an equivalence between tropical polyhedra and mean payoff games, linking tropical convexity problems to zero-sum game theory and providing new insights into tropical rank and linear independence.

Contribution

It introduces a novel equivalence between tropical convex sets and zero-sum stochastic games, connecting tropical geometry with game theory and proving properties of tropical rank.

Findings

Tropical polyhedra correspond to deterministic finite-action games.

Winning positions in these games can be derived from tropical polyhedra.

Tropical rank equals the size of the largest linearly independent submatrix.

Abstract

We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. In particular, we set up an equivalence between the external representation of tropical convex sets and zero-sum stochastic games, in which tropical polyhedra correspond to deterministic games with finite action spaces. Then, we show that the winning initial positions can be determined from the associated tropical polyhedron. We obtain as a corollary a game theoretical proof of the fact that the tropical rank of a matrix, defined as the maximal size of a submatrix for which the optimal assignment problem has a unique solution, coincides with the maximal number of rows (or columns) of the matrix which are linearly independent in the tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius theory.