## 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 PerronFrobenius theory.

Original language | English |
---|---|

Article number | 1250001 |

Journal | International Journal of Algebra and Computation |

Volume | 22 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2012 |

Externally published | Yes |

### Bibliographical note

Funding Information:The two first authors were partially supported by the joint RFBR-CNRS grant 05-01-02807, by a MSRI Research membership for the Fall 2009 Semester on Tropical Geometry, and by a grant from LEA (Laboratoire Européen Associé) Math-Mode. The second author was also partially supported by the Arpege programme of the French National Agency of Research (ANR), project “ASOPT”, number ANR-08-SEGI-005 and by the Digiteo project DIM08 “PASO” number 3389. The third author was partially supported by the invited professors program from INRIA Paris-Rocquencourt and by the grants MD-2535.2009.1 and RFBR 09-01-00303.

## Keywords

- Assignment problem
- Linear independence
- Mean-payoff games
- Nonexpansive maps
- PerronFrobenius theory
- Tropical algebra
- Tropical polyhedra