Entropy Games and Matrix Multiplication Games

Eugene Asarin (1) , Julien Cervelle (2) , Aldric Degorre (1) , Catalin Dima (2) , Florian Horn (1) , Victor Kozyakin (3)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
2 LACL - Laboratoire d'Algorithmique Complexité et Logique
3 IITP - Institute for Information Transmission Problems

Two intimately related new classes of games are introduced and studied: entropy games (EGs) and matrix multiplication games (MMGs). An EG is played on a finite arena by two-and-a-half players: Despot, Tribune and the non-deterministic People. Despot wants to make the set of possible People's behaviors as small as possible, while Tribune wants to make it as large as possible. An MMG is played by two players that alternately write matrices from some predefined finite sets. One wants to maximize the growth rate of the product, and the other to minimize it. We show that in general MMGs are undecidable in quite a strong sense. On the positive side, EGs correspond to a subclass of MMGs, and we prove that such MMGs and EGs are determined, and that the optimal strategies are simple. The complexity of solving such games is in NP&coNP.

Mots clés: game theory; entropy; joint spectral radius

Domaines: Informatique et théorie des jeux [cs.GT]; Théorie de l'information [cs.IT]; Autre [cs.OH]

Eugene Asarin, Julien Cervelle, Aldric Degorre, Catalin Dima, Florian Horn, et al.. Entropy Games and Matrix Multiplication Games. 2015. ⟨hal-01164086v3⟩