### March 21, 2016

**Catalin Dima**(LACL)

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 player wants to maximize the growth rate of the product, while the other wants 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 cup coNP.

Based on joint work with Eugene Asarin, Aldric Degorre and Florian Horn (IRIF-Paris Diderot), and Victor Kozyakin (IITP Moscow).