  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.
