  Entropy games and matrix multiplication games have been recently introduced
by Asarin et al. They model the situation in which one player (Despot) wishes
to minimize the growth rate of a matrix product, whereas the other player
(Tribune) wishes to maximize it. We develop an operator approach to entropy
games. This allows us to show that entropy games can be cast as stochastic mean
payoff games in which some action spaces are simplices and payments are given
by a relative entropy (Kullback-Leibler divergence). In this way, we show that
entropy games with a fixed number of states belonging to Despot can be solved
in polynomial time. This approach also allows us to solve these games by a
policy iteration algorithm, which we compare with the spectral simplex
algorithm developed by Protasov.
