Abstract
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.
Highlights
1.1 Entropy games and matrix multiplication gamesEntropy games have been introduced by Asarin et al [5]
Whereas general matrix multiplication games are hard in general, entropy games correspond to a tractable subclass of multiplication games, in which the matrix sets have the property of being invariant by row interchange, the so called independent row uncertainty (IRU) assumption
We report experiments showing that when specialized to one player problems, policy iteration yields a speedup by one order of magnitude by comparison with the “spectral simplex” method recently introduced by Protasov [23]
Summary
Entropy games have been introduced by Asarin et al [5]. They model the situation in which two players with conflicting interests, called “Despot” and “Tribune”, wish to minimize or to maximize a topological entropy representing the freedom of a half-player, “People”. The Operator Approach to Entropy Games in [5] that the problem of comparing the value of an entropy game to a given rational number is in NP ∩ coNP, giving to entropy games a status somehow comparable to other important classes of games with an unsettled complexity, including mean payoff games, simple stochastic games, or stochastic mean payoff games, see [4] for background Another motivation to study entropy games arises from risk sensitive control [13, 14, 3]: as we shall see, essentially the same class of operators arise in the latter setting. Further motivations originate from symbolic dynamics [21, Chapter 1.8.4]
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