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The Operator Approach to Entropy Games

Authors Marianne Akian, Stéphane Gaubert, Julien Grand-Clément, Jérémie Guillaud

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Marianne Akian
Stéphane Gaubert
Julien Grand-Clément
Jérémie Guillaud

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Marianne Akian, Stéphane Gaubert, Julien Grand-Clément, and Jérémie Guillaud. The Operator Approach to Entropy Games. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.STACS.2017.6

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.
Keywords
  • Stochastic games
  • Shapley operators
  • policy iteration
  • Perron eigenvalues
  • Risk sensitive control

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