Theory of Computing Systems

, Volume 63, Issue 5, pp 1089–1130 | Cite as

The Operator Approach to Entropy Games

  • Marianne Akian
  • Stéphane GaubertEmail author
  • Julien Grand-Clément
  • Jérémie Guillaud
Part of the following topical collections:
  1. Special Issue on Theoretical Aspects of Computer Science (STACS 2017)


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.


Stochastic games Shapley operators Policy iteration Perron eigenvalues Risk sensitive control 



An announcement of the present results appeared in the proceedings of STACS, [4]. We are very grateful to the referees of this STACS paper and also to the referees of the present extended version, for their detailed comments which helped us to improve this manuscript.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Inria and CMAPÉcole polytechnique, CNRSPalaiseauFrance
  2. 2.IEOR DepartmentColumbia UniversityNew YorkUSA
  3. 3.Inria ParisParisFrance

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