Théorie de l'information, jeux répétés avec observation imparfaite et réseaux de communication décentralisés

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This thesis is devoted to the study of mutual contributions between games theory and information theory and their applications to decentralized communication networks. First, game theory provides answers to optimization problems in which agents interact. In a game, players choose actions and obtain gains called utilities. Assumptions about the information possessed by players before play are fundamental to determine the outcome a game, also called equilibrium. When the same game is repeated from stage to stage and the players do not observe the past actions perfectly, then the equilibrium utilities are not known. On the other hand, information theory studies the performance of a communicating system. Nowadays, communication networks are so dense that they cannot organize around a single central operator. Game theory is appropriate to explore new organizations of communication networks in which decisions are taken locally. At first, in Chapter 3, we study the game of power control in terms of energy efficiency, thanks to the existing results for repeated games. Transmitters are regarded as players and choose the transmission power of the signal, considered as their action. The objective of a player is to choose an optimal power for the quality of its own communication. The players do not observe the past actions perfectly, but we show that the observation of the "signal over interference plus noise ratio" is sufficient to ensure optimal equilibrium results for the communication network. In a second step, we use the tools of the information theory for further study of the flow of information among the players. In Chapter 4, an encoder sends an extra signal to the players so that they perfectly observe the actions chosen in the previous stage-game. The observation of players is sufficiently precise to characterize the set of equilibrium utilities of the repeated game. These results are, in turn, used to model new communication networks and to provide more realistic solutions. In Chapter 5, we deepen the study of equilibrium utilities when players observe the past actions to through an arbitrary observation channel. We show a rate region is achievable for the multi-user channel with states which includes an encoder, two legitimate receivers and an eavesdropper. This result allows us to study the correlations over the sequences of actions a group of players can implement while keeping it secret from an opponent player. The study of multi-user channels is a step towards the characterization of equilibrium utilities in a repeated game with imperfect monitoring.

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