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Section: Scientific Foundations

Applications to the theory of repeated games

The individual sequences mentioned above can be thought of as being chosen by an opponent player, in which case a repeated two-player game is at hand. We study two fundamental tools in this context: calibration and approachability.

Calibration is the ability to forecast well, on average, the opponent's actions. It is often used as the property of some auxiliary strategy on which some main strategy can be built. The latter will turn to be efficient since it can accurately reconstruct the opponent's behavior.

Approachability is the ability to control random walks. At each round, a vector payoff is obtained by the first player, depending on his action and on the action of the opponent player. The aim is to ensure that the average of the vector payoffs converges to some convex set. Necessary and sufficient conditions were obtained by Blackwell and others to ensure that such strategies exist, both in the full information and in the bandit cases. We want to extend the result to the case of games with signals (games with partial monitoring), where at each round the only feedback obtained by the first player is a random signal drawn according to a distribution that depends on the action profile taken by the two players, while the opponent player still has a full monitoring.


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