Differential games with asymmetric and correlated information

Differential games with asymmetric information were introduced by Cardaliaguet (2007). As in repeated games with lack of information on both sides (Aumann and Maschler (1995)), each player receives a private signal (his type) before the game starts and has a prior belief about his opponent's type. Then, a differential game is played in which the dynamic and the payoff function depend on both types: each player is thus partially informed about the differential game that is played. The existence of the value function and some characterizations have been obtained under the assumption that the signals are drawn independently. In this paper, we drop this assumption and extend these two results to the general case of correlated types. This result is then applied to repeated games with incomplete information: the characterization of the asymptotic value obtained by Rosenberg and Sorin (2001) and Laraki (2001) for the independent case is extended to the general case.Comment: 22 page

Duality and optimal strategies in the finitely repeated zero-sum games with incomplete information on both sides

URL des Cahiers : https://halshs.archives-ouvertes.fr/CAHIERS-MSECahiers de la MSE 2005.27 - Série Bleue - ISSN : 1624-0340The recursive formula for the value of the zero-sum repeated games with incomplete information on both sides is known for a long time. As it is explained in the paper, the usual proof of this formula is in a sense non constructive: it just claims that the players are unable to guarantee a better payoff than the one prescribed by formula, but it does not indicates how the players can guarantee this amount. In this paper we aim to give a constructive approach to this formula using duality techniques. This will allow us to recursively describe the optimal strategies in those games and to apply these results to games with infinite action spaces.La formule de récurrence pour la valeur d'un jeu répété à somme nulle avec asymétrie bilatérale d'information a été établie depuis longtemps dans la littérature existante. Comme nous le remarquons dans ce papier, la preuve usuelle de cette formule est dans un sens non constructive. Elle démontre en particulier que les joueurs ne peuvent pas se garantir un paiement supérieur à celui décrit par la formule, mais cette analyse n'indique pas comment les joueurs parviennent à garantir cette quantité. Dans cet article, en utilisant des techniques de dualité, nous aborderons une approche constructive de cette formule. Cette analyse nous permettra d'apporter une description récursive des stratégies optimales dans ces jeux et également d'étendre les résultats aux jeux avec des espaces d'actions infinis

Advances in Zero-Sum Dynamic Games

International audienceThe survey presents recent results in the theory of two-person zero-sum repeated games and their connections with differential and continuous-time games. The emphasis is made on the following(1) A general model allows to deal simultaneously with stochastic and informational aspects.(2) All evaluations of the stage payoffs can be covered in the same framework (and not only the usual Cesàro and Abel means).(3) The model in discrete time can be seen and analyzed as a discretization of a continuous time game. Moreover, tools and ideas from repeated games are very fruitful for continuous time games and vice versa.(4) Numerous important conjectures have been answered (some in the negative).(5) New tools and original models have been proposed. As a consequence, the field (discrete versus continuous time, stochastic versus incomplete information models) has a much more unified structure, and research is extremely active