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

The 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.Repeated games, dual games, incomplete information, recurrence formula.

Differential games with asymmetric information and without Isaacs condition

We investigate a two-player zero-sum differential game with asymmetric information on the payoff and without Isaacs condition. The dynamics is an ordinary differential equation parametrised by two controls chosen by the players. Each player has a private information on the payoff of the game, while his opponent knows only the probability distribution on the information of the other player. We show that a suitable definition of random strategies allows to prove the existence of a value in mixed strategies. Moreover, the value function can be characterised in term of the unique viscosity solution in some dual sense of a Hamilton-Jacobi-Isaacs equation. Here we do not suppose the Isaacs condition which is usually assumed in differential games

Continuous versus Discrete Market Games

De Meyer and Moussa Saley [4] provide an endogenous justification for the appearance of Brownian Motion in Finance by modeling the strategic interaction between two asymmetrically informed market makers with a zero-sum repeated game with one-sided information. The crucial point of this justification is the appearance of the normal distribution in the asymptotic behavior of Vn(P)//n. In De Meyer and Moussa Saley’s model [4], agents can fix a price in a continuous space. In the real world however, the market compels the agents to post prices in a discrete set. The previous remark raises the following question: Does the normal density still appear in the asymptotic of Vn//n for the discrete market game? The main topic of this paper is to prove that for all discretization of the price set, Vn(P)//n converges uniformly to 0. Despite of this fact, we do not reject De Meyer, Moussa analysis: when the size of the discretization step is small as compared to n-1/2, the continuous market game is a good approximation of the discrete one.Insider trading, game of incomplete information, Brownian Motion

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