Extensions of the Cav(u) theorem for repeated games with one-sided information

This work is devoted to extend several asymptotic results concerning repeated games with incomplete information on one side. The model we consider is a generalization of the classical model of Aumann and Maschler [4] to infinite action spaces and partial information. We prove an extension of the classical “Cav(u)” Theorem in this model for both the lower and upper value functions using two different methods: respectively a probabilistic method based on martingales and a functional one based on approximation schemes for viscosity solutions of Hamilton Jacobi equations similar to the dual differential approach of Laraki [25]. Moreover, we show that solutions of these two asymptotic problems provide O(1/ √ n)-optimal strategies for both players in any game of length n. All these results are based on a compact approach, which consists in identifying a continuous-time problem defined on the time interval [0, 1] representing the “limit” of a sequence of finitely repeated games, as the number of repetitions is going to infinity. Finally, our results imply the existence of the uniform value of the infinitely repeated game whenever the value of the non-revealing game exists. This work is devoted to generalize and extend several asymptotic results concerning zero-sum repeated games with incomplete information on one side. Our approach, known as compact approach, consists in identifying a continuous-time problem (in most known examples a differential game or a control problem) defined on the time interval [0, 1] which represents the “limit” of a sequence of finitely repeated games. These games are seen as discrete-time models defined on the time interval [0, 1], being piecewise constant on a partition whose mesh is going to zero as the number of repetitions is going to infinity. The model we consider is an extension of the classical Aumann-Maschler model [4], where the informed player receives only a partial information about the state variable (i.e. a noisy signal) and where action spaces may be infinite. As a consequence, and since no regularity of the payoff functions is assumed, these games have in general no value. Our main result is the extension of the classical “Cav(u)” Theorem for both the lower and upper value functions in terms of the lower and upper value functions of the non-revealing game. Moreover, we identify two limit continuous-time problems, generalizing the compact problems mentioned in Sorin [29] (section 3.7.2 p.50). All the results are presented within the framework of finitely repeated games, but the extension to time-dependent payoff functions and to general evaluations (including discounted games) is possible (precise statements and details needed in order to adapt the proofs are given in section 5). These extensions are typical of the compact approach (see Cardaliaguet-Laraki-Sotin [14] for general results in this direction) and also relate our results to differential games with incomplete information (see Cardaliaguet-Rainer [11, 13]). The analysis of the asymptotic behavior of the lower value functions (or maxmin) of these games is based on probabilistic tools. The proofs rely on a representation formula for the lower value functions as the value functions of some discrete-time stochastic optimization problem over martingales that we call maximal variation problem. The maximizers of this problem represent the optimal processes of revelation for the informed player. This formula extends the one introduced in De Meyer [19] in a multidimensional setting and without assuming the existence of the value, since the class of financial exchange games considered in [19] is formally contained in the model described here. The concavification operator appearing in the limit is then expressed as the value function of a continuous-time stochastic control problem and we deduce from this representation that any limit point of a sequence of maximizers for the maximal variation problem is a maximizer of the control problem. Conversely, as for the finite case, any maximizer of the limit problem induces an O( 1 √ n ) + ε -optimal strategy for the informed player in the n-stages game for any n (optimal strategies in this context being strategies for the players guaranteeing respectively the lower and upper value). Although some classical arguments are used, such as the splitting Lemma and the bound on the L1-variation of a martingale, our proof also introduce new arguments in order to consider infinite action spaces. Moreover, we provide precise properties of the value functions and optimal strategies, some of them being new even for finite games, which may be interesting for their own sake. The analysis of the upper value functions relies on the introduction of a dual game similar to the game introduced in De Meyer [18] for finite games and appearing more recently in De Meyer-Lehrer-Rosenberg [20] for a finite non-repeated game with partial information. We show that the idea of considering a PDE introduced in De Meyer [17] and already used in De Meyer-Rosenberg [21] and Laraki [25] to prove the classical “Cav(u)” Theorem for finite games can be adapted in our case. In particular, we extend the dual variational characterization in the viscosity sense of the concavification operator obtained in Laraki [25] on an infinitedimensional space of probabilities. This allows us to deduce from the asymptotic behavior of the upper value