Extensions of the Cav(u) Theorem for Repeated Games with Incomplete Information on One Side

Abstract

Abstract. 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 13]).The analysis of the asymptotic behavior of the lower value functions (or max...

On Repeated Zero-Sum Games with Incomplete Information and Asymptotically Bounded Values

“…The scheme resembles the one used by De Meyer and Marino (2005) [8] to derive an upper bound for the value of the discrete market model. We start from the recurrent equation V N +1 (p) = T [V N ](p) in the form from Gensbittel (2015) [18] and construct an explicit non-negative invariant function h of the Shapley operator T using the Kantorovich metric. Monotonicity ideas of De Meyer and Marino imply that h is an upper bound for V N .…”

“…The first two items immediately follow from (3.1). To prove the third item we take in (3.1) a constant martingale p (n) ≡ p. For such martingale P = δ p , where δ p is the Dirac δ-measure at p. The game G 1 (δ p ) can be identified with the non-revealing game Γ 1 (p) (see Gensbittel (2015) [18]). Hence by almost-fairness V 1 (δ p ) = 0 that implies the third item.…”

“…Throughout the paper we assume that the total payoff equals to the sum of stage payoffs (i.e., there is neither discounting nor normalization). Then, as it was shown by Aumann and Maschler, under some technical assumptions V N grows linearly up to an error term bounded by a constant times √ N (Aumann and Maschler (1995) [1]; Gensbittel (2015) [18] and Neyman (2013) [27] for recent extensions).…”

“…Similar problems of dynamic strategic use of information also demonstrate √ N -behavior of the value. Mertens and Zamir (1977) [24] and De Meyer (1998) [4] studied martingale optimization problem of the maximal L 1variation of a bounded martingale which is connected to the optimal speed of information revelation in games with √ N-behavior (Gensbittel (2015) citeCavU showed that any game can be reduced to a generalized problem of the maximal variation). Problems arising as continuous-time limits of √ N -games were discussed by De Meyer (1999) [5] and Gensbittel (2013) [19] (working with such limit problems is known as "compact approach", see Sorin (2002) [31]).…”

Value of Persistent Information

“…As we explain below, our characterization is a generalization of that result. Similar ideas have been used elsewhere in the literature on dynamic games with incomplete information; see, for example, De Meyer (2010) or Gensbittel (2015). (We are grateful to a referee for these examples.…”