Variational Inequalities, System of Functional Equations, and Incomplete Information Repeated Games

We consider a pair of functional equations obtained by Mertens and Zamir (Internat. J. Game Theory, 1 (1971--72), pp. 39--64; J. Math. Anal. Appl., 60 (1977), pp. 550--558) to characterize the asymptotic value of a two person zero sum repeated with lack of information on both sides (Aumann and Maschler Repeated Games with Incomplete Information, MIT Press, Cambridge, MA 1995)). We give a new proof for the convergence of the discounted values of the repeated game and a new characterization of the limit using variational inequalities. The same idea allows us to prove existence and uniqueness of a Lipschitz solution for the pair of functional equations in a general framework using an auxiliary game: "the splitting game," introduced by Sorin (A First Course on Zero-Sum Repeated Games, preprint).