The splitting game and applications

First we define the splitting operator, which is related to the Shapley operator of the splitting game introduced by Sorin (2002). It depends on two compact convex sets C and D and associates to a function defined on C ×D a saddle function, extending the usual convexification or concavification operators. We first prove general properties on its domain and its range. Then we give conditions on C and D allowing to preserve continuity or Lipschitz properties, extending the results in Laraki (2001a) obtained for the convexification operator. These results are finally used, through the analysis of the asymptotic behavior of the splitting game, to prove the existence of a continuous solution for the Mertens-Zamir system of functional equations (Mertens and Zamir (1971-72) and (1977)) in a quite general framework.