Splitting games over finite sets

This paper studies zero-sum splitting games with finite sets of states. Players dynamically choose a pair of martingales {p(t) , q(t)}(t), in order to control a terminal payoff u(p(infinity),q(infinity)). A first part introduces the notion of "Mertens-Zamir transform" of a real-valued matrix and use it to approximate the solution of the Mertens-Zamir system for continuous functions on the square [0, 1](2). A second part considers the general case of finite splitting games with arbitrary correspondences containing the Dirac mass on the current state: building on Laraki and Renault (Math Oper Res 45:1237-1257, 2020), we show that the value exists by constructing non Markovian epsilon-optimal strategies and we characterize it as the unique concave-convex function satisfying two new conditions.