This paper studies zero-sum splitting games with finite sets of states. Players dynamically choose a pair of martingales {pt,qt}t, in order to control a terminal payoff u(p∞,q∞). A firstpartintroduces 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 (2020), we show that the value exists by constructing non Markovian ε-optimal strategies and we characterize it as the unique concave-convex function satisfying two new conditions
