Splitting games over finite sets

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