Splitting games over finite sets

Abstract

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 \(u(p_\infty ,q_\infty )\). 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 \(\varepsilon \)-optimal strategies and we characterize it as the unique concave-convex function satisfying two new conditions.