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.

Automated summary

This paper studies zero-sum splitting games with finite sets of states. Players dynamically choose a pair of martingales {p(t), q(t)}(t) to control a terminal payoff. The first part introduces the concept of the Mertens-Zamir transform and uses it to approximate the solution of the Mertens-Zamir system for continuous functions. The second part considers the general case of finite splitting games with arbitrary correspondences containing the Dirac mass on the current state, and proves the existence of its value by constructing non-Markovian epsilon-optimal strategies and characterizing it as the unique concave-convex function satisfying two new conditions.