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 {pt,qt}t, in order to control a terminal payoff u(p∞,q∞). 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 ε-optimal strategies and we characterize it as the unique concave-convex function satisfying two new conditions.

Replaces

Frédéric Koessler, Marie Laclau, Jérôme Renault, and Tristan Tomala, “Splitting games over finite sets”, TSE Working Paper, n. 22-1321, March 2022.

Reference

Frédéric Koessler, Marie Laclau, Jérôme Renault, and Tristan Tomala, “Splitting games over finite sets”, Mathematical Programming, vol. 203, n. 1-2, January 2024, p. 477–498.