Splitting games over finite sets

Frédéric Koessler (1, 2) , Marie Laclau (3, 4) , Jérôme Renault (5) , Tristan Tomala (3, 4)

1 PSE - Paris School of Economics
2 PJSE - Paris Jourdan Sciences Economiques
3 HEC Paris - Ecole des Hautes Etudes Commerciales
4 GREGHEC - Groupement de Recherche et d'Etudes en Gestion
5 TSE-R - Toulouse School of Economics

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.

Mots clés: Splitting games; Mertens-Zamir system; Repeated games with incomplete information; Bayesian persuasion; Information design

Domaines: Economies et finances; Probabilités [math.PR]

Frédéric Koessler, Marie Laclau, Jérôme Renault, Tristan Tomala. Splitting games over finite sets. Mathematical Programming, In press, ⟨10.1007/s10107-022-01806-7⟩. ⟨halshs-03672222⟩