Publications des chercheurs de PSE

We study the existence of different notions of values in two-person zero-sum repeated games where the state evolves and players receive signals. We provide some examples showing that the limsup value and the uniform value may not exist in general. Then, we show the existence of the value for any Borel payoff function if the players observe a public signal including the actions played. We prove also two other positive results without assumptions on the signaling structure: the existence of the $sup$-value and the existence of the uniform value in recursive games with non-negative payoffs.

We study long-term Markov Decision Processes and Gambling Houses, with applications to any partial observation MDPs with finitely many states and zero-sum repeated games with an informed controller. We consider a decision-maker which is maximizing the weighted sum t≥1 θtrt, where rt is the expected reward of the t-th stage. We prove the existence of a very strong notion of long-term value called general uniform value, representing the fact that the decision-maker can play well independently of the evaluations (θt) t≥1 over stages, provided the total variation (or impatience) t≥1 |θt+1 − θt| is small enough. This result generalizes previous results of Rosenberg, Solan and Vieille [35] and Renault [31] that focus on arithmetic means and discounted evaluations. Moreover, we give a variational characterization of the general uniform value via the introduction of appropriate invariant measures for the decision problems, generalizing the fundamental theorem of gambling or the Aumann-Maschler cavu formula for repeated games with incomplete information. Apart the introduction of appropriate invariant measures, the main innovation in our proofs is the introduction of a new metric d * such that partial observation MDP's and repeated games with an informed controller may be associated to auxiliary problems that are non-expansive with respect to d *. Given two Borel probabilities over a compact subset X of a normed vector space, we define d * (u, v) = sup f ∈D 1 |u(f) − v(f)|, where D1 is the set of functions satisfying: ∀x, y ∈ X, ∀a, b ≥ 0, af (x) − bf (y) ≤ ax − by. The particular case where X is a simplex endowed with the L 1-norm is particularly interesting: d * is the largest distance over the probabilities with finite support over X which makes every disintegration non-expansive. Moreover, we obtain a Kantorovich-Rubinstein type duality formula for d * (u, v) involving couples of measures (α, β) over X × X such that the first marginal of α is u and the second marginal of β is v. MSC Classification: Primary: 90C40 ; Secondary: 60J20, 91A15.

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.

We analyze information design games between two designers with opposite preferences and a single agent. Before the agent makes a decision, designers repeatedly disclose public information about persistent state parameters. Disclosure continues until no designer wishes to reveal further information. We consider environments with general constraints on feasible information disclosure policies. Our main results characterize equilibrium payoffs and strategies of this long information design game and compare them with the equilibrium outcomes of games where designers move only at a single predetermined period. When information disclosure policies are unconstrained, we show that at equilibrium in the long game, information is revealed right away in a single period; otherwise, the number of periods in which information is disclosed might be unbounded. As an application, we study a competition in product demonstration and show that more information is revealed if each designer could disclose information at a predetermined period. The format that provides the buyer with most information is the sequential game where the last mover is the ex-ante favorite seller.