Long information design

Abstract

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.

Splitting games over finite sets

“…Given any pair of pure strategies, by the martingale property, the sequence of states (i t , j t ) t almost surely converges to a random variable (i ∞ , j ∞ ) and the payoff for Player 1 is defined as the expectation of u(i ∞ , j ∞ ). As in Koessler et al (2021a), the value of Γ(U ) is V .…”

“…(One can think for instance that, in each state, players have access to exogenous sets of statistical experiments to affect posterior beliefs as in Koessler et al (2021a).) A payoff function u :…”

“…The recent literature on information design and Bayesian persuasion (Kamenica and Gentzkow, 2011;Bergemann and Morris, 2019;Koessler, Laclau, and Tomala, 2021b) has given new interpretation and applications to splitting games: choosing a splitting amounts to choosing an information structure, and information design considers optimization problems or games over information structures. In Koessler, Laclau, Renault, and Tomala (2021a), we consider a zero-sum splitting game between two information designers who provide information to a decision maker, and where u(p, q) represents the expected payoff induced by the action chosen by the decision maker who has belief (p, q). MZ(u) is also the value of the splitting game with terminal payoff (long information design game) where the payoff of the splitting game is simply the expectation of u(p ∞ , q ∞ ).…”

“…The approach lim λ→0 v λ = MZ(u) is particularly useful to prove the existence of MZ(u) (by considering a uniform limit point of (v λ )). This approach has been followed in Laraki and Renault (2020) and Koessler et al (2021a) to prove the existence of the MZ transform for general classes of games (strongly acyclic gambling games, information design games with experiments).…”

“…In section 4, we introduce a general model of splitting games over finite sets, where the splittings available at a given p always allow to stay at p but may be arbitrarily constrained otherwise. Contrary to the existing litterature (Laraki and Renault, 2020;Koessler et al, 2021a), we make no further assumption such as acyclicity or closedness under iteration. In Theorem 2, we prove the existence of the value via the construction of non Markovian ε-optimal pure strategies and provide a new characterization of the value with conditions replacing (C1) and (C2) above.…”