Hypothetical Beliefs Identify Information

Abstract

After observing the outcome of a Blackwell experiment, a Bayesian decisionmaker can form (a) posterior beliefs over the state, as well as (b) posterior beliefs she would observe any given signal (assuming an independent draw from the same experiment). I call the latter her contingent hypothetical beliefs. I show geometrically how contingent hypothetical beliefs relate to information structures. Specifically, the information structure can (generically) be derived by regressing contingent hypothetical beliefs on posterior beliefs over the state. Her prior is the unit eigenvector of a matrix determined from her posterior beliefs over the state and her contingent hypothetical beliefs. Thus, all aspects of a decisionmaker's information acquisition problem can be determined using ex-post data (i.e., beliefs after having received signals). I compare my results to similar ones obtained in cases where information is modeled deterministically; the focus on single-agent stochastic information distinguishes my work.

Information Payoffs: An Interim Perspective

“…Indeed, when the payoff function is linear in beliefs as in Section 4, an interim payoff profile can be seen as a profile of second-order expectations. Starting with Samet (1998), a body of work uses Markov matrices to represent such higher-order beliefs and expectations of higher-order beliefs for a given information structure (see, for instance, Cripps et al, 2008;Golub and Morris, 2017;Libgober, 2021). Instead, our result in Theorem 3 identifies the set of matrices that can correspond to some information structure.…”