The central question we address in this paper is: what can an analyst infer from choice data about what a decision maker has learned? The key constraint we impose, which is shared across models of Bayesian learning, is that any learning must be rationalizable. To implement this constraint, we introduce two conditions, one of which refines the mean preserving spread of Blackwell (1953) to take account for optimality, and the other of which generalizes the NIAC condition (Caplin and Dean 2015) and the NIAS condition (Caplin and Martin 2015) to allow for arbitrary learning. We apply
our framework to show how identification of what was learned can be strengthened with additional assumptions on the form of Bayesian learning.

This paper studies costly information acquisition and transmission. An expert can communicate with a decision-maker about a state of nature by sending a cheap-talk message. I establish a version of the recommendation principle, meaning that the sender generally reveals all acquired information to the decision-maker in efficient equilibria. Furthermore, I show existence of efficient equilibria under general conditions. For the class of posterior separable cost structures, I derive properties of efficient experiments. Under posterior-mean preferences, any cheap-talk problem is solved by a convex combination of two bi-pooling policies. Beyond that, I characterize the best bi-pooling policies for the uniform-quadratic case. Contrary to existing cheap-talk models, monotone partitions are not always optimal.

Evidence suggests that consumers do not perfectly optimize, contrary to a critical assumption of classical consumer theory. We propose a model in which consumer types can vary in both their preferences and their choice behavior. Given data on demand and the distribution of prices, we identify the set of possible values of the consumer surplus based on minimal rationality conditions: every type of consumer must be no worse off than if they either always bought the good or never did. We develop a procedure to narrow the set of surplus values using richer datasets and provide bounds on counterfactual demands.

We develop a method for making robust predictions in games with flexible information acquisition (i.e., rational inattention Sims, 2003). In games with exogenous information, one can describe the set of attainable outcomes using the Bayes correlated equilibrium (BCE) concept (Bergemann and Morris, 2016). We introduce a refinement of BCE, Blackwell correlated equilibrium (BKE), and prove that it spans all outcomes attainable under some flexible learning technology whose costs increase in Blackwell’s (1951,1953) information order. We show the BKE set is either dense or nowhere dense in the BCE set, with the former being true for generic games. We also characterize the set of outcomes attainable under almost-free learning. We conclude by exploring the implications of BKE on a Bertrand competition game, where we show the best BCE for consumers may not be approximable by BKEs.