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An analyst observes the frequency with which a decision maker (DM) takes actions, but does not observe the frequency of actions conditional on the payoff-relevant state. We ask when can the analyst rationalize the DM's choices as if the DM first learns something about the state before taking action. We provide a support function characterization of the triples of utility functions, prior beliefs, and (marginal) distributions over actions such that the DM's action distribution is consistent with information given the agent's prior and utility function. Assumptions on the cardinality of the state space and the utility function allow us to refine this characterization, obtaining a sharp system of finitely many inequalities the utility function, prior, and action distribution must satisfy. We apply our characterization to study comparative statics and ring-network games, and to identify conditions under which a data set is consistent with a public information structure in first-order Bayesian persuasion games. We characterize the set of distributions over posterior beliefs that are consistent with the DM's choices. Assuming the first-order approach applies, we extend our results to settings with a continuum of actions and/or states.%

We study the allocation of deceased-donor lungs to patients in need of a transplant. Patients make sequential decisions in an order dictated by a priority policy. Using data from a prominent Organ Procurement Organization in the United States, we provide reduced-form evidence of social learning: because patients accept or reject organs in sequence, their decisions exhibit herding behavior, often rejecting an organ that would otherwise be accepted. We develop and estimate a structural model to quantify the impact of various policy proposals and informational regimes. Our results show that blinding patients to their position in the sequence\textemdash thereby eliminating social learning\textemdash boosts organ allocation but reduces average utility per patient. In contrast, prioritizing patients by their likelihood of acceptance exacerbates social learning, leading to fewer organ allocations. Nevertheless, it raises utility per accepted organ and expedites the allocation process.

We consider search problems with nonobligatory inspection and single-item or combinatorial selection. A decision maker is presented with a number of items, each of which contains an unknown price, and can pay an inspection cost to observe the item's price before selecting it. Under single-item selection, the decision maker must select one item; under combinatorial selection, the decision maker must select a set of items that satisfies certain constraints. In our nonobligatory inspection setting, the decision maker can select items without first inspecting them. It is well-known that search with nonobligatory inspection is harder than the well-studied obligatory inspection case, for which the optimal policy for single-item selection (Weitzman, 1979) and approximation algorithms for combinatorial selection (Singla, 2018) are known. We introduce a technique, local hedging, for constructing policies with good approximation ratios in the nonobligatory inspection setting. Local hedging transforms policies for the obligatory inspection setting into policies for the nonobligatory inspection setting, at the cost of an extra factor in the approximation ratio. The factor is instance-dependent but is at most 4/3. We thus obtain the first approximation algorithms for a variety of combinatorial selection problems, including matroid basis, matching, and facility location.

We provide a framework to study stability notions for two-sided dynamic matching markets in which matching is one-to-one and irreversible. The framework gives center stage to the set of matchings an agent anticipates would ensue should they remain unmatched, which we refer to as the agent's conjectures. A collection of conjectures, together with a pairwise stability and individual rationality requirement given the conjectures, defines a solution concept for the economy. We identify a sufficient condition--consistency--for a family of conjectures to lead to a nonempty solution (cf. Hafalir, 2008). As an application, we introduce two families of consistent conjectures and their corresponding solution concepts: continuation-value-respecting dynamic stability, and the extension to dynamic markets of the solution concept in Hafalir (2008), sophisticated dynamic stability.

Thick two-sided matching platforms, such as the room-rental market, face the challenge of showing relevant objects to users to reduce search costs. Many platforms use ranking algorithms to determine the order in which alternatives are shown to users. Ranking algorithms may depend on simple criteria, such as how long a listing has been on the platform, or incorporate more sophisticated aspects, such as personalized inferences about users' preferences. Using rich data on a room rental platform, we show how ranking algorithms can be a source of unnecessary congestion, especially when the ranking is invariant across users. Invariant rankings induce users to view, click, and request the same rooms in the platform we study, greatly limiting the number of matches it creates. We estimate preferences and simulate counterfactuals under different ranking algorithms varying the degree of user personalization and variation across users. In our case, increased personalization raises both user match utility and congestion, which leads to a trade-off. We find that the current outcome is inefficient as it lies below the possibility frontier, and propose alternatives that improve upon it.

An analyst observes the frequency with which an agent takes actions, but not the frequency with which she takes actions conditional on a payoff relevant state. In this setting, we ask when the analyst can rationalize the agent's choices as the outcome of the agent learning something about the state before taking action. Our characterization marries the obedience approach in information design (Bergemann and Morris, 2016) and the belief approach in Bayesian persuasion (Kamenica and Gentzkow, 2011) relying on a theorem by Strassen (1965) and Hall's marriage theorem. We apply our results to ring-network games and to identify conditions under which a data set is consistent with a public information structure in first-order Bayesian persuasion games.

Information policies such as scores, ratings, and recommendations are increasingly shaping society's choices in high-stakes domains. We provide a framework to study the welfare implications of information policies on a population of heterogeneous individuals. We define and characterize the Bayes welfare set, consisting of the population's utility profiles that are feasible under some information policy. The Pareto frontier of this set can be recovered by a series of standard Bayesian persuasion problems, in which a utilitarian planner takes the role of the information designer. We provide necessary and sufficient conditions under which an information policy exists that Pareto dominates the no-information policy. We illustrate our results with applications to data leakage, price discrimination, and credit ratings.

Product personalization opens the door to price discrimination. A rich product line allows firms to better tailor products to consumers' tastes, but the mere choice of a product carries valuable information about consumers that can be leveraged for price discrimination. We study this trade-off in an upstream-downstream model, where a consumer buys a good of variable quality upstream, followed by an indivisible good downstream. The downstream firm's use of the consumer's purchase history for price discrimination introduces a novel distortion: The upstream firm offers a subset of the products that it would offer if, instead, it could jointly design its product line and downstream pricing. By controlling the degree of product personalization the upstream firm curbs ratcheting forces that result from the consumer facing downstream price discrimination., Comment: Replaced to correct figure placement

I introduce a stability notion, dynamic stability, for two-sided dynamic matching markets where (i) matching opportunities arrive over time, (ii) matching is one-to-one, and (iii) matching is irreversible. The definition addresses two conceptual issues. First, since not all agents are available to match at the same time, one must establish which agents are allowed to form blocking pairs. Second, dynamic matching markets exhibit a form of externality that is not present in static markets: an agent's payoff from remaining unmatched cannot be defined independently of what other contemporaneous agents' outcomes are. Dynamically stable matchings always exist. Dynamic stability is a necessary condition to ensure timely participation in the economy by ensuring that agents do not strategically delay the time at which they are available to match.

A buyer wishes to purchase a durable good from a seller who in each period chooses a mechanism under limited commitment. The buyer's valuation is binary and fully persistent. We show that posted prices implement all equilibrium outcomes of an infinite-horizon, mechanism selection game. Despite being able to choose mechanisms, the seller can do no better and no worse than if he chose prices in each period, so that he is subject to Coase's conjecture. Our analysis marries insights from information and mechanism design with those from the literature on durable goods. We do so by relying on the revelation principle in Doval and Skreta (2020).

We provide tools to analyze information design problems subject to constraints. We do so by extending the insight in Le Treust and Tomala (2019) to the case of multiple inequality and equality constraints. Namely, that an information design problem subject to constraints can be represented as an unconstrained information design problem with a additional states, one for each constraint. Thus, without loss of generality, optimal solutions induce as many posteriors as the number of states and constraints. We provide results that refine this upper bound. Furthermore, we provide conditions under which there is no duality gap in constrained information design, thus validating a Lagrangian approach. We illustrate our results with applications to mechanism design with limited commitment (Doval and Skreta, 2022a) and persuasion of a privately informed receiver (Kolotilin et al., 2017).

We develop a tool akin to the revelation principle for dynamic mechanism-selection games in which the designer can only commit to short-term mechanisms. We identify a canonical class of mechanisms rich enough to replicate the outcomes of any equilibrium in a mechanism-selection game between an uninformed designer and a privately informed agent. A cornerstone of our methodology is the idea that a mechanism should encode not only the rules that determine the allocation, but also the information the designer obtains from the interaction with the agent. Therefore, how much the designer learns, which is the key tension in design with limited commitment, becomes an explicit part of the design. Our result simplifies the search for the designer-optimal outcome by reducing the agent's behavior to a series of participation, truthtelling, and Bayes' plausibility constraints the mechanisms must satisfy., Comment: Added an omitted assumption in Section 4 (see footnote 21 and the proof of Proposition 4.1)