We consider an agent who chooses an option after receiving some private information. This information, however, is unobserved by an analyst, so from the latter's perspective, choice is probabilistic or random. We provide a theory in which information can be fully identified from random choice. In addition, the analyst can perform the following inferences even when information is unobservable: (1) directly compute ex ante valuations of menus from random choice and vice versa, (2) assess which agent has better information by using choice dispersion as a measure of informativeness, (3) determine if the agent's beliefs about information are dynamically consistent, and (4) test to see if these beliefs are well‐calibrated or rational.

We propose a novel model of stochastic choice: the single‐crossing random utility model (SCRUM). This is a random utility model in which the collection of preferences satisfies the single‐crossing property. We offer a characterization of SCRUMs based on two easy‐to‐check properties: the classic Monotonicity property and a novel condition, Centrality. The identified collection of preferences and associated probabilities is unique. We show that SCRUMs nest both single‐peaked and single‐dipped random utility models and establish a stochastic monotone comparative result for the case of SCRUMs.

Buyers often search across sellers to learn which product best fits their needs. We study how sellers manage these search incentives through their disclosure strategies (e.g. product trials, reviews and recommendations), and ask how competition affects information provision. If sellers can observe the beliefs of buyers or can coordinate their strategies, then there is an equilibrium in which sellers provide the "monopoly level" of information. In contrast, if buyers' beliefs are private, then there is an equilibrium in which sellers provide full information as search costs vanish. Anonymity and coordination thus play important roles in understanding how advice markets work.

We provide a theory of random intertemporal choice. Agents exhibit stochastic choice over consumption due to preference shocks to discounting attitudes. We first demonstrate how the distribution of these preference shocks can be uniquely identified from random choice data. We then provide axiomatic characterizations of some common random discounting models, including exponential and quasi-hyperbolic discounting. In particular, we show how testing for exponential discounting under stochastic choice involves checking for both a stochastic version of stationarity and a novel axiom characterizing decreasing impatience.

We provide a revealed preference methodology for identifying beliefs and utilities that can vary across states. A notion of comparative informativeness is introduced that is weaker than the standard Blackwell ranking. We show that beliefs and state-dependent utilities can be identified using stochastic choice from two informational treatments, where one is strictly more informative than another. Moreover, if the signal structure is known, then stochastic choice from a single treatment is enough for identification. These results illustrate novel identification methodologies unique to stochastic choice. Applications include identifying biases in job hiring, loan approvals, and medical advice. (JEL D11, D82, D83, J23, M51)