We study the "reduced-form" indirect cost of information arising from flexible sequential minimization of a "primitive" direct cost function. Indirect cost functions are characterized by a simple recursive condition, sequential learning-proofness (SLP). Under a smoothness condition, (i) SLP is equivalent to uniform posterior separability and (ii) the mapping from direct to indirect costs is tractably characterized by the cost of incrementally informative "diffusion" signals. We apply this framework to establish-and resolve-a trilemma among SLP and two other natural properties of information costs: prior invariance and constant marginal cost. Our analysis provides foundations for two new indirect cost functions: Total Information and the Minimal Likelihood Ratio (MLR) cost.

Consider a canonical bargaining problem: a buyer makes a take-it-or-leave-it offer to a seller for a single object. The two parties’ values may be interdependent. We study the set of payoff vectors that can be implemented (in sequential equilibria) using joint information design. We establish, in part constructively, that the set is a triangle characterized by simple feasibility and individual-rationality constraints. We also investigate what is implementable only using information structures in which the seller is more informed than the buyer, or more generally, under a “no signaling” equilibrium restriction. We show that there is then no loss in providing the buyer with no information and only varying the seller’s information; i.e., familiar adverse-selection structures emerge. Our model encompasses monopoly pricing, for which our results augment those of Bergemann, Brooks, and Morris (2015) and Roesler and Szentes (2017).

We consider a decentralized market where buyers search to trade with sellers of unknown quality. Each buyer targets sellers based on their ratings — a coarse summary (e.g. average) of the seller’s quality collected from previous transactions involving these sellers. We study the implication of a novel informational externality in the rating-guided market: the informational content of the sellers’ ratings is endogenous, depending on the frequency of their trading, but buyers make trading decisions not taking into account their informational effects. First, we show that an improvement in the ratings technology may exacerbate the informational externality, and hence can be welfare-worsening. Second, we extend the baseline model to allow for two ex ante identical demographic groups, and show that the informational externality endogenously generates statistical discrimination. In a stable equilibrium, highly-rated sellers (or workers) in the advantaged group receive more attention than highly-rated sellers (or workers) in the disadvantaged group, leading to discrimination against the latter group in a self-fulfilling fashion. Our analysis implies that an affirmative action policy restores equality, but only in the short run, as the non-discriminative equilibrium is unstable.

We consider the problem of a Bayesian agent receiving signals over time and then taking an action. The agent chooses when to stop and take an action based on her current beliefs, and prefers (all else equal) to act sooner rather than later. The signals received by the agent are determined by a principal, whose objective is to maximize engagement (the total attention paid by the agent to the signals). We show that engagement maximization by the principal minimizes the agent's welfare; the agent does no better than if she gathered no information. Relative to a benchmark in which the agent chooses the signals, engagement maximization induces excessive information acquisition and extreme beliefs. An optimal strategy for the principal involves "suspensive signals" that lead the agent's belief to become "less certain than the prior" and "decisive signals" that lead the agent's belief to jump to the stopping region.

We propose a combinatorial ascending auction that is "approximately" optimal, requiring minimal rationality to achieve this level of optimality, and is robust to strategic and distributional uncertainties. Specifically, the auction is rankguaranteed, meaning that for any menu M and any valuation profile, the ex-post revenue is guaranteed to be at least as high as the highest revenue achievable from feasible allocations, taking the (|M| + 1)th-highest valuation for each bundle as the price. Our analysis highlights a crucial aspect of combinatorial auction design, namely, the design of menus. We provide simple and approximately optimal menus in various settings

This paper explores statistical discrimination within two-sided matching markets, focusing on the entrepreneurial financing market. Through an experiment involving US startup founders, we identify statistical discrimination against female investors, whose signals are also perceived as less informative than those of male investors. This discrimination is predominantly driven by male founders and disproportionately affects high-quality female investors. We then develop a novel search-and-matching model with endogenous information aggregation and belief formation. The model explains how statistical discrimination can arise endogenously within two-sided matching markets, leading to the observed glass ceiling distributional effect and perpetuating a low female participation rate in equilibrium.

"We fully characterize the possible outcomes of exploration and stopping: all state-time distributions corresponding to stopping some martingale process with bounded variation. Utilizing this characterization, we provide a general methodology for solving an optimal exploration-stopping problem where the stopping utility depends on state and time arbitrarily. We reveal the close relation between the pattern of exploration and time preference and apply it to study competitive exploration contests."

"We provide a unified analysis of how dynamic information should be designed in optimal stopping problems: a principal controls the flow of information about a payoff relevant state to persuade an agent to stop at the right time, in the right state, and choose the right action. We further show that for arbitrary preferences, intertemporal commitment is unnecessary: optimal dynamic information designs can always be made dynamically consistent."