In many auction settings the auctioneer must disclose the identity of the winner and the price he pays. We characterize the auction that minimizes the winner's privacy loss among those that maximize total surplus or the seller's revenue, and are strategy-proof. Privacy loss is measured with respect to what an outside observer learns from the disclosed price, and is quantified by the mutual information between the price and the winner's willingness to pay. When only interim individual-rationality is required, the most privacy preserving auction involves stochastic ex-post payments. Under ex-post individual rationality, and assuming the bidders' type distribution exhibits a monotone hazard rate, privacy loss is minimized by the second-price auction with deterministic payments.

An analyst observes the frequency with which a decision maker (henceforth, DM) takes actions, but not the frequency with which she takes actions conditional on a payoff relevant state. In this setting, we characterize the information structures that rationalize the marginal distribution over actions as the result of the DM first observing the information structure's realization and then choosing actions optimally. Our main result identifies a core condition that must be satisfied between the marginal action distribution and the distribution over the DM's posterior beliefs for the information structure to rationalize the DM's choices.

There are many economic situations in which players compete to identify an uncertain event, or an "opportunity." Once an opportunity appears, seizing it quickly is critical. For example, two firms may consider the introduction of a new technology into a market, which may either be ripe for the technology or not. The first firm to identify that the market is ripe and act benefits greatly. However, acting if the market is not ripe generates large losses, and checking market conditions is costly. Other natural contexts in which competition for opportunities occurs include innovation, patenting, launching new products, selling assets, head hunting, and dating. What is common to all these examples is that the right moment to act depends on identifying an unobservable event, while taking the other player's strategy into account.