In the standard herding model, privately informed individuals sequentially see prior actions and then act. An identical action herd eventually starts and public beliefs tend to “cascade sets” where social learning stops. What behaviour is socially efficient when actions ignore informational externalities? We characterize the outcome that maximizes the discounted sum of utilities. Our four key findings are: (1) cascade sets shrink but do not vanish, and herding should occur but less readily as greater weight is attached to posterity. (2) An optimal mechanism rewards individuals mimicked by their successor. (3) Cascades cannot start after period one under a signal log-concavity condition. (4) Given this condition, efficient behaviour is contrarian, leaning against the myopically more popular actions in every period. We make two technical contributions: as value functions with learning are not smooth, we use monotone comparative statics under uncertainty to deduce optimal dynamic behaviour. We also adapt dynamic pivot mechanisms to Bayesian learning.

This paper analyzes how asset prices in a binary market react to information when traders have heterogeneous prior beliefs. We show that the competitive equilibrium price underreacts to information when there is a bound to the amount of money traders are allowed to invest. Underreaction is more pronounced when prior beliefs are more heterogeneous. Even in the absence of exogenous bounds on the amount that traders can invest, prices underreact to information provided that traders become less risk averse as their wealth increases. In a dynamic setting, underreaction results in initial momentum and then reversal in the long run.

Empirical analyses of parimutuel betting markets have documented that market probabilities of favorites (longshots) tend to underestimate (overestimate) the corresponding empirical probabilities. We argue that this favorite-longshot bias is consistent with bettors taking simultaneous positions on the basis of private information about the likelihood of different outcomes. The ex post realization of a high market probability indicates favorable information about the occurrence of an outcome—and the opposite is true for longshots. This explanation for the bias relies on the bettors' inability to incorporate the surprise revealed by the final odds.

Are the highest sample realizations selected from a larger presample more or less informative than the same amount of random data? Developing multivariate accuracy for interval dominance ordered preferences, we show that sample selection always benefits (or always harms) a decision maker if the reverse hazard rate of the data distribution is log‐supermodular (or log‐submodular), as in location experiments with normal noise. We find nonpathological conditions under which the information contained in the winning bids of a symmetric auction decreases in the number of bidders. Exploiting extreme value theory, we quantify the limit amount of information revealed when the presample size (number of bidders) goes to infinity. In a model of equilibrium persuasion with costly information, we derive implications for the optimal design of selected experiments when selection is made by an examinee, a biased researcher, or contending sides with the peremptory challenge right to eliminate a number of jurors.