We develop a strategic theory of counterfeiting as a multi-market large game. Bad guys choose whether to counterfeit, and what quality to produce. Opposing them is a continuum of good guys who select a costly verification effort. In equilibrium, counterfeiters produce better quality at higher notes, but verifiers try sufficiently harder that verification still improves. We develop a graphical framework for deducing comparative statics. Passed and counterfeiting rates vanish for low and high notes. Our predictions are consistent with time series and cross-sectional patterns in a unique data set assembled largely from the Secret Service.

We introduce and solve a new class of “downward-recursive” static portfolio choice problems. An individual simultaneously chooses among ranked stochastic options, and each choice is costly. In the motivational application, just one may be exercised from those that succeed. This often emerges in practice, such as when a student applies to many colleges or when a firm simultaneously tries several technologies. We show that such portfolio choice problems quite generally entail maximizing a submodular function of finite sets—which is NP-hard in general. Still, we show that a greedy algorithm finds the optimal set, finding first the best singleton, then the best single addition to it, and so on. We show that the optimal choices are “less aggressive” than the sequentially optimal ones, but “more aggressive” than the best singletons. Also, the optimal set in general contains gaps. We provide some comparative statics results on the chosen set.

This paper explores how Bayes-rational individuals learn sequentially from the discrete actions of others. Unlike earlier informational herding papers, we admit heterogeneous preferences. Not only may type-specific ‘herds’ eventually arise, but a new robust possibility emerges: confounded learning. Beliefs may converge to a limit point where history offers no decisive lessons for anyone, and each type's actions forever nontrivially split between two actions. To verify that our identified limit outcomes do arise, we exploit the Markov-martingale character of beliefs. Learning dynamics are stochastically stable near a fixed point in many Bayesian learning models like this one.

Toward understanding assortative matching, this is a self-contained introduction to research on search and matching. We first explore the nontransferable and perfectly transferable utility matching paradigms, and then a unifying imperfectly transferable utility matching model. Motivated by some unrealistic predictions of frictionless matching, we flesh out the foundational economics of search theory. We then revisit the original matching paradigms with search frictions. We finally allow informational frictions that often arise, such as in college-student sorting.

The timing of elections is flexible in many countries. We study this optimization by first creating a Bayesian learning model of a mean-reverting political support process. We then explore optimal electoral timing, modelling it as a renewable American option with interacting waiting and stopping values. Inter alia, we show that the expected longevity is a convex, then concave, function of the support. Finally, we calibrate our model to the post-1945 Labour-Tory U.K. rivalry. Our story quite well explains when the elections were called. We also show that election options approximately double the expected time in power in the current streak.

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.

We explore costly deliberation by two differentially informed and possibly biased jurors: A hawk Lones and a dove Moritz alternately insist on a verdict until one concedes. Debate assumes one of two genres, depending on bias: A juror, say Lones, is intransigent if he wishes to prevail and reach a conviction for any type of Moritz next to concede. In contrast, Lones is ambivalent if he wants the strongest conceding types of Moritz to push for acquittal. Both jurors are ambivalent with small bias or high delay costs. As Lones grows more hawkish, he argues more forcefully for convictions, mitigating wrongful acquittals. If dovish Moritz is intransigent, then he softens (strategic substitutes), leading to more wrongful convictions. Ambivalent debate is new, and yields a novel dynamic benefit of increased polarization. For if Moritz is ambivalent, then he toughens (strategic complements), and so, surprisingly, a more hawkish Lones leads to fewer wrongful acquittals and convictions. So more polarized but balanced debate can improve communication, unlike in static cheap talk. We also show that patient and not too biased jurors vote against their posteriors near the end of the debate, optimally playing devil’s advocate. We shed light on the adversarial legal system, peremptory challenges, and cloture rules.