We extend search theory to multiple indivisible units and perfectly divisible assets, solving them respectively with induction and recursion. Buyer demands and prices are random, and the seller can partially exercise orders. With divisible assets, the Bellman value function is increasing and strictly concave, and the optimal reservation price falls in the position, reflecting increasing holding costs (opportunity cost of delaying optionality for inframarginal units). The marginal value exists, and is strictly convex with a falling purchase cap density. Our model is amenable to price-quantity bargaining; e.g., greater buyer bargaining power is tantamount to greater search frictions. (JEL C61, C78, D25, D83, G31)

WE ARE CONCERNED here with perfect for infinitely repeated games with complete information. Folk theorems assert that any feasible and individually rational payoff vector of the stage game is a (subgame perfect) equilibrium payoff in the associated infinitely repeated game with little or no discounting (where payoff streams are evaluated as average discounted or average values respectively). It is obvious that feasibility and individual rationality are necessary conditions for a payoff vector to be an equilibrium payoff. The surprising content of the folk theorems is that these conditions are also (almost) sufficient. Perhaps the first folk theorem type result is due to Friedman (1971) who showed that any feasible payoff which Pareto dominates a equilibrium payoff of the stage game will be an equilibrium payoff in the associated repeated game with sufficiently patient players. This kind of result is sometimes termed a Nash threats folk theorem, a reference to its method of proof. For the more permissive kinds of folk theorems considered here, the seminal results are those of Aumann and Shapley (1976) and Rubinstein (1977, 1979). These authors assume that payoff streams are undiscounted.2 Fudenberg and Maskin (1986) establish an analogous result for discounted repeated games as the discount factor goes to 1. Their result uses techniques of proof rather different from those used by Aumann-Shapley and Rubinstein, respectively. See their paper for an insightful discussion of this point, and quite generally for more by way of background. It is a key reference for subsequent work in this area, including our own. For the two-player case, the result of Fudenberg and Maskin (1986) is a complete if and only if characterization (modulo the requirement of strict rather than weak individual rationality, which we retain in this note) and does not employ additional conditions. For three or more players Fudenberg and Maskin introduced a full dimensionality condition: The convex hull F, of the set of feasible payoff vectors of the stage game must have dimension n (where n is the number of players), or equivalently a nonempty interior. This condition has been widely adopted in proving folk theorems for related environments such as finitely repeated games (Benoit and Krishna (1985)), and overlapping generations games (Kandori (1992), Smith (1992)). Full dimensionality is a sufficient condition. Fudenberg and Maskin present an example of a three-player stage game in which the conclusion of the folk theorem is false. In this example all players receive the same payoffs in all contingencies; the (convex hull of the) set of feasible payoffs is one-dimensional. This example violates full dimensionality in a rather extreme way. Less extreme violations may also lead to

An unresolved problem in Bayesian decision theory is how to value and price information. This paper resolves both problems assuming inexpensive information. Building on Large Deviation Theory, we produce a generically complete asymptotic order on samples of i.i.d. signals in finite–state, finite–action models. Computing the marginal value of an additional signal, we find it is eventually exponentially falling in quantity, and higher for lower quality signals. We provide a precise formula for the information demand, valid at low prices: asymptotically a constant times the log price, and falling in the signal quality for a given price.

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. (JEL C78, D82, D83, I23, J12)

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.

We develop a continuum player timing game that subsumes standard wars of attrition and pre‐emption games, and introduces a new rushes phenomenon. Payoffs are continuous and single‐peaked functions of the stopping time and stopping quantile. We show that if payoffs are hump‐shaped in the quantile, then a sudden “rush” of players stops in any Nash or subgame perfect equilibrium. Fear relaxes the first mover advantage in pre‐emption games, asking that the least quantile beat the average; greed relaxes the last mover advantage in wars of attrition, asking just that the last quantile payoff exceed the average. With greed, play is inefficiently late: an accelerating war of attrition starting at optimal time, followed by a rush. With fear, play is inefficiently early: a slowing pre‐emption game, ending at the optimal time, preceded by a rush. The theory predicts the length, duration, and intensity of stopping, and the size and timing of rushes, and offers insights for many common timing games.

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.