We study the joint design of dynamic incentives and performance feedback for an environment with a coarse (all‐or‐nothing) measure of performance, and show that hiding information from the agent can be an optimal way to motivate effort. Using a novel approach to incentive compatibility, we derive a two‐phase solution that begins with a “silent phase” where the agent is given no feedback and is asked to work non‐stop, and ends with a “full‐transparency phase” where the agent stops working as soon as a performance threshold is met. Hiding information leads to greater effort, but an ignorant agent is also more expensive to motivate. The two‐phase solution—where the agent's ignorance is fully frontloaded—stems from a “backward compounding effect” that raises the cost of hiding information as time passes.

Standard game-theoretic analysis yields highly incomplete descriptions of behavior in complex games of complete information. A key part of the reason is that standard models do not account for the role of complexity in shaping players' strategic behavior. We investigate the implications of complexity empirically and theoretically, focusing on the game of chess---a good setting for our study because it is a rare example of an extensive-form game that is played by experienced, motivated players, and for which we have vast amounts of data. First, using a large database of chess play, we quantify the significant margin by which machine-learning predictions trained on human play outperform a Zermelo's Theorem benchmark prediction of outcomes under perfectly rational play. Comparing this prediction to the actual empirical average outcome in a given position, we find that the game-theoretic model achieves 72% completeness. This demonstrates that while classical game-theoretic forces are predictive, they leave a significant amount of variation in outcomes unexplained. To assess how much of the remaining variation is predictable, we train a machine learning algorithm using additional features of the positions beyond the minimax value. This algorithm achieves 93% completeness, indicating that positional information beyond standard game theory is crucial for predicting behavior. We then focus on the role of complexity in explaining deviations from game-theoretic predictions. We define a heuristic measure of complexity based on whether a position's minimax value can be predicted using easily observable features that are commonly noticed by human players, or requires more complicated analysis. This measure of complexity predicts deviations from minimax predictions, with actual play closer to game-theoretic predictions in simple positions than in complex ones. An important cross-sectional finding also emerges: complex positions occur more frequently in games between higher-rated players. Specifically, complex positions are 73% more common in games between top-tercile players compared to bottom-tercile players. To understand the mechanisms behind this finding, we develop a tractable model capturing key strategic incentives for (and against) complex play in chess. The model features two players who take turns moving, with an evolving position state variable that encodes whether the position is simple or complex. Players can choose between simple moves that aim to produce simple states, or complex moves that aim to produce complex states for the opponent. However, players may unwittingly make blunders, defined as moves that reduce the value of the game for them. The key tradeoff in the model is that a more complex move creates a harder situation for the opponent when it succeeds, but increases the probability of one's own blunder. Players receive a stochastic signal called an "insight" before each move, informing them of how likely they are to avoid blundering if they attempt a complex move. We show theoretically that increasing player skill has theoretically ambiguous effects on the prevalence of complex positions because there are competing forces at play. More skilled players are more likely to execute complex moves without blundering, but more skilled opponents are more likely to successfully respond and thereby punish a complex move with a complex ensuing position. We structurally estimate a version of our model, recovering estimates of players' willingness to make complex moves at different rating levels. Surprisingly, we find that strong players are more cautious than weak players given the same information. We conclude that high-rated players spend more time in complex positions not because they are more willing to take risks, but because they maintain complex positions for many moves without blundering.

We derive an optimal dynamic contest for environments where effort can be monitored only through a coarse, binary performance measure and the principal chooses prize-allocation and termination rules together with a real-time feedback policy for the contestants. The optimal contest takes a stark cyclical form: contestants are kept fully apprised of their own successes, and at the end of each fixed-length cycle, if at least one agent has succeeded, the contest ends and the prize is shared equally among all successful agents irrespective of when they succeeded; otherwise, the designer informs all contestants that nobody has yet succeeded and the contest resets. Applications include promotions, innovation contests, and proof-of-work protocols.

Two players with common interests exchange information to make a decision. But they fear scrutiny. Their unencrypted communications will be observed by another agent with different interests who can object to their decision. We show how the players can implement their ideal decision rule using a back and forth conversation. Such a subversive conversation reveals enough information for the players to determine their best decision but not enough information for the observer to determine whether the decision was against his interest. Our results show how conversations can maintain deniability even in the face of leaks, hacks

A health authority chooses a binary action for each of several individuals that differ in their pretest probabilities of being infectious and in the additive losses associated with two types of decision errors. The authority is endowed with a portfolio of tests that differ in their sensitivities and specificities. We derive a simple necessary condition for optimality of test allocation. In special cases, precision parameters of the allocated test are monotone in the individuals' types. We characterize the marginal benefit of a test, provide an algorithmic solution for the test-allocation problem and consider the benefits of confirmatory testing. (c) 2021 Elsevier Inc. All reserved.

To prevent the spread of an infection, an organization obeys social distancing restrictions and thus limits the number of its members physically present on a given day. We study rotation schemes in which mutually exclusive groups are active on different days. The frequency of rotation affects risk over the duration of diffusion prior to the time the organization is able to react to the infection. If this reaction time is speedy, then such risk is undesirable because prevalence is initially convex in time. In this case, frequent rotation acts as insurance against exposure-time risk and is optimal. Infrequent rotation becomes optimal if the organization reacts slowly. Cross-mixing of the rotating subpopulations is detrimental because it increases contacts between sick and healthy individuals. However, the effect of mixing is small if the terminal prevalence is low in the absence of mixing. This paper was accepted by Joshua Gans, business strategy.

We study games of incomplete information as both the information structure and the extensive form vary. An analyst may know the payoff‐relevant data but not the players' private information, nor the extensive form that governs their play. Alternatively, a designer may be able to build a mechanism from these ingredients. We characterize all outcomes that can arise in an equilibrium of some extensive form with some information structure. We show how to specialize our main concept to capture the additional restrictions implied by extensive‐form refinements.

This supplement contains results omitted from the main text of the paper. ∗Division of the Humanities and Social Sciences, ldoval@caltech.edu †Department of Economics, Northwestern University. jeffely@northwestern.edu.

We study information as an incentive device in a dynamic moral hazard framework. An agent works on a task of uncertain difficulty, modeled as the duration of required effort. The principal knows the task difficulty and provides information over time. The optimal mechanism features moving goalposts: an initial disclosure makes the agent sufficiently optimistic that the task is easy. If the task is indeed difficult, the agent is told this only after working long enough to put the difficult task within reach. The agent then completes the difficult task even though he never would have chosen to at the outset.

We define Perfect Bayesian Equilibrium as an assessment (a behavioral strategy profile and system of beliefs) that is consistent with a conditional probability system over the set of strategies. Consistency with a conditional probability system captures the idea of Bayes’ rule where possible. We compare this definition with almost Perfect Bayesian equilibrium (Mailath (2018)) and show that it is stronger.

I propose a mechanism for redistricting inspired by cake-cutting mechanisms for fair division. The majority party proposes a partition of a state into districts. The minority party can accept it or undo any partisan disadvantage caused by irregular boundaries. Thus without imposing any requirement of regularity, the mechanism ensures that to the extent that irregular districts result from the process, the minority is never harmed by them.

In auction environments in which agents have private values, the Vickrey auction induces agents to truthfully reveal their preferences and selects the efficient allocation accordingly. When the agents' valuations are interdependent, various generalizations of the Vickrey auction have been found which provide incentives for truthful revelation of all private information and preserve efficiency. However, these mechanisms generally do not provide the bidders with dominant strategies. The existing literature has therefore used a stronger equilibrium solution concept. In this paper we show that while the generalized VCG mechanism admits a multiplicity of equilibria, many of which are inefficient. We give conditions under which the efficiency equilibrium is the unique outcome of iterative elimination of ex post weakly dominated strategies. With two bidders, the standard single-crossing condition is sufficient. With more than two bidders, we show by example that a strengthening of the single-crossing condition is necessary.

We consider optimal pricing policies for airlines when passengers are uncertain at the time of ticketing of their eventual willingness to pay for air travel. Auctions at the time of departure efficiently allocate space and a profit maximizing airline can capitalize on these gains by overbooking flights and repurchasing excess tickets from those passengers whose realized value is low. Nevertheless profit maximization entails distortions away from the efficient allocation. Under regularity conditions, we show that the optimal mechanism can be implemented by a modified double auction. In order to encourage early booking, passengers who purchase late are disadvantaged. In order to capture the information rents of passengers with high expected values, ticket repurchases at the time of departure are at a subsidized price, sometimes leading to unused capacity. (JEL: D42, D44, D82).

We investigate whether expert players with high incentives are able to optimally determine their degree of risk taking in contest. We use a large dataset on tennis matches and look at players' risk taking on first and second serves. We isolate a specific situation, let serves, where second serves and first serves occur in a way which is as good as random. This creates the setting of a quasi-experiment which we can use to study players' serving strategies on first and second serves in comparable serving situations. We find that players, both men and women, are able to adopt serving strategies which meet the requirements of optimality arising from simple assumptions about risk-return trade-offs in serves. (C) 2017 Published by Elsevier B.V.

We study optimal price discrimination when a monopolist faces a continuum of consumers with reference-dependent preferences. A consumer's valuation for product quality consists of an intrinsic valuation affected by a private state signal (type), and a gain-loss valuation that depends on deviations of purchased quality from a reference point. Following Kőszegi and Rabin (2006), we consider loss-averse buyers who evaluate gains and losses in terms of changes in the consumption valuation, but in our model each buyer evaluates consumption outcomes relative to his own state-contingent reference quality level. We capture the process by which reference qualities are formed via a reference consumption plan, and use a generalization of the Mirrlees representation of the indirect utility to fully characterize optimal contracts for loss-averse consumers. We find that, depending on the reference plan, optimal price discrimination may exhibit (i) downward distortions beyond the standard downward distortions due to screening; (ii) efficiency gains relative to second best contracts without loss aversion; (iii) upward distortions above first best quality levels without loss aversion. We consider ex-ante and ex-post consistent contracts in which quality offers by the firm coincide, in expectations or at every state realization, respectively, with the reference quality levels. We find the firm's unique preferred ex-ante and ex-post consistent contract menu and specify conditions under which, for the second case, it also constitutes the consumers' preferred menu.

We study torture as a mechanism for extracting information from a suspect who may or may not be informed. We show that a standard rationale for torture generates two commitment problems. First, the principal would benefit from a commitment to torture a suspect he knows to be innocent. Secondly, the principal would benefit from a commitment to limit the amount of torture faced by the guilty. We analyse a dynamic model of torture in which the credibility of these threats and promises is endogenous. We show that these commitment problems dramatically reduce the value of torture and can even render it completely ineffective. We use our model to address questions such as the effect of enhanced interrogation techniques, rights against indefinite detention, and delegation of torture to specialists.

We model demand for noninstrumental information, drawing on the idea that people derive entertainment utility from suspense and surprise. A period has more suspense if the variance of the next period's beliefs is greater. A period has more surprise if the current belief is further from the last period's belief. Under these definitions, we analyze the optimal way to reveal information over time so as to maximize expected suspense or surprise experienced by a Bayesian audience. We apply our results to the design of mystery novels, political primaries, casinos, game shows, auctions, and sports.

We study torture as a mechanism for extracting information from a suspect who may or may not be informed. We show that a standard rationale for torture generates two commitment problems. First, the principal would benefit from a commitment to torture a suspect he knows to be innocent. Second, the principal would benefit from a commitment to limit the amount of torture faced by the guilty. We analyze a dynamic model of torture in which the credibility of these threats and promises is endogenous. We show that these commitment problems dramatically reduce the value of torture and can even render it completely ineffective. We use our model to address questions such as the effect of enhanced interrogation techniques, rights against indefinite detention, and delegation of torture to specialists.

Consider two agents who learn the value of an unknown parameter by observing a sequence of private signals. Will the agents commonly learn the value of the parameter, i.e., will the true value of the parameter become approximate common-knowledge? If the signals are independent and identically distributed across time (but not necessarily across agents), the answer is yes (Cripps et al., Econometrica, 76(4):909–933, 2008). This paper explores the implications of allowing the signals to be dependent over time. We present a counterexample showing that even extremely simple time dependence can preclude common learning, and present sufficient conditions for common learning.