This paper studies the maximum level of cooperation that can be sustained in perfect Bayesian equilibrium in repeated games with network monitoring, where players observe each other's actions either perfectly or not at all. The foundational result is that the maximum level of cooperation can be robustly sustained in grim trigger strategies. If players are equally well monitored, comparative statics on the maximum level of cooperation are highly tractable and depend on the monitoring technology only through a simple statistic, its effective contagiousness. Typically, cooperation in the provision of pure public goods is greater in larger groups, while cooperation in the provision of divisible public goods is greater in smaller groups, and making monitoring less uncertain in the second-order stochastic dominance sense increases cooperation. For fixed monitoring networks, a new notion of network centrality is developed, which determines which players cooperate more in a given network, as well as which networks support greater cooperation. Copyright , Oxford University Press.

Two players announce bargaining postures to which they may become committed and then bargain over the division of a surplus. The share of the surplus that a player can guarantee herself under first-order knowledge of rationality is determined (as a function of her probability of becoming committed), as is the bargaining posture that she must announce in order to guarantee herself this much. This “maxmin” share of the surplus is large relative to the probability of becoming committed (e.g., it equals 30% if the commitment probability is 1 in 10 and equals 13% if the commitment probability is 1 in 1000), and the corresponding bargaining posture simply demands this share plus compensation for any delay in reaching agreement.

I study ultimatum bargaining with imperfectly observed offers. Imperfectly observed offers must be rejected with positive probability, even when the players’ preferences are common knowledge. Noisier observations imply a greater risk of rejection. In repeated ultimatum bargaining, the responding party can obtain a positive payoff if his signal of the opponent’s offer is also observed by the opponent herself, but not if his signal is private. In alternating-offers bargaining, a player is better off when her own offers are observed more precisely and her opponent’s offers are observed less precisely. Possible applications include international relations, regulation, principal-agency, and product quality provision. (JEL C73, C78, D82)

I develop a simple model of social learning in which players observe others’ outcomes but not their actions. A continuum of players arrives continuously over time, and each player chooses once-and-for-all between a safe action (which succeeds with known probability) and a risky action (which succeeds with fixed but unknown probability, depending on the state of the world). The actions also differ in their costs. Before choosing, a player observes the outcomes of K earlier players. There is always an equilibrium in which success is more likely in the good state, and this alignment property holds whenever the initial generation of players is not well informed about the state. In the case of an outcome-improving innovation (where the risky action may yield a higher probability of success), players take the correct action as K → ∞. In the case of a cost-saving innovation (where the risky action involves saving a cost but accepting a lower probability of success), inefficiency persists as K → ∞ in any aligned equilibrium. Whether inefficiency takes the form of under-adoption or over-adoption also depends on the nature of the innovation. Convergence of the population to equilibrium may be nonmonotone. (JEL D81, D83, O32, Q12, Q16)

We study the problem of a partisan gerrymanderer who assigns voters to equipopulous districts to maximize his party's expected seat share. The designer faces both aggregate, district‐level uncertainty (how many votes his party will receive) and idiosyncratic, voter‐level uncertainty (which voters will vote for his party). Segregate‐pair districting , where weaker districts contain one type of voter, while stronger districts contain two, is optimal for the gerrymanderer. The optimal form of segregate‐pair districting depends on the designer's popularity and the relative amounts of aggregate and idiosyncratic uncertainty. When idiosyncratic uncertainty dominates, a designer with majority support pairs all voters, while a designer with minority support segregates opposing voters and pairs more favorable voters; these plans resemble uniform districting and “packing‐and‐cracking,” respectively. When aggregate uncertainty dominates, the designer segregates moderate voters and pairs extreme voters; this “matching slices” plan has received some attention in the literature. Estimating the model using precinct‐level returns from recent U.S. House elections shows that, in practice, idiosyncratic uncertainty dominates. We discuss implications for redistricting reform, political polarization, and detecting gerrymandering. Methodologically, we exploit a formal connection between gerrymandering—partitioning voters into districts—and information design—partitioning states of the world into signals.

We study reputation formation where a long‐run player repeatedly observes private signals and takes actions. Short‐run players observe the long‐run player's past actions but not her past signals. The long‐run player can thus develop a reputation for playing a distribution over actions, but not necessarily for playing a particular mapping from signals to actions. Nonetheless, we show that the long‐run player can secure her Stackelberg payoff if distinct commitment types are statistically distinguishable and the Stackelberg strategy is confound‐defeating . This property holds if and only if the Stackelberg strategy is the unique solution to an optimal transport problem. If the long‐run player's payoff is supermodular in one‐dimensional signals and actions, she secures the Stackelberg payoff if and only if the Stackelberg strategy is monotone. Applications include deterrence, delegation, signaling, and persuasion. Our results extend to the case where distinct commitment types may be indistinguishable, but the Stackelberg type is salient under the prior.

The majority of labor transactions throughout much of history and a significant fraction of such transactions in many developing countries today are “coercive,” in the sense that force or the threat of force plays a central role in convincing workers to accept employment or its terms. We propose a tractable principal–agent model of coercion, based on the idea that coercive activities by employers, or “guns,” affect the participation constraint of workers. We show that coercion and effort are complements, so that coercion increases effort, but coercion always reduces utilitarian social welfare. Better outside options for workers reduce coercion because of the complementarity between coercion and effort: workers with a better outside option exert lower effort in equilibrium and thus are coerced less. Greater demand for labor increases coercion because it increases equilibrium effort. We investigate the interaction between outside options, market prices, and other economic variables by embedding the (coercive) principal–agent relationship in a general equilibrium setup, and studying when and how labor scarcity encourages coercion. General (market) equilibrium interactions working through the price of output lead to a positive relationship between labor scarcity and coercion along the lines of ideas suggested by Domar, while interactions those working through the outside option lead to a negative relationship similar to ideas advanced in neo-Malthusian historical analyses of the decline of feudalism. In net, a decline in available labor increases coercion in general equilibrium if and only if its direct (partial equilibrium) effect is to increase the price of output by more than it increases outside options. Our model also suggests that markets in slaves make slaves worse off, conditional on enslavement, and that coercion is more viable in industries that do not require relationship-specific investment by workers.

This paper studies optimal dynamic tax policy under the threat of political reform. A policy will be reformed ex post if a large enough coalition of citizens supports reform; thus, sustainable policies are those that will continue to attract enough political support in the future. We find that optimal marginal capital taxes are either progressive or U-shaped, so that savings are subsidized for the poor and/or the middle class but are taxed for the rich. U-shaped capital taxes always emerge when individuals' political behavior is purely determined by economic motives. (JEL D12, D14, D31, D72, H21, H25)

We propose a model of cycles of conflict and distrust. Overlapping generations of agents from two groups sequentially play coordination games under incomplete information about whether the other side consists of bad types who always take bad actions. Good actions may be misperceived as bad and information about past actions is limited. Conflict spirals start as a result of misperceptions but also contain the seeds of their own dissolution: Bayesian agents eventually conclude that the spiral likely started by mistake, and is thus uninformative of the opposing group's type. The agents then experiment with a good action, restarting the cycle. (JEL D72, D74, D83, Z13)

The communication revelation principle (RP) of mechanism design states that any outcome that can be implemented using any communication system can also be implemented by an incentive-compatible direct mechanism. In multistage games, we show that in general the communication RP fails for the solution concept of sequential equilibrium (SE). However, it holds in important classes of games, including single-agent games, games with pure adverse selection, games with pure moral hazard, and a class of social learning games. For general multistage games, we establish that an outcome is implementable in SE if and only if it is implementable in a canonical Nash equilibrium in which players never take codominated actions. We also prove that the communication RP holds for the more permissive solution concept of conditional probability perfect Bayesian equilibrium.