Session 3: Dynamic Games, Contracts, and Markets

We study optimal regulation of a dynamic job market in which firms are matched with workers over time, and can offer long-term employment contracts. Firms and workers have idiosyncratic match productivity, which is privately observed by firms. The regulator can design the set of permitted contracts from which firms can make offers, and aims to maximize a weighted sum of worker and firm welfare. We derive an optimal regulatory policy in which all contracts take a simple structure, comprising a signing bonus, a constant effort level, a flat wage, and a termination fee if the worker wants to leave the relationship. The policy links the permitted effort, wage and termination fee to incentivize firms to offer higher wages to workers with better match productivity. Properly designed termination fees introduce frictions into the market which improve worker welfare: firms share surplus with workers as they are protected from poaching by other firms. Wage restrictions result in a binding minimum wage which creates early-career unemployment, but improves worker welfare by reducing firm information rents.

We study learning on social media with an equilibrium model of users interacting with shared news stories. Rational users arrive sequentially, observe an original story (i.e., a private signal) and a sample of predecessors’ stories in a news feed, and then decide which stories to share. The observed sample of stories depends on what predecessors share as well as the sampling algorithm, which represents a design choice of the platform. We focus on how much the algorithm relies on virality (how many times a story has been previously shared) when generating news feeds. Showing users more viral stories can increase information aggregation, but it can also generate steady states where most shared stories are wrong. Such misleading steady states self-perpetuate, as users who observe these wrong stories develop wrong beliefs, and thus rationally continue to share them. We find that these bad steady states appear discontinuously, and platform designers either accept these misleading steady states or induce fragile learning outcomes in the optimal design.

We consider an agent who posits a set of probabilistic models for the payoff- relevant outcomes. The agent has a prior over this set but fears the actual model is omitted and hedges against this possibility. The concern for misspecification is endogenous: If a model explains the previous observations well, the concern attenuates. We show that different static preferences under uncertainty (subjective expected utility, maxmin, robust control) arise in the long run, depending on how quickly the agent becomes unsatisfied with unexplained evidence and whether they are misspecified. The misspecification concern’s endogeneity naturally induces behavior cycles, and we characterize the limit action frequency. This model is consistent with the empirical evidence on monetary policy cycles and choices in the face of complex tax schedules. Finally, we axiomatize in terms of observable choices this decision criterion and how quickly the agent adjusts their misspecification concern.

We consider a continuous-time game between a buyer and a seller. The buyer privately knows how often he needs to trade. When he does, he can choose to either engage with the seller, who chooses what utility to supply, or search for an alternative. Because time is informative, the
seller learns and adjusts her behavior over time. Without commitment, in the Markov perfect equilibrium, the seller starts with a pooling offer, before experimenting with occasional separating offers. Her payoff is non-monotone –in fact, quasi-convex– in her belief about the buyer’s type. With commitment, the seller can take advantage of limited-time offers to extract all the buyer’s surplus.

I analyze a novel reputation game between a patient player and a sequence of myopic short-run players. The short-run players have limited memories and cannot observe the exact sequence of the patient player’s actions. I focus on the case in which every short-run player only observes the number of times that the patient player took each of his actions in the last K periods. When players have monotone-super modular payoffs, I show that the patient player can approximately secure his commitment payoff in all equilibria as long as K is at least one. I also show that the short-run players can approximately attain their highest feasible payoff in all equilibria if and only if their memory length K is lower than some cutoff. Although a longer memory enables more short-run players to punish the patient player once the patient player deviates from his commitment action, it weakens the short-run players’ incentives to execute the punishment.

This paper characterizes the set of perfect public equilibrium payoffs, the set of Markov perfect equilibrium payoffs, and a simple class of state-order dependent equilibrium payoffs in continuous-time stochastic games with finitely many states and a publicly observable Brownian signal about past actions. Contrary to many discrete-time methods, the characterization does not rely on a convergence to a stationary distribution of the underlying state process. As a consequence, the correspondence of initial state to equilibrium payoffs is preserved, the characterization is possible for any level of discounting, and the characterization is applicable to games that are not irreducible.

At the height of the Covid-19 pandemic, consumers stockpiled common household items in expectation of shortages and rising prices. In this paper, we explore the inter-action of consumer stockpiling with monopoly pricing in the face of a supply disruption. If consumers can store the good, they purchase stockpiles in anticipation of a price hike. This causes the firm to raise the price sooner in expectation and erodes profit. Consumers are better off (worse off) than in a world without consumer stockpiles if their storage costs are lower (higher) than a critical level. Social welfare, however, is often lower when consumers can stockpile. We explore potential remedies to resolve the distortions, including rationing, price controls, and strategic reserves. Despite conflicting objectives between the firm and the government, there exists a government strategic reserve policy that achieves the social optimum.

An asset owner designs an asset-backed security and a signal about its value. After privately observing the signal, he sells the security to the monopolistic liquidity supplier. Any optimal signal structure guarantees the security sale and commits the issuer not to learn too positive private information about the security value. The optimal security design is pure equity – the most informationally sensitive security. It is risky debt under additional liquidity requirements akin to Basel III regulation. Thus, the standard intuition behind debt optimality as the least informationally sensitive security holds only under additional restrictions (e.g., regulatory) on security or information design.

This paper studies durable goods monopoly without commitment under an informationally robust objective. A seller cannot commit to future prices and does not know the information arrival process according to which a representative buyer learns about her valuation. To avoid known conceptual difficulties associated with formulating a dynamically-consistent maxmin objective, we posit the seller’s uncertainty is resolved by an explicit player (nature) who chooses the information arrival process adversarially and sequentially. Under a simple transformation of the buyer’s value distribution, the solution (in the gap case) is payoff-equivalent to a classic environment where the buyer knows her valuation at the beginning. This result immediately delivers a sharp characterization of the equilibrium price path. Furthermore, we provide a (simple to check and frequently satisfied) sufficient condition which guarantees that no arbitrary (even dynamically-inconsistent) information arrival process can lower the seller’s profit against this equilibrium price path. We call a price path with this property a reinforcing solution, and suggest this concept may be of independent interest as a way of tractably analyzing limited commitment robust objectives. We consider alternative ways of specifying the robust objective, and also show that the analogy to known-values in the no-gap case need not hold in general.

A principal needs the expertise of a biased agent in order to assess applicants who are available for hire. The principal can commit to a mechanism in order to make use of the agentís expertise, while mitigating the consequences of the agentís bias. The optimal mechanism is such that the agent is rewarded or punished in real time for his hiring decisions. If the agent hires an applicant, the mechanism rewards him with an increase in value. If the agent does not hire an applicant, the mechanism punishes him with a decrease in value. The system of punishments and rewards moves the agentís reservation quality towards the one preferred by the principal and, hence, reduces the impact of the agentís bias on the hiring. The punishments and rewards are ultimately delivered by the mechanism as changes in probability with which the hiring decision is taken away from the agent. Once the hiring decision is taken away from the agent, the principal permanently takes over and hires every applicant.

I study how organizations assign tasks to identify the best candidate to promote among a pool of workers. Task allocation and workers’ motivation interact through the organization’s promotion decisions. The organization designs the workers’ careers to both screen and develop talent. When only non-routine tasks are informative about a worker’s type and non-routine tasks are scarce, the organization’s preferred promotion system is an index contest. Each worker is assigned a number that depends only on his own type. The principal delegates the non-routine task to the worker whose current index is the highest and promotes the first worker whose type exceeds a threshold. Each worker’s threshold is independent of the other workers’ types. Competition is mediated by the allocation of tasks: who gets the opportunity to prove themselves is a determinant factor in promotions. Finally, features of the optimal promotion contest rationalize the prevalence of fast-track promotion, the role of seniority, or when a group of workers is systemically advantaged.

We consider a team-production environment where all participants are motivated by career concerns, and where a team’s joint productive outcome may have different reputational implications for different team members. In this context, we characterize equilibrium disclosure of team-outcomes when team-disclosure choices aggregate individual decisions through some deliberation protocol. In contrast with individual disclosure problems, we show that equilibria often involve partial disclosure. Furthermore, we study the effort-incentive properties of equilibrium disclosure strategies implied by different deliberation protocols; and show that the partial disclosure of team outcomes may improve individuals’ incentives to contribute to the team. Finally, we study the design of deliberation protocols, and characterize productive environments where effort-incentives are maximized by unilateral decision protocols or more consensual deliberation procedures.

When does society eventually learn the truth, or take the correct action, via observational learning? In a general model of sequential learning over social networks, we identify a simple sufficient—and, in a sense, necessary—condition for learning dubbed excludability. Excludability is a joint property of agents’ preferences and their information. When required to hold for all preferences, it is equivalent to information having 'unbounded beliefs', which demands that any agent can individually identify the truth, even if only with small probability. But unbounded beliefs may be untenable with more than two states: e.g., it is incompatible with the monotone likelihood ratio property. Excludability reveals that what is crucial for learning, instead, is that a single agent must be able to rule out any wrong action, even if she cannot take the correct action. Consequently, excludability helps study classes of preferences and information that mutually ensure learning. We develop two such pairs: (i) for a one-dimensional state, preferences with single-crossing differences and a new informational condition, directionally unbounded beliefs; and (ii) for a multi-dimensional state, Euclidean preferences and subexponential location-shift information.

We consider a dynamic informed principal problem, where a buyer privately gets lump-sum payoffs from the service provided by a seller at an exponential rate if and only if the service is good. The seller is informed about the exponential rate but not the lump-sum payoff of the service. The buyer is initially uninformed about both, but she privately and perfectly learns both upon the first arrival of the lump-sum payoff. The dynamic nature of the service enables learning-based discrimination, which mitigates the adverse selection on the seller’s side but creates a multi-dimensional screening problem on the buyer’s side. The optimal mechanism for the good-type seller is a two-phase trial mechanism. In the first phase (the trial), the seller sells access to the service at a discounted price such that any buyer purchases. In the second phase, the seller sells the remainder of the service at a price lower than the Myerson price, and only buyers who experience a high payoff from the trial purchase. Varying prices and the length of two phases can also achieve the Pareto frontier of the equilibrium payoff set.

We develop a continuous-time model of dynamic predatory behavior à la Milgrom and Roberts (1982) and Matthews and Mirman (1983). An incumbent who is privately informed about whether the demand is strong or weak faces a potential entrant who decides when, if ever, to pay an entry cost to become an incumbent’s competitor or take an outside option. The incumbent chooses its output so to affect the market price, which is a noisy signal of the market demand. We provide a complete characterization of Markov perfect equilibria as a solution to a system of differential equations and show how to leverage its tractability to investigate the welfare effects of predation. We show that even when limit pricing is unsuccessful in preventing entry, entry may occur too late from the point of view of social welfare: depending on the strategy of the incumbent, the potential entrant may anticipate being able to observe informative signals about the demand and hence has incentives to wait longer before entering.

This paper studies quid-pro-quo dynamic agreements between an interest group and two political parties. Political parties repeatedly compete for office. Before each election, the interest group decides which party to support. When in power, parties choose the rent they transfer to the interest group to buy its support. Yet, binding agreements are not possible, so agreements must be self-enforcing. When electoral uncertainty is low, the interest group favors an opportunistic agreement in which it always supports the current incumbent. As electoral uncertainty increases, the interest group prefers an exclusive agreement in which it supports a single party even when it is in opposition. An interest group with more inefficient rents is also less likely to favor an opportunistic agreement. The model offers a novel explanation for why studies on the impact of campaign contributions on policymaking find mixed evidence. Besides, my results shed new light on existing empirical findings by showing that interest groups' long-term loyalty does not necessarily imply an ideological alignment, and interest groups' opportunism can be a signal of effective institutions. Lastly, I study the impact of emergencies, weak political parties, and the interest group's entry costs on the interest group's best agreement.

As emphasized in Weitzman (1998), the search for novel ideas, technologies, and products is often combinatorial: familiar ideas are combined in unfamiliar ways to advance the stock of knowledge. We develop a model of directed Bayesian search over a rich multi-dimensional landscape of available ideas, some within a single field of knowledge and others combining distinct fields. The success of combinations is modeled through the sample paths of the Brownian staple, a natural extension of the Brownian motion framework of Callander (2011) to higher dimensions. We characterize the optimal search dynamics by a sequence of short-lived researchers. Recombinant search is fundamentally different from search within a single field. It is gradual and proceeds in a grid of existing knowledge: it pushes the frontier of at most one field at a time, combining a familiar idea from one field with an unfamiliar one in the other. The analysis illuminates the important role of derivative research—that is, intensive search that does not aim to advance the frontier of knowledge—in facilitating innovative research at the frontier. Such dynamics of optimal recombinant search resonate with stylized facts about the evolution of technology codes in patent innovation.

This project analyzes how a principal can motivate an agent to conserve rather than exploit a depletable resource. This dynamic problem is relevant for tropical deforestation as well as for other environmental problems. It is shown that the smaller is the agents discount factor (e.g., because of political instability), the more the principal benefits from debt-for-nature contracts compared to flow payments (in return for lower deforestation). The debt-for-nature contract combines a loan to the agent with repayments that are contingent on the forest cover.