Session 9: Dynamic Games, Contracts, and Markets

Motivated by recent concerns surrounding the use of third-party pricing algorithms by competing firms, we study repeated Bertrand competition where market demand or the cost of serving the market is observed by an intermediary (or "algorithm") that optimally discloses demand or cost information to maximize firms' collusive profit. We assume that profit is affine in the unknown state, so expected profit is determined by the expected state. We show that an upper censorship disclosure policy is opti-mal, which leads to price rigidity and supra-monopoly prices at some states. Under a concavity condition, improving the algorithm's accuracy reduces expected consumer surplus. When the state is positively correlated over time, the algorithm discloses more information when recent demand was lower or costs were higher. The analysis extends to a generalized model that accommodates product differentiation and capacity constraints.

I study repeated games with anonymous random matching where players can add or remove sig-nals from their records. The ability to manipulate records introduces monotonicity constraints on players’ continuation values, under which sufficiently long-lived players will almost never cooperate. When players’ expected lifespans are intermediate, their ability to sustain cooperation depends on (i) whether their actions are complements or substitutes and (ii) whether manipulation takes the form of adding or removing signals.

Cartels often form among heterogeneous firms, with large firms acting as leaders and coordinating collusion. Motivated by these instances, I study optimal repeated-game equilibria in a market with a cartel leader and several less patient competitors. Optimal collusive equilibria may involve an initial phase with gradually increasing prices, followed by a stationary phase in which prices are set at the monopoly level and market shares remain constant. These dynamics align with the behavior of several detected cartels, which initially exhibited gradual price increases. Methodologically, I use Lagrangian techniques to characterize Pareto optimal repeated-game equilibria for fixed (and possibly heterogeneous) discount factors.

This paper studies a dynamic screening model in which a principal hires an agent with limited liability. The agent’s private cost of working is an i.i.d. draw from a continuous distribution. His working status is publicly observable. The limited liability constraint requires that payments remain nonnegative at all times. In this setting, despite costs being i.i.d. and the payoffs being additively separable across periods, the optimal mechanism does not treat each period independently. Instead, it features back- loading payments and requires the agent to work in consecutive periods. Specifically, I characterize conditions under which the optimal mechanism either grants the agent flexibility to start working in any period or restricts the starting period to the first. In either case, once the agent begins working, he is incentivized to work consecutively until the end.

We study a dynamic moral hazard problem involving initial exploration followed by exploitation, merging experimentation with dynamic corporate finance. We show how the methods and conclusions of the experimentation literature change when considering the exploitation phase’s non-monotonic payoff structure that arises naturally in the presence of moral hazard and limited liability. In particular, the agent’s incentive constraints may be slack during the exploration phase, which affects compensation dynamics and can reduce inefficiencies from under-experimentation.

We analyze how uncertain technologies should be robustly regulated, and how regulation should respond to evolving information. An adaptive sandbox comprising of a zero marginal tax on R\&D up to an evolving hard limit is (i) robust: it delivers optimal payoff guarantees when nature chooses the agent's learning process and/or preferences adversarially; (ii) dominant: it outperforms other robust mechanisms evaluated at any learning process and any agent preference; (iii) time-consistent: it is the unique mechanism that can be implemented without commitment; and (iv) important: absent hard limits, worst-case payoffs can be arbitrarily poor and is induced by weak but growing optimism that encourages excessive risk-taking. Our results offer optimality foundations for existing policy as well as guidance for implementation.

This paper studies dynamic mechanisms with Markovian private information both in transferable utility (TU) and non-transferable utility (NTU) settings. We propose the Guaranteed Utility Mechanism (GUM) that is efficient and collusion-proof in the TU setting, and is approximately so in the NTU setting (independently of the type space size). Moreover, GUM does not suffer from key shortcomings of existing (approximately) efficient mechanisms that we iden-tify: (i) in the TU setting, we show that in canonical efficient mechanisms, such as the Balanced Team Mechanism and the Dynamic Pivot Mechanism, none of efficient equilibria might survive iterative elimination of weakly dominated strategies; (ii) in the NTU setting, approaches from “linking” and “review” mechanisms cannot be used for infinite type spaces. In both settings, GUM satisfies participation constraints, permits observability of past types, and accommodates private actions.

Moral universalists have moral principles that do not depend on whom they interact with (e.g., they are caring toward everyone). Moral particularists have moral principles that do depend on whom they interact with (e.g., they are caring only toward members of their in-group). This paper provides a framework grounded in evolutionary game theory to determine whether universalism or particularism is more likely to prevail in a society. The success of a moral principle depends on whether the principle possesses either or both of the following attributes: (i) resistance to invasion from mutant (outsider) moral principles and (ii) the ability to invade incumbent moral principles. Most of the existing literature has focused on the first attribute, which is captured by the standard solution concept of evolutionary stability. This paper’s innovation is a solution concept that strengthens evolutionary stability by accommodating both attributes. I find that whether universalism or particularism is more likely to prevail depends on the probability with which interactions are repeated. In environments in which the probability of repetition is low, universalism is more likely to emerge, whereas in environments in which this probability is high, particularism is more likely. I further show that societies with a low probability of repeated interactions exhibit more cooperation, more moral diversity, and lesser satisfaction from prosocial behavior. My theoretical results are consistent with existing empirical findings, which I discuss.

This paper develops a dynamic model of information acquisition in which the decision maker lacks control over the timing of her actions. Dynamic experiments are flexible, con-strained only by the quantity of information they can generate. The optimal experiment produces a single piece of breakthrough evidence and follows a cost-efficiency principle: willingness to pay for information continuously decreases over time. We explore the dy-namics of learning in two important environments: a known deadline, and a random de-cision time with a constant hazard rate. The presence of timing risk forces the decision maker to generate contrarian breakthroughs, pitting the same set of alternatives against each other until sufficient evidence accumulates to rule out one. This limits the breadth of learning relative to the case where the timing is known. Over time, interim-optimal actions become monotonically more extreme, while breakthroughs become increasingly rare and more impactful.

We study dynamic buyer-seller interactions in an imperfectly competitive market with search frictions. Prices are publicly observable, but consumers must engage in costly search to discover their firm-specific match value. In equilibrium, firms initially sell for free to attract consumers, but as search frictions gradually weaken outside options, firms gain market power and randomly hike their prices. This process generates persistent price dispersion and firm size heterogeneity, even when firms and consumers are ex ante identical. Our model explains staggered price hikes, long-run price differences, and endogenous firm growth patterns.

An agent engages in sequential search. He does not directly observe the quality of the goods he samples, but he can purchase signals designed by profit maximizing principal(s). We formulate the principal-agent relationship as a repeated contracting problem within a stopping game, and characterize the set of equilibrium payoffs. We show that when the agent’s search cost falls below a given threshold, competition does not impact how much surplus is generated in equilibrium nor how the surplus is divided. In contrast, competition benefits the agent at the expense of total surplus when the search cost exceeds that threshold. Our results challenge the view that monopoly decreases market efficiency, and moreover, suggest that it leads to more information provision than does competition.

We consider an alternating-offers redistributive bargaining model where an affected third party can protest against proposals under review. Protests are costly and only stochastically successful. When successful, they secure the status quo. Stationary equilibria feature either inefficient protests or excessive accommodation to the third party. In both cases, the bargainers do not extract the full surplus. Strategic delay is necessary and sufficient to curb this issue: By delaying a harmful agreement with positive probability only after acquiescence, the bargainers create an endogenous punishment device that allows them to extract more surplus from the third party without triggering protests. The bargainers’ misaligned interests are key for this result: If they internalized each other’s payoff, strategic delay would not be credible.

We analyze the dynamic problem of decision makers with quasi-hyperbolic discounting and random shocks to temptation. We show that this problem is equivalent to that of a standard consumer-saver who assigns biased weights to future shocks. This equivalence provides a straightforward methodology for finding, theoretically and numerically, the Markov equilibrium with hyperbolic agents. Through this equivalent problem, we provide conditions for the existence and uniqueness of the equilibrium. If the weights constitute a probability measure, the decision maker can be interpreted as optimistically biased, ensuring a unique equilibrium with continuous decision rules and implying no value for commitment devices. Otherwise, there is intertemporal “conflict” between present and future selves: if the conflict is limited, uniqueness is guaranteed.

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 play-ing 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. An application of our results provides a reputational foundation for a class of Bayesian persuasion solutions when the sender has a small lying cost. Our results extend to the case where distinct commitment types may be indistinguishable but the Stackelberg type is salient under the prior.

In many settings, individuals imitate their peers’ public decisions for one or both of two reasons: to adapt to a common fundamental state, and to conform to their peers’ preferences. In this model, the fundamental state and peers’ preferences are unknown, and the players learn these random variables by observing others’ decisions. With each additional decision, the public beliefs about these unknowns become more precise. This increased precision endogenously increases the desire to conform and can result in decisions that are uninformative about a player’s preferences or perceptions of the fundamental state. When this occurs, social learning about peers’ preferences and fundamentals ceases prematurely, resulting in inefficient decisions. In line with findings from social psychology, I identify settings where interventions aimed at correcting misperceptions of the fundamental state have no effect but interventions aimed at correcting misperceptions of peers’ preferences lead to more efficient decision-making.

We introduce a dynamic model in which a developer incrementally improves a product of uncertain quality over time, with the quality evolving as a controlled Brownian motion. At each moment in time, the developer can continue exploring by paying a flow cost, restart from a previously attained quality level by paying a fixed cost, or terminate the process by either freely abandoning the project or by incurring a cost to launch the highest quality observed so far. The optimal strategy is characterized by a free boundary of an impulse-controlled Brownian motion, reflecting how the developer’s tolerance for setbacks evolves over time in three distinct stages. In the early stage, setbacks lead to project termination; in the intermediate stage, unsuccessful paths serve as learning opportunities, leading to strategic restarts; and in the final stage, the product is inevitably launched, with further improvements enhancing its final quality in a final push. Our model captures the role of active decision-making in managing uncertainty through a true process of trial and error that can be reversed when necessary. The results highlight the essential interplay of persistence and chance, demonstrating that success hinges not only on avoiding prolonged failure but also on the precise timing of interventions.

This paper investigates the impact of artificial intelligence on the interaction between firms and consumers. It focuses on the use of learning algorithms in environments with strategic consumers — where learning must occur in the face of consumers who best-respond and adapt their behavior. An algorithm is transparent if consumers observe its inputs, whereas it is opaque if consumers do not observe its inputs. The main results are as follows. First, opaque algorithms perform better for the firm than transparent ones. In contrast to a transparent algorithm, an opaque algorithm learns the optimal policy and maximizes long-run profits. Second, opaque algorithms outperform transparent ones in terms of consumer welfare in important applications. That is, consumers may benefit from having less information about the algorithm’s inputs. Third, whether the firm benefits from using an algorithm instead of behaving strategically depends on consumers’ information about the algorithm’s inputs. When the algorithm is opaque, it yields higher payoffs than a fully strategic firm.

We study the optimal placement of advertisements for interactive platforms like conversational AI assistants. Importantly, conversations add a feature absent in canon-ical search markets — time. The evolution of a conversation is informative about ad qualities, thus a platform could delay ad delivery to improve selection. However, de-lay endogenously shapes the supply of quality ads, possibly affecting revenue. We characterize the equilibria of first- and second-price auctions where the platform can commit to the auction format but not to its timing. We document sharp differences in the mechanisms’ outcomes: first-price auctions are efficient but delay ad delivery, while second-price auctions avoid delay but allocate inefficiently. Revenue may be ar-bitrarily larger in a second-price auction than in a first-price auction. Optimal reserve prices alleviate these differences but flip the revenue ordering.