Session 5: Dynamic Games, Contracts, and Markets

An organization is autonomous if it has the right or power of self-government. Self-government implies that autonomous organizations cannot rely on outside parties for monitoring or contract enforcement. We present a model of the optimal power allocation in such an organization. The organization commits to a governance structure that allocates managerial power to agents. Members with power (“managers”) can punish members without power (“subordinates”). This power is, however, limited by the subordinates’ right to exit the organization. There are three main results. First, the goals of autonomy, decentralization, and efficiency conflict with one another. We call this result the Organizational Trilemma. Second, there is a Paradox of Power: an agent can be made worse off by their own power. Third, optimal governance structures in autonomous organizations are centralized and populist: the powerful party shows restraint in early periods, only to abuse their power in later periods.

We model search in unfamiliar territories, where agents know what can be found but not where to find it. A searcher faces a set of choice arranged by an observable attribute. Each period, she either selects a choice and pays a cost to learn about its quality, or she concludes search to take her best discovery to date. She knows that similar choices have similar qualities and uses this to guide her search. We identify robustly optimal search policies with a simple structure. Search is ordered, perfect recall is never invoked, stopping follows a threshold rule, and the policy at each history depends only on a simple index.

We study a bilateral trade problem where a principal has private information that is revealed with delay, such as a seller who does not yet know her production cost. Postponing the contracting process incurs a costly delay, while early contracting with limited information can create incentive issues, as the principal might misrepresent private information that will be revealed later. We show that the optimal mechanism can effectively address these challenges by leveraging the sequential nature of the problem. The optimal mechanism is a menu of two-part tariffs, where the variable part is determined by the principal's incentives and the fixed part by the agent's incentives. As two-part tariffs might be impractical in some applications, we also study price mechanisms. We show that the optimal price mechanism often entails trade at both the ex-ante and ex-post stages. Dynamic price mechanisms can lower the cost of delay by transacting with high-type agents early and relax the incentive constraints by postponing contracts with lower-type agents. We also generalize our analysis to costly learning and study ex-post efficiency in our context.

Pricing decisions are crucial for managing a firm’s reputation and maximizing profits. Consumer reviews reflect both the product   and its price, with more favorable reviews being left when a product is priced lower. We study whether such review behavior can induce a firm to manipulate the review process by underpricing its product, or pricing it below current consumers’ willingness to pay. We introduce an equilibrium model with a privately informed firm repeatedly selling its product to uninformed but rational consumers who learn about the quality of the product from past reviews and current prices. We show that underpricing can arise only when the firm reputation is low and then only under a specific condition on consumers’ taste shock distribution, which we fully characterize. Rating manipulation unambiguously benefits consumers, because it operates via underpricing.

We analyze the dynamic tradeoff between the generation and the disclosure of evidence.

Agents are tempted to delay investing in a new technology in order to learn from information generated by the experiences of others. This informational free-riding is collectively harmful as it slows down innovation adoption. A welfare-maximizing designer can delay the disclosure of previously generated information in order to speed up adoption.

The optimal policy transparently discloses bad news and delays good news. This finding resonates with regulation demanding that fatal breakdowns be reported promptly. Remarkably, the designer's intervention makes all agents better off.

I study a dynamic disclosure game in which an agent controls the time window over which information flows to the decision maker, but does not control the content of that information. In equilibrium, the agent has incentives to delay the start of disclosure to continue to learn privately for some time. This delay exacerbates the information asymmetry between the agent and the decision maker as the agent is learning while the decision maker is not. This information asymmetry is then (partially) resolved during the disclosure window. The length of the disclosure window is determined by the degree of information asymmetry at the beginning of the window, with longer windows associated with greater information asymmetry.

We examine the strategic interaction between an expert (principal) maximizing engagement and an agent seeking swift information. Our analysis reveals: When priors align, relative patience determines optimal disclosure—impatient agents induce gradual revelation, while impatient principals cause delayed, abrupt revelation. When priors disagree, catering to the bias often emerges, with the principal initially providing signals aligned with the agent’s bias. With private agent beliefs, we observe two phases: one engaging both agents, followed by catering to one type. Comparing personalized and non-personalized strategies, we find faster information revelation in the non-personalized case, but higher quality information in the personalized case.

The allocation of services and goods often involves both stochastic supply and demand, a feature not captured by the classic auction model. Motivated by applications such as cloud computing, gig platforms, and blockchain auctions, we study the optimal design of mechanisms in an M/M/1 queueing model where buyers have private valuations and incur waiting costs. We derive the dynamically optimal screening mechanism, which strategically manages buyer participation and competition through a reserve price that increases with queue length and an auction to allocate the good. The mechanism balances efficiency and revenue, offering insights into the design of queueing systems in various settings where supply and demand fluctuate over time.

We consider the fixed points of nonlinear operators that naturally arise in games and general equilibrium models with endogenous networks, dynamic stochastic games, in models of opinion dynamics with stubborn agents, and financial networks. We study limit cases that correspond to high coordination motives, infinite patience, vanishing stubbornness and small exposure to the real sector in the applications above. Under monotonicity and continuity assumptions, we provide explicit expressions for the limit fixed points. We show that, under differentiability, the limit fixed point is linear in the initial conditions and characterized by the Jacobian of the operator at any constant vector with an explicit and linear rate of convergence. Without differentiability, but under additional concavity properties, the multiplicity of Jacobians is resolved by a representation of the limit fixed point as a maxmin functional evaluated at the initial conditions. In our applications, we use these results to characterize the limit equilibrium actions, prices, and endogenous networks, show the existence and give the formula of the asymptotic value in a class of zero-sum stochastic games with a continuum of actions, compute a nonlinear version of the eigenvector centrality of agents in networks, and the characterize the equilibrium loss evaluations in financial networks.

For common knowledge to arise in dynamic settings, knowledge that it has arisen must be attained simultaneously by all players. Consequently, common knowledge cannot arise in many realistic settings with timing frictions. This counterintuitive observation of Halpern and Moses (1990) was discussed by Arrow et al. (1987) and Aumann (1989), was called a paradox by Morris (2014), and has evaded satisfactory resolution for four decades. We resolve this paradox by proposing a new definition for common knowledge, which coincides with the traditional one in static settings but is more permissive in dynamic settings. Under our definition, common knowledge can arise without simultaneity, particularly in canonical examples of the Haplern–Moses paradox. We demonstrate its usefulness by deriving for it an agreement theorem `a la Aumann (1976), showing it arises in the setting of Geanakoplos and Polemarchakis (1982) with timing frictions added, and applying it to characterize equilibrium behavior in a dynamic coordination game.

Consider a population of agents who repeatedly compete for awards, as in the case of researchers annually applying for grants. Noise in the selection process may encourage entry of low quality proposals, forcing the principal to commit large resources to reviewing applications and further increasing award misallocation. A temporary exclusion policy prohibits an agent from applying in the current period if they were rejected in the previous, encouraging self-selection. We compare the steady-state equilibria of the games with and without exclusion. Whenever the benefit from winning is large, exclusion leads to fewer low-quality applications and higher welfare for agents.

We examine the strategic interaction between an expert (principal) maximizing engagement and an agent seeking swift information. Our analysis reveals: When priors align, relative patience determines optimal disclosure—impatient agents induce gradual revelation, while impatient principals cause delayed, abrupt revelation. When priors disagree, catering to the bias often emerges, with the principal initially providing signals aligned with the agent’s bias. With private agent beliefs, we observe two phases: one engaging both agents, followed by catering to one type. Comparing personalized and non-personalized strategies, we find faster information revelation in the non-personalized case, but higher quality information in the personalized case.

A worker interacts with a sequence of clients under a manager’s supervision. I highlight a novel role of this manager’s mediation in addressing the worker’s moral hazard, namely to intertemporally reduce suspensions of the worker’s service that are surplus-depleting but crucially serve as punishments to motivate her costly effort. I show that, to best address moral hazard, the manager at times secretly asks a high-performing worker to scale down her effort against a current client and implements dynamic correlation by telling the worker that current underperformance will not be punished. These occasions are frequent in the short run and eventually disappear.

We develop a dynamic contracting theory of asset- and cash flow-based financing that demonstrates how firm, intermediary, and capital market characteristics shape firms’ financing constraints. A firm with imperfect access to equity financing covers financing needs through costly sources — an intermediary and retained cash. The firm’s financing capacity is endogenously determined by either the liquidation value of assets (asset-based) or the intermediary’s going-concern valuation of the firm’s cash flows (cash flow-based). We implement the optimal contract between the firm and intermediary with both unsecured and secured debt (credit lines) in an overlapping pecking order: the firm simultaneously finances cash flow shortfalls with unsecured debt and either cash reserves (if available) or secured debt (otherwise). Improved access to equity financing increases debt capacity, thus debt and equity are dynamic complements. When the firm does well, it repays its debt in full, while when in distress, repayment dynamics mirror U.S. bankruptcy procedures (Chapter 7 vs 11).

We introduce a tractable methodology for analyzing dynamic decision problems and games involving irreversible decisions under uncertainty. By leveraging regularity properties, our approach offers an intuitive method for solving these problems using equivalent certain values, and derives properties and comparative statics of the solution. We show that irreversibility is analogous to information loss, leading agents to act as if they had worse information than with reversible actions. We use our methodology to analyze design features of previously intractable long auctions, establish revenue equivalence, and show that increasing bidding opportunities or allowing bid retraction can harm bidders and benefit the auctioneer.

Collective action is a dynamic process where individuals in a group assess over time the benefits and costs of participating toward the success of a collective goal. Early participation improves the expectation of success and thus stimulates the subsequent participation of other individuals who might otherwise be unwilling to engage. On the other hand, a slow start can depress expectations and lead to failure for the group. Individuals have an incentive to procrastinate, not only in the hope of free riding, but also in order to observe the flow of participation by others, which allows them to better gauge whether their own participation will be useful or simply wasted. How do these phenomena affect the probability of success for a group? As the size of the group increases, will a “power of large numbers” prevail producing successful outcomes, or will a “curse of large numbers” lead to failure? In this paper, we address these questions by studying a dynamic collective action problem in which n individuals can achieve a collective goal if a share αn of them takes a costly action (e.g., participate in a protest, join a picket line, or sign an environmental agreement). Individuals have privately known participation costs and decide over time if and when to participate. We characterize the equilibria of this game and show that under general conditions the eventual success of collective action is necessarily probabilistic. The process starts for sure, and hence there is always a positive probability of success; however, the process “gets stuck” with positive probability, in the sense that participation stops short of the goal. Equilibrium outcomes have a simple characterization in large populations: welfare converges to either full efficiency or zero depending on whether αn converges to zero faster or slower than the cube root of 1/n as n → ∞. Whether success is achievable or not, delays are always irrelevant: in the limit, either success is achieved either instantly or never.