# Research

## Working Papers

I study the optimal design of performance or product ratings to motivate agents’ performance or investment in product quality. The principal designs a rating that maps their quality (performance) to possibly stochastic scores. Agents have private information about their abilities (costs of effort/quality) and choose their quality. The market observes the scores and offers a wage equal to the agent’s expected quality [resp. ability].

I first show that an incentive-compatible interim wage function can be induced by a rating (i.e., feasible) if and only if it is a mean-preserving spread of quality [resp. ability] in the quantile space. Thus, I reduce the principal's rating design problem to the design of a feasible interim wage. When restricted to deterministic ratings, the optimal rating design is equivalent to the optimal delegation with participation constraints (Amador and Bagwell, 2022). Using optimal control theory, I provide necessary and sufficient conditions under which lower censorship, especially a simple pass/fail test, is optimal within deterministic ratings. In particular, when the principal elicits maximal effort (quality), lower censorship [resp. pass/fail] is optimal if the density is unimodal [resp. increasing]. I also solve for the optimal deterministic ratings beyond lower censorship for general distributions and preferences. For general ratings, I provide sufficient conditions under which lower censorship remains optimal. In the effort maximization case, a pass/fail test remains optimal if the density is increasing.

### Allocating Positional Goods: A Mechanism Design Approach

Consumers of positional goods care about their relative positions in the consumption of the goods, so allocating an item to one buyer has externalities on others. Using a mechanism design approach, I characterize the externalities by a feasibility condition. Under Myerson’s regularity, the revenue-maximizing mechanism excludes some low types and fully separates the rest. The seller can guarantee at least half the maximal revenue by offering one level of positional goods, and the approximation can be arbitrarily close if the buyer’s type distribution is sufficiently concave. Moreover, total pooling (full separation) and no exclusion maximize the consumer surplus if the distribution has an increasing (decreasing) failure rate. Applications include educational arms races, priority services, and luxury goods.

We study the optimal design of a two-player tournament in which one player has discretion over hiring the other. The manager hires an agent of a certain ability and competes with him in a Lazer-Rosen-style tournament. In the tournament, both players produce at heterogeneous marginal costs (abilities), and the one with higher output wins a fraction of the total output. The principal determines the payout ratio and the head start (or handicap) to the manager—an advantage (or disadvantage) when comparing output. We find the optimal contract offers just enough head start to induce the manager to hire the best candidate. However, in a two-period model where the first-period winner is retained for the future, the principal with succession concerns may allow hiring sabotage to prevail in equilibrium but will ensure the new hire has a higher ability than the manager.

## Notes and Comments

I reformulate Gershkov and Winter’s (2023) model of priority services as a mechanism design problem under a feasibility condition. Thus, I provide the necessary and sufficient conditions for their Propositions 1, 2, 7, and 8, while allowing for stochastic priority levels. Under the weaker conditions, adding more priority levels can increase both the provider’s revenue and consumer welfare if the cost distribution has a decreasing failure rate but satisfies Myerson’s regularity. Full separation can be implemented by an all-pay auction. I also show that the provider can guarantee at least half the maximal revenue by offering one priority service in addition to a (free) regular service, and the approximation can be arbitrarily close if the distribution is sufficiently concave. The approach can also be applied to Moldovanu, Sela, and Shi's (2007) model of status contests.

I apply (hybrid) Pontryagin’s maximum principles in the optimal control theory to solve delegation problems (in particular, Amador and Bagwell 2013, 2022), as an alternative to (cumulative) Lagrangian methods developed by Amador, Werning, and Angeletos (2006). In delegation problems with voluntary participation (Amador and Bagwell 2022), where the participation constraint leads to a jump in allocation, this approach makes it possible to study the global problem (instead of the truncated problem) and thus provide necessary and sufficient conditions that allow a bang-bang solution (either a given action or opt-out) to be optimal.

I study how a monopoly certifier designs and prices the quality certification to maximize revenue when agents’ quality is endogenous. The certifier designs the quality certification (i.e., Blackwell experiment) and charges agents a flat certification fee. Agents with heterogeneous costs choose their quality (and whether to take the test) and receive a payment equal to the expected quality conditional on the test result by the competitive market. By characterizing the feasibility condition for interim posterior means, I show the certifier can maximize revenue by committing to a noisy test. The interim approach applies to a more general class of problems, revenue maximization being a special case where feasibility is equivalent to Bayesian plausibility.