# Research

## Working Papers

### Incentivizing Agents through Ratings

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 (cost 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]. For example, a school incentivizes learning through a grading policy that reveals students’ quality to the job market.

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]. 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 is optimal within deterministic ratings and solve for the optimal deterministic ratings in general. In particular, when the principal elicits maximal effort (quality), lower censorship [resp. pass/fail] is optimal if the density is unimodal [resp. increasing]. For general ratings, I provide sufficient conditions under which lower censorship remains optimal. In the effort-maximizing case, a pass/fail test remains optimal if the density is increasing.

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 as a mechanism design problem under a feasibility condition. Thus, I rewrite the revenue and consumer welfare maximization problems as linear optimization problems under majorization constraints. Using the techniques in Kleiner, Moldovanu, and Strack (2021), I provide the necessary and sufficient conditions for GW’s Propositions 1, 2, 7, and 8, while allowing for stochastic priority levels. In particular, the trade-off between the provider's revenue and consumer welfare is less stark—if the distribution has a decreasing failure rate (e.g., Weibull distribution) but satisfies a weaker regularity condition, adding more priority levels can increase both. Infinitely many priority levels can be implemented by an all-pay auction.

I revisit Albano and Lizzeri (2001) using an interim information design approach: I model certifications (tests) as Blackwell experiments and characterize the feasibility condition for interim posterior means to be inducible by Blackwell experiments. Thus, I consider an equivalent reduced-form problem where the intermediary designs a feasible interim posterior mean rather than the experiment itself. Transfers are limited to a flat certification fee, so the test is the only channel to provide incentives. I show that the intermediary can maximize revenue by committing to a noisy test. Moreover, the interim approach applies to a more general class of problems, revenue maximization being a special case where the feasibility condition is equivalent to Bayesian plausibility.

I apply (hybrid) Pontryagin’s maximum principles in the optimal control theory to solve delegation problems (e.g., Amador and Bagwell 2013, 2022; Guo 2016), as an alternative to (cumulative) Lagrangian methods developed by Amador, Werning, and Angeletos (2006). In delegation problems with voparticipation (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.