# Contracts with Private Cost per Unit-of-Effort

arXiv: https://arxiv.org/abs/2111.09179 · HTML: https://mlsanity.com/paper/2111.09179/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Nov 2021.

## Built on by (most cited first, 12 of 12)

- [The Sample Complexity of Online Contract Design](https://mlsanity.com/paper/2211.05732/) (2022)
- [Principal-Agent Hypothesis Testing](https://mlsanity.com/paper/2205.06812/) (2022)
- [Delegating Data Collection in Decentralized Machine Learning](https://mlsanity.com/paper/2309.01837/) (2023)
- [Learning to Incentivize Information Acquisition: Proper Scoring Rules Meet Principal-Agent Model](https://mlsanity.com/paper/2303.08613/) (2023)
- [Are Bounded Contracts Learnable and Approximately Optimal?](https://mlsanity.com/paper/2402.14486/) (2024)
- [Learning Optimal Contracts: How to Exploit Small Action Spaces](https://mlsanity.com/paper/2309.09801/) (2023)
- [Contractual Reinforcement Learning: Pulling Arms with Invisible Hands](https://mlsanity.com/paper/2407.01458/) (2024)
- [Fair Contracts in Principal-Agent Games with Heterogeneous Types](https://mlsanity.com/paper/2506.15887/) (2025)
- [Optimal Pricing for Data-Augmented AutoML Marketplaces](https://mlsanity.com/paper/2310.17843/) (2023)
- [Provably Efficient Algorithm for Best Scoring Rule Identification in Online Principal-Agent Information Acquisition](https://mlsanity.com/paper/2505.17379/) (2025)
- [Scalable Principal-Agent Contract Design via Gradient-Based Optimization](https://mlsanity.com/paper/2510.21177/) (2025)
- [The Optimal Sample Complexity of Linear Contracts](https://mlsanity.com/paper/2601.01496/) (2026)

## Abstract (arXiv)

Economic theory distinguishes between principal-agent settings in which the agent has a private type and settings in which the agent takes a hidden action. Many practical problems, however, involve aspects of both. For example, brand X may seek to hire an influencer Y to create sponsored content to be posted on social media platform Z. This problem has a hidden action component (the brand may not be able or willing to observe the amount of effort exerted by the influencer), but also a private type component (influencers may have different costs per unit-of-effort). This "effort" and "cost per unit-of-effort" perspective naturally leads to a principal-agent problem with hidden action and single-dimensional private type, which generalizes both the classic principal-agent hidden action model of contract theory `a la Grossman and Hart [1983] and the (procurement version) of single-dimensional mechanism design `a la Myerson [1981]. A natural goal in this model is to design an incentive-compatible contract, which consist of an allocation rule that maps types to actions, and a payment rule that maps types to payments for the stochastic outcomes of the chosen action. Our main contribution is a linear programming (LP) duality based characterization of implementable allocation rules for this model, which applies to both discrete and continuous types. This characterization shares important features of Myerson's celebrated characterization result, but also departs from it in significant ways. We present several applications, including a polynomial-time algorithm for finding the optimal contract with a constant number of actions. This is in sharp contrast to recent work on hidden action problems with multi-dimensional private information, which has shown that the problem of computing an optimal contract for constant numbers of actions is APX-hard.