# Bayesian Agency: Linear versus Tractable Contracts

arXiv: https://arxiv.org/abs/2106.00319 · HTML: https://mlsanity.com/paper/2106.00319/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2021.

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

- [The Sample Complexity of Online Contract Design](https://mlsanity.com/paper/2211.05732/) (2022)
- [Contracting with a Learning Agent](https://mlsanity.com/paper/2401.16198/) (2024)
- [Principal-Agent Hypothesis Testing](https://mlsanity.com/paper/2205.06812/) (2022)
- [Deep Contract Design via Discontinuous Networks](https://mlsanity.com/paper/2307.02318/) (2023)
- [Online Learning in a Creator Economy](https://mlsanity.com/paper/2305.11381/) (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)
- [Repeated Contracting with Multiple Non-Myopic Agents: Policy Regret and Limited Liability](https://mlsanity.com/paper/2402.17108/) (2024)
- [The Pseudo-Dimension of Contracts](https://mlsanity.com/paper/2501.14474/) (2025)
- [Optimal Pricing for Data-Augmented AutoML Marketplaces](https://mlsanity.com/paper/2310.17843/) (2023)
- [The Optimal Sample Complexity of Linear Contracts](https://mlsanity.com/paper/2601.01496/) (2026)

## Abstract (arXiv)

We study principal-agent problems in which a principal commits to an outcome-dependent payment scheme (a.k.a. contract) so as to induce an agent to take a costly, unobservable action. We relax the assumption that the principal perfectly knows the agent by considering a Bayesian setting where the agent's type is unknown and randomly selected according to a given probability distribution, which is known to the principal. Each agent's type is characterized by her own action costs and action-outcome distributions. In the literature on non-Bayesian principal-agent problems, considerable attention has been devoted to linear contracts, which are simple, pure-commission payment schemes that still provide nice approximation guarantees with respect to principal-optimal (possibly non-linear) contracts. While in non-Bayesian settings an optimal contract can be computed efficiently, this is no longer the case for our Bayesian principal-agent problems. This further motivates our focus on linear contracts, which can be optimized efficiently given their single-parameter nature. Our goal is to analyze the properties of linear contracts in Bayesian settings, in terms of approximation guarantees with respect to optimal contracts and general tractable contracts (i.e., efficiently-computable ones). First, we study the approximation guarantees of linear contracts with respect to optimal ones, showing that the former suffer from a multiplicative loss linear in the number of agent's types. Nevertheless, we prove that linear contracts can still provide a constant multiplicative approximation ρ of the optimal principal's expected utility, though at the expense of an exponentially-small additive loss 2-Omega(ρ). Then, we switch to tractable contracts, showing that, surprisingly, linear contracts perform well among them.