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Linear Contracts in Multitasking: Robustness, Uniformity, and Learning

Shiliang Zuo

TL;DR

This paper analyzes linear contracts in multitask principal–agent problems from robustness, uniformity, and learning perspectives. It proves that linear contracts are worst-case optimal under first-moment ambiguity, derives a uniform optimal contract form when costs are homogeneous and principal utility is linear, and develops instrumental-regression methods (GMM) to learn contracts from observational data in offline and online settings. It establishes regret bounds for online learning, including $\widetilde{O}(d\sqrt{T})$ in general and potential improvements under repeated observations and agent diversity. The work bridges contract theory with econometric learning, offering practical guidance for designing uniform, learnable contracts in platforms and franchising contexts. Overall, it advances the understanding of when simple linear contracts suffice and how to learn them from data in multitask environments.

Abstract

In this work, we study the multitasking principal-agent problem. The agent performs several task for the principal, and the principal posts a contract incentivizing the agent to exert effort. The principal can observe a signal for each task, and the contract is a mapping from the space of possible signals to a payment. We study the special class of linear contracts from three perspectives: robustness, uniformity, and learning. Firstly, we show a robustness result: in an ambiguous setting when only first moment information is known, there is a linear contract maximizing the principal's payoff in a worst-case scenario. Secondly, we show a uniformity result: when the agent's cost function is homogeneous to a certain degree and the the principal's utility takes a linear form across tasks, then the optimal contract depends on the agent's cost function only through its homogeneuity degree. Thirdly, we study the problem of learning an optimal linear contract through observational data. We identify this as an measurement error model, and propose instrumental regression methods to estimate the optimal contract parameters in an offline setting, or to learn the optimal contract in an online setting.

Linear Contracts in Multitasking: Robustness, Uniformity, and Learning

TL;DR

This paper analyzes linear contracts in multitask principal–agent problems from robustness, uniformity, and learning perspectives. It proves that linear contracts are worst-case optimal under first-moment ambiguity, derives a uniform optimal contract form when costs are homogeneous and principal utility is linear, and develops instrumental-regression methods (GMM) to learn contracts from observational data in offline and online settings. It establishes regret bounds for online learning, including in general and potential improvements under repeated observations and agent diversity. The work bridges contract theory with econometric learning, offering practical guidance for designing uniform, learnable contracts in platforms and franchising contexts. Overall, it advances the understanding of when simple linear contracts suffice and how to learn them from data in multitask environments.

Abstract

In this work, we study the multitasking principal-agent problem. The agent performs several task for the principal, and the principal posts a contract incentivizing the agent to exert effort. The principal can observe a signal for each task, and the contract is a mapping from the space of possible signals to a payment. We study the special class of linear contracts from three perspectives: robustness, uniformity, and learning. Firstly, we show a robustness result: in an ambiguous setting when only first moment information is known, there is a linear contract maximizing the principal's payoff in a worst-case scenario. Secondly, we show a uniformity result: when the agent's cost function is homogeneous to a certain degree and the the principal's utility takes a linear form across tasks, then the optimal contract depends on the agent's cost function only through its homogeneuity degree. Thirdly, we study the problem of learning an optimal linear contract through observational data. We identify this as an measurement error model, and propose instrumental regression methods to estimate the optimal contract parameters in an offline setting, or to learn the optimal contract in an online setting.
Paper Structure (37 sections, 18 theorems, 70 equations, 1 figure, 2 algorithms)

This paper contains 37 sections, 18 theorems, 70 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Robustness.
  3. Uniformity.
  4. Learning
  5. Additional Related Work
  6. Contract Theory
  7. Instrumental Regression and Measurement Error Models
  8. Bandits and Exploration-Exploitation Tradeoff
  9. The Model
  10. Affine Contracts
  11. A Preliminary Lemma
  12. Robustness in Ambiguous settings
  13. Proof Sketch
  14. From a Non-affine Contract to an Affine Contract.
  15. A Family of Affine Contract via Upper Supporting Hyperplanes.
  16. ...and 22 more sections

Key Result

Lemma 1

Suppose the agent's action set is downward-closed, meaning if $a\in \mathcal{A}$, then $\rho a \in \mathcal{A}$ for some $\rho \in (0,1)$. Among all non-negative linear or affine contracts that induce some non-zero action $a$, the linear contract $\beta = \nabla c(a)$ induces $a$ at least expected p

Figures (1)

  • Figure 1: Causal relationship between variables. The agent's response is shaded indicating it is unobserved by the principal. The dashed item is a second set of observed signals, corresponding the the scenario studied in \ref{['sec:repeated']} when repeated observations are available.

Theorems & Definitions (36)

  • Lemma 1: Lemma 9 in roth2016watch
  • Definition 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 26 more