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Approximate Optimality of Linear Contracts Under Uncertainty

Tal Alon, Paul Dütting, Yingkai Li, Inbal Talgam-Cohen

TL;DR

It is shown that linear contracts are near-optimal whenever there is enough uncertainty, and other simple contract formats such as debt contracts may suffer from a loss linear in the number of possible actions, even when there is sufficient uncertainty.

Abstract

We consider a hidden-action principal-agent model, in which actions require different amounts of effort, and the agent privately knows his ability that determines his cost of effort. We show that linear contracts admit approximation guarantees that improve with a natural metric that captures the degree of uncertainty in the contracting setting. We thus show that linear contracts are near-optimal whenever there is enough uncertainty. In contrast, other simple contract formats such as debt contracts may suffer from a loss linear in the number of possible actions, even when there is sufficient uncertainty.

Approximate Optimality of Linear Contracts Under Uncertainty

TL;DR

It is shown that linear contracts are near-optimal whenever there is enough uncertainty, and other simple contract formats such as debt contracts may suffer from a loss linear in the number of possible actions, even when there is sufficient uncertainty.

Abstract

We consider a hidden-action principal-agent model, in which actions require different amounts of effort, and the agent privately knows his ability that determines his cost of effort. We show that linear contracts admit approximation guarantees that improve with a natural metric that captures the degree of uncertainty in the contracting setting. We thus show that linear contracts are near-optimal whenever there is enough uncertainty. In contrast, other simple contract formats such as debt contracts may suffer from a loss linear in the number of possible actions, even when there is sufficient uncertainty.
Paper Structure (34 sections, 16 theorems, 108 equations, 1 figure)

This paper contains 34 sections, 16 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. A Principal-Agent Model
  5. Types and utilities.
  6. Contracts
  7. Incentives.
  8. The principal's objective.
  9. Near-Optimality of Linear Contracts
  10. Approximation guarantee.
  11. Proof outline.
  12. Approximation for Slowly-Increasing Distributions
  13. Additional Discussions
  14. Discussion of the thin-tail parameterization.
  15. Interpretation of approximation results.
  16. ...and 19 more sections

Key Result

Theorem 1

Let $q,\alpha\in(0,1),\eta\in(0,1-q)$ and suppose the agent's cost for quantile $q$ is $c_q$. For any principal-agent instance with $(\frac{c_q}{\alpha},\eta)$-thin-tail, a linear contract with parameter $\alpha$ provides expected revenue that is an $\frac{1}{(1-\alpha)\eta q}$-approximation of the

Figures (1)

  • Figure 1: For fixed $\alpha$, say $\alpha = 1/2$, we plot (black curve) the fraction $\eta$ of welfare from types above $c_q/\alpha = 2 c_q$ for $q \in [0,1]$. This curve is decreasing, and the intercept with the $y$-axis is typically smaller than $1$ because of the division of $c_q$ by $\alpha$. The best approximation guarantee that can be obtained with a given $\alpha$ is proportional to the area $\eta q$ of the largest box that fits under the curve (gray rectangle). The larger this area the further away the setting from being point mass-like, and the better the approximation.

Theorems & Definitions (63)

  • Definition 1
  • Definition 2
  • Definition 3: Thin-tail parameterization
  • Theorem 1
  • Definition 4: Slowly-increasing distribution
  • Theorem 2
  • proof : Proof of \ref{['thm:universal']}
  • Lemma 1
  • Corollary 1
  • Example 1
  • ...and 53 more