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Optimal Robust Contract Design

Bo Peng, Zhihao Gavin Tang

Abstract

We consider the robust contract design problem when the principal only has limited information about the actions the agent can take. The principal evaluates a contract according to its worst-case performance caused by the uncertain action space. Carroll (AER 2015) showed that a linear contract is optimal among deterministic contracts. Recently, Kambhampati (JET 2023) showed that the principal's payoff can be strictly increased via randomization over linear contracts. In this paper, we characterize the optimal randomized contract, which remains linear and admits a closed form of its cumulative density function. The advantage of randomized contracts over deterministic contracts can be arbitrarily large even when the principal knows only one non-trivial action of the agent. Furthermore, our result generalizes to the model of contracting with teams, by Dai and Toikka (Econometrica 2022).

Optimal Robust Contract Design

Abstract

We consider the robust contract design problem when the principal only has limited information about the actions the agent can take. The principal evaluates a contract according to its worst-case performance caused by the uncertain action space. Carroll (AER 2015) showed that a linear contract is optimal among deterministic contracts. Recently, Kambhampati (JET 2023) showed that the principal's payoff can be strictly increased via randomization over linear contracts. In this paper, we characterize the optimal randomized contract, which remains linear and admits a closed form of its cumulative density function. The advantage of randomized contracts over deterministic contracts can be arbitrarily large even when the principal knows only one non-trivial action of the agent. Furthermore, our result generalizes to the model of contracting with teams, by Dai and Toikka (Econometrica 2022).
Paper Structure (21 sections, 11 theorems, 82 equations, 1 figure)

This paper contains 21 sections, 11 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model
  3. Robust Contract Design.
  4. Tie-breaking.
  5. Non-triviality Assumption.
  6. Linear Program for Robust Contract Design
  7. Mechanism design counterpart.
  8. Optimal Randomized Contracts
  9. Proof of Lemma \ref{['lem:degenerate']}
  10. Proof of Lemma \ref{['lem:lower']}
  11. Proof of Lemma \ref{['lem:upper']}
  12. Comparison between Optimal Randomized and Deterministic Contracts
  13. Optimal Randomized Linear Contracts for Teams
  14. Productive known technology.
  15. Degenerated case: $\bm{\alpha}^*=\bm{0}$.
  16. ...and 6 more sections

Key Result

Theorem 4.1

The optimal randomized contract achieves an expected payoff of $\frac{\underline{u}(\alpha^*)}{- \ln (1-\alpha^*)}~.$ And it is achieved by a randomized linear contract $w_\alpha(\cdot)$, where $\alpha$ is drawn according to the cumulative distribution function $G^*$.

Figures (1)

  • Figure 1: Ratio between the optimal randomized payoff and the optimal deterministic payoff

Theorems & Definitions (30)

  • Theorem 4.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Theorem : Carroll carroll2015robustness
  • Claim 4.1
  • proof
  • Claim 4.2
  • proof
  • Remark
  • ...and 20 more