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Regret-Minimizing Contracts: Agency Under Uncertainty

Martino Bernasconi, Matteo Castiglioni, Alberto Marchesi

TL;DR

The paper studies principal-agent contracts under cost uncertainty without assuming a prior, modeling uncertainty via an uncertainty set \\mathcal{C} and optimizing contracts to minimize additive regret against all costs. It establishes that deterministic contracts suffice for worst-case regret minimization with tight \\mathcal{O}(\\sqrt{d(\\mathcal{C})}) bounds and a 1/2-\\text{H\\ölder} continuity in the uncertainty level, while randomized constructs can outperform deterministic ones in some instances. Computationally, it proves APX-hardness in general and introduces a general epsilon-cover template that yields polynomial-time algorithms for finite, single-dimensional, and L_p-uncertainty cases, by solving reduced RM-contract problems and applying a linearization step. The framework unifies existence and computational results across contract classes (deterministic, randomized, and menus) and provides practical instantiations to design regret-aware contracts in robust principal-agent settings with no distributional information, with potential applications to crowdsourcing, blockchain, and healthcare.

Abstract

We study the fundamental problem of designing contracts in principal-agent problems under uncertainty. Previous works mostly addressed Bayesian settings in which principal's uncertainty is modeled as a probability distribution over agent's types. In this paper, we study a setting in which the principal has no distributional information about agent's type. In particular, in our setting, the principal only knows some uncertainty set defining possible agent's action costs. Thus, the principal takes a robust (adversarial) approach by trying to design contracts which minimize the (additive) regret: the maximum difference between what the principal could have obtained had them known agent's costs and what they actually get under the selected contract.

Regret-Minimizing Contracts: Agency Under Uncertainty

TL;DR

The paper studies principal-agent contracts under cost uncertainty without assuming a prior, modeling uncertainty via an uncertainty set \\mathcal{C} and optimizing contracts to minimize additive regret against all costs. It establishes that deterministic contracts suffice for worst-case regret minimization with tight \\mathcal{O}(\\sqrt{d(\\mathcal{C})}) bounds and a 1/2-\\text{H\\ölder} continuity in the uncertainty level, while randomized constructs can outperform deterministic ones in some instances. Computationally, it proves APX-hardness in general and introduces a general epsilon-cover template that yields polynomial-time algorithms for finite, single-dimensional, and L_p-uncertainty cases, by solving reduced RM-contract problems and applying a linearization step. The framework unifies existence and computational results across contract classes (deterministic, randomized, and menus) and provides practical instantiations to design regret-aware contracts in robust principal-agent settings with no distributional information, with potential applications to crowdsourcing, blockchain, and healthcare.

Abstract

We study the fundamental problem of designing contracts in principal-agent problems under uncertainty. Previous works mostly addressed Bayesian settings in which principal's uncertainty is modeled as a probability distribution over agent's types. In this paper, we study a setting in which the principal has no distributional information about agent's type. In particular, in our setting, the principal only knows some uncertainty set defining possible agent's action costs. Thus, the principal takes a robust (adversarial) approach by trying to design contracts which minimize the (additive) regret: the maximum difference between what the principal could have obtained had them known agent's costs and what they actually get under the selected contract.
Paper Structure (42 sections, 55 theorems, 111 equations, 3 figures, 1 algorithm)

This paper contains 42 sections, 55 theorems, 111 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Part One: Worst-Case Regret Guarantees
  4. Part Two: Computing Regret-Minimizing Contracts
  5. Related Works
  6. Works on Principal-Agent Problems Without Uncertainty
  7. Works on Principal-Agent Problems With Uncertainty
  8. Related Results in Different Settings
  9. Preliminaries
  10. Principal-Agent Problems Under Uncertainty
  11. Regret-Minimizing Contracts
  12. On the Existence of Regret-Minimizing Contracts
  13. When Do Regret-Minimizing Contracts Exist?
  14. When Regret-Minimizing Contracts Do Not Exist?
  15. Worst-Case Regret Guarantees of Deterministic Contracts
  16. ...and 27 more sections

Key Result

Theorem 1

Deterministic contracts achieve worst-case-optimal regret. More formally, $\inf_{\bm{p} \in \mathcal{P}} \mathcal{R}(\bm{p})$ is at most $O(\sqrt{\delta} )$ as $\delta \to 0$, while there are PAPU instances where the minimum possible achievable regret is at least $\Omega(\sqrt{\delta})$ as $\delta \

Figures (3)

  • Figure 1: Uncertainty set $\mathcal{C}$ and two of its "scalings".
  • Figure 2: (Left) Diagram depicting the "inclusion" relationship between different classes of contracts. (Right) Diagram depicting the relationship between different classes of contracts in terms of "regret gap". An arrow from class ${\mathcal{X}}$ to class $\mathcal{Y}$ means that there exists a family of PAPU instances parametrized by $\delta$ in which the uncertainty level is $d(\mathcal{C}) = \delta$, the regret attained by contracts of class ${\mathcal{X}}$ is at most $R_\delta$, while all the contracts of class $\mathcal{Y}$ achieve regret at least $R_\delta+\Omega(\sqrt\delta)$ as $\delta \to 0$.
  • Figure 3: Behavior of ${\textnormal{OPT}\xspace}(p)-U^{\mathfrak{p}}(p,\bm{c})$ as a function of $c_{a_2}$ in the instance described in \ref{['def:nonexist_inst']}.

Theorems & Definitions (86)

  • Theorem : Informal
  • Theorem : Informal
  • Theorem : Informal
  • Definition 2.1: Regret-minimizing contracts
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Definition 3.5: "Non-existence" instance of the PAPU
  • Proposition 3.5
  • ...and 76 more