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Are Bounded Contracts Learnable and Approximately Optimal?

Yurong Chen, Zhaohua Chen, Xiaotie Deng, Zhiyi Huang

TL;DR

This paper investigates the learnability and approximate optimality of contracts with bounded payments in the hidden-action principal–agent model. Under two standard assumptions (FOSD and CDFP), it develops two polynomial-query learning algorithms operating in the action-query and contract-query models, leveraging a complementary CDF representation and a robustification technique to output near-optimal $H$-bounded contracts. It also demonstrates fundamental limits: bounded contracts cannot achieve constant multiplicative or additive approximations to the unrestricted optimum, yet they offer a meaningful mixed-approximation via linear contracts, with an asymptotically tight trade-off. The approach introduces threshold contracts and subgradient-based learning to handle potentially infinite action spaces, providing polynomial-time learnability and robust performance on nearby instances. Overall, bounded contracts can be learned efficiently under standard assumptions, but their proximity to the true optimum depends on the structure of the outcome distributions and cost–utility tradeoffs, as captured by the mixed approximation framework.

Abstract

This paper considers the hidden-action model of the principal-agent problem, in which a principal incentivizes an agent to work on a project using a contract. We investigate whether contracts with bounded payments are learnable and approximately optimal. Our main results are two learning algorithms that can find a nearly optimal bounded contract using a polynomial number of queries, under two standard assumptions in the literature: a costlier action for the agent leads to a better outcome distribution for the principal, and the agent's cost/effort has diminishing returns. Our polynomial query complexity upper bound shows that standard assumptions are sufficient for achieving an exponential improvement upon the known lower bound for general instances. Unlike the existing algorithms, which relied on discretizing the contract space, our algorithms directly learn the underlying outcome distributions. As for the approximate optimality of bounded contracts, we find that they could be far from optimal in terms of multiplicative or additive approximation, but satisfy a notion of mixed approximation.

Are Bounded Contracts Learnable and Approximately Optimal?

TL;DR

This paper investigates the learnability and approximate optimality of contracts with bounded payments in the hidden-action principal–agent model. Under two standard assumptions (FOSD and CDFP), it develops two polynomial-query learning algorithms operating in the action-query and contract-query models, leveraging a complementary CDF representation and a robustification technique to output near-optimal -bounded contracts. It also demonstrates fundamental limits: bounded contracts cannot achieve constant multiplicative or additive approximations to the unrestricted optimum, yet they offer a meaningful mixed-approximation via linear contracts, with an asymptotically tight trade-off. The approach introduces threshold contracts and subgradient-based learning to handle potentially infinite action spaces, providing polynomial-time learnability and robust performance on nearby instances. Overall, bounded contracts can be learned efficiently under standard assumptions, but their proximity to the true optimum depends on the structure of the outcome distributions and cost–utility tradeoffs, as captured by the mixed approximation framework.

Abstract

This paper considers the hidden-action model of the principal-agent problem, in which a principal incentivizes an agent to work on a project using a contract. We investigate whether contracts with bounded payments are learnable and approximately optimal. Our main results are two learning algorithms that can find a nearly optimal bounded contract using a polynomial number of queries, under two standard assumptions in the literature: a costlier action for the agent leads to a better outcome distribution for the principal, and the agent's cost/effort has diminishing returns. Our polynomial query complexity upper bound shows that standard assumptions are sufficient for achieving an exponential improvement upon the known lower bound for general instances. Unlike the existing algorithms, which relied on discretizing the contract space, our algorithms directly learn the underlying outcome distributions. As for the approximate optimality of bounded contracts, we find that they could be far from optimal in terms of multiplicative or additive approximation, but satisfy a notion of mixed approximation.
Paper Structure (49 sections, 27 theorems, 96 equations, 5 figures)

This paper contains 49 sections, 27 theorems, 96 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Our Results
  3. Further Related Work
  4. Learning Optimal Contract.
  5. Simple and Approximately Optimal Contract.
  6. Combinatorial Contracts
  7. Preliminaries
  8. Notations.
  9. Basic Model
  10. Bounded Contracts
  11. Query Complexity Models
  12. Action Query Model.
  13. Contract Query Model.
  14. Learning Optimal (Bounded) Contracts.
  15. Assumptions and Alternative Representation of an Instance
  16. ...and 34 more sections

Key Result

Lemma 3.1

Suppose that $\tilde{D}$ is the uniform distribution over $N = \Omega \left( \frac{m+\log1/\delta}{\varepsilon^2} \right)$ IID samples from $D$. With probability at least $1-\delta$, we have

Figures (5)

  • Figure 1: Representation of a problem instance under FOSD and CDFP assumptions in the table form and in the complementary CDF form.
  • Figure 2: Psuedo-code for the learning algorithm in the action query model.
  • Figure 3: Psuedo-code for the learning algorithm in the contract query model.
  • Figure 4: An illustration on the proof of \ref{['lmm: approximate convex']}.
  • Figure 5: Two key scenarios in the proof of \ref{['lmm: approximate concave']}. (a) The first subcase of the Main Case. (b) Boundary Case I.

Theorems & Definitions (40)

  • Definition 1
  • Definition 2: Total Variation Distance
  • Definition 3: Kolmogorov Distance
  • Lemma 3.1: e.g., canonne2020short
  • Lemma 3.2
  • Lemma 3.3
  • Definition 4
  • Lemma 3.4: Robustification
  • proof
  • Theorem 3.5
  • ...and 30 more