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Contract Design Under Approximate Best Responses

Francesco Bacchiocchi, Jiarui Gan, Matteo Castiglioni, Alberto Marchesi, Nicola Gatti

TL;DR

This paper studies principal-agent problems with hidden actions under approximate best responses, introducing a δ-robust contract framework where the agent may select any action within δ of the best. It proves a polynomial-time algorithm for computing an optimal δ-robust contract by decomposing the problem into a union of linear programs across action pairs and subspaces, and it analyzes the price of robustness via tight bounds on OPT(δ) that depend only on δ, OPT, and social welfare SW. The authors also develop a no-regret online learning algorithm for robust contracts in a multi-type setting, leveraging a continuity lemma to discretize the contract space and attain sublinear regret without prior knowledge of rewards. These results separate hidden-action contract design from Stackelberg robustness, show robustness can be efficiently achieved, and have implications for algorithmic contract design in domains like crowdsourcing, blockchain, and delegated ML tasks.

Abstract

Principal-agent problems model scenarios where a principal incentivizes an agent to take costly, unobservable actions through the provision of payments. Such problems are ubiquitous in several real-world applications, ranging from blockchain to the delegation of machine learning tasks. In this paper, we initiate the study of hidden-action principal-agent problems under approximate best responses, in which the agent may select any action that is not too much suboptimal given the principal's payment scheme (a.k.a. contract). Our main result is a polynomial-time algorithm to compute an optimal contract under approximate best responses. This positive result is perhaps surprising, since, in Stackelberg games, computing an optimal commitment under approximate best responses is computationally intractable. We also investigate the learnability of contracts under approximate best responses, by providing a no-regret learning algorithm for a natural application scenario where the principal has no prior knowledge about the environment.

Contract Design Under Approximate Best Responses

TL;DR

This paper studies principal-agent problems with hidden actions under approximate best responses, introducing a δ-robust contract framework where the agent may select any action within δ of the best. It proves a polynomial-time algorithm for computing an optimal δ-robust contract by decomposing the problem into a union of linear programs across action pairs and subspaces, and it analyzes the price of robustness via tight bounds on OPT(δ) that depend only on δ, OPT, and social welfare SW. The authors also develop a no-regret online learning algorithm for robust contracts in a multi-type setting, leveraging a continuity lemma to discretize the contract space and attain sublinear regret without prior knowledge of rewards. These results separate hidden-action contract design from Stackelberg robustness, show robustness can be efficiently achieved, and have implications for algorithmic contract design in domains like crowdsourcing, blockchain, and delegated ML tasks.

Abstract

Principal-agent problems model scenarios where a principal incentivizes an agent to take costly, unobservable actions through the provision of payments. Such problems are ubiquitous in several real-world applications, ranging from blockchain to the delegation of machine learning tasks. In this paper, we initiate the study of hidden-action principal-agent problems under approximate best responses, in which the agent may select any action that is not too much suboptimal given the principal's payment scheme (a.k.a. contract). Our main result is a polynomial-time algorithm to compute an optimal contract under approximate best responses. This positive result is perhaps surprising, since, in Stackelberg games, computing an optimal commitment under approximate best responses is computationally intractable. We also investigate the learnability of contracts under approximate best responses, by providing a no-regret learning algorithm for a natural application scenario where the principal has no prior knowledge about the environment.

Paper Structure

This paper contains 25 sections, 15 theorems, 38 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Results and Techniques
  3. The Price of Robustness
  4. Computing Robust Contracts
  5. Learning Robust Contracts
  6. Related Works
  7. Robustness to approximate best responses
  8. Contract design
  9. Other forms of robustness in contract design
  10. Learning in principal-agent problems
  11. Preliminaries
  12. Hidden-Action Principal-Agent Problems
  13. Robust Contracts and Approximate Best Responses
  14. The Price of Robustness
  15. Comparison with Stackelberg games
  16. ...and 10 more sections

Key Result

Proposition 3.0

Given an instance of hidden-action principal-agent problem:

Figures (1)

  • Figure 1: The blue area corresponds to the region in which $\textnormal{OPT}(\delta)$ is bounded as a function of $\delta \in (0,1)$ in an instance of hidden-action principal-agent problem with $\textnormal{SW}=0.9$ and $\textnormal{OPT}=0.7$.

Theorems & Definitions (25)

  • Proposition 3.0: Upper and lower bounds
  • Proposition 3.0
  • Proposition 3.0
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • Remark 4.3
  • Theorem 4.4
  • proof : Proof Sketch
  • Lemma 5.0
  • ...and 15 more