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Learning Optimal Contracts: How to Exploit Small Action Spaces

Francesco Bacchiocchi, Matteo Castiglioni, Alberto Marchesi, Nicola Gatti

TL;DR

The paper tackles learning optimal contracts in repeated hidden-action principal-agent problems with small action spaces, where the agent’s actions are unobserved and outcomes are the feedback signal. It introduces the Discover-and-Cover framework, leveraging meta-actions to partition best-response regions and a trio of subroutines (Action-Oracle, Try-Cover, Find-Contract) to learn a bounded contract efficiently. The main theoretical gains are a polynomial-round sample complexity when $n$ is fixed and a cumulative regret bound that scales as $ ilde{O}(T^{4/5})$, thereby resolving an open problem for general settings with many outcomes. The construction also yields a no-regret online learning algorithm, with practical implications for sequential contract design in digital economics and Stackelberg-like settings where actions are unobserved.

Abstract

We study principal-agent problems in which a principal commits to an outcome-dependent payment scheme -- called contract -- in order to induce an agent to take a costly, unobservable action leading to favorable outcomes. We consider a generalization of the classical (single-round) version of the problem in which the principal interacts with the agent by committing to contracts over multiple rounds. The principal has no information about the agent, and they have to learn an optimal contract by only observing the outcome realized at each round. We focus on settings in which the size of the agent's action space is small. We design an algorithm that learns an approximately-optimal contract with high probability in a number of rounds polynomial in the size of the outcome space, when the number of actions is constant. Our algorithm solves an open problem by Zhu et al.[2022]. Moreover, it can also be employed to provide a $\tilde{\mathcal{O}}(T^{4/5})$ regret bound in the related online learning setting in which the principal aims at maximizing their cumulative utility, thus considerably improving previously-known regret bounds.

Learning Optimal Contracts: How to Exploit Small Action Spaces

TL;DR

The paper tackles learning optimal contracts in repeated hidden-action principal-agent problems with small action spaces, where the agent’s actions are unobserved and outcomes are the feedback signal. It introduces the Discover-and-Cover framework, leveraging meta-actions to partition best-response regions and a trio of subroutines (Action-Oracle, Try-Cover, Find-Contract) to learn a bounded contract efficiently. The main theoretical gains are a polynomial-round sample complexity when is fixed and a cumulative regret bound that scales as , thereby resolving an open problem for general settings with many outcomes. The construction also yields a no-regret online learning algorithm, with practical implications for sequential contract design in digital economics and Stackelberg-like settings where actions are unobserved.

Abstract

We study principal-agent problems in which a principal commits to an outcome-dependent payment scheme -- called contract -- in order to induce an agent to take a costly, unobservable action leading to favorable outcomes. We consider a generalization of the classical (single-round) version of the problem in which the principal interacts with the agent by committing to contracts over multiple rounds. The principal has no information about the agent, and they have to learn an optimal contract by only observing the outcome realized at each round. We focus on settings in which the size of the agent's action space is small. We design an algorithm that learns an approximately-optimal contract with high probability in a number of rounds polynomial in the size of the outcome space, when the number of actions is constant. Our algorithm solves an open problem by Zhu et al.[2022]. Moreover, it can also be employed to provide a regret bound in the related online learning setting in which the principal aims at maximizing their cumulative utility, thus considerably improving previously-known regret bounds.
Paper Structure (23 sections, 28 theorems, 53 equations, 5 figures, 6 algorithms)

This paper contains 23 sections, 28 theorems, 53 equations, 5 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Original contributions
  3. Relation to repeated Stackelberg games
  4. Preliminaries on hidden-action principal-agent problems
  5. Learning optimal contracts
  6. The Discover-and-Cover algorithm
  7. Action-Oracle
  8. Try-Cover
  9. Find-Contract
  10. Putting it all together
  11. Connection with online learning in principal-agent problems
  12. Related works
  13. Learning in principal-agent problems
  14. Learning in Stackelberg games
  15. Assumptions relaxed compared to Stackelberg games
  16. ...and 8 more sections

Key Result

Theorem 1

For any number of rounds $N \in \mathbb{N}$, there is no algorithm that is guaranteed to find a $\kappa$-optimal contract with probability greater than or equal to $1 - \delta$ by using less than $N$ rounds, where $\kappa, \delta > 0$ are some suitable absolute constants.

Figures (5)

  • Figure : Discover-and-Cover
  • Figure : Action-Oracle
  • Figure : Try-Cover
  • Figure : Find-Contract
  • Figure : No-regret algorithm

Theorems & Definitions (51)

  • Theorem 1
  • Definition 1: Learning an optimal bounded contract
  • Definition 2: Clean event
  • Definition 3: Associated actions
  • Lemma 1
  • Lemma 2
  • Definition 4: Cost of a meta-action
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 41 more