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Distributionally Robust Principal-Agent Problems and Optimality of Contracts

Peter Zhang

Abstract

We propose a distributionally robust principal agent formulation, which generalizes some common variants of worst-case and Bayesian principal agent problems. We construct a theoretical framework to certify whether any surjective contract family is optimal, and bound its sub-optimality. We then apply the framework to study the optimality of affine contracts. We show with geometric intuition that these simple contract families are optimal when the surplus function is convex and there exists a technology type that is simultaneously least productive and least efficient. We also provide succinct expressions to quantify the optimality gap of any surplus function, based on its concave biconjugate. This new framework complements the current literature in two ways: invention of a new toolset; understanding affine contracts' performance in a larger landscape. Our results also shed light on the technical roots of this question: why are there more positive results in the recent literature that show simple contracts' optimality in robust settings rather than stochastic settings? This phenomenon is related to two technical facts: the sum of quasi-concave functions is not quasi-concave, and the maximization and expectation operators do not commute.

Distributionally Robust Principal-Agent Problems and Optimality of Contracts

Abstract

We propose a distributionally robust principal agent formulation, which generalizes some common variants of worst-case and Bayesian principal agent problems. We construct a theoretical framework to certify whether any surjective contract family is optimal, and bound its sub-optimality. We then apply the framework to study the optimality of affine contracts. We show with geometric intuition that these simple contract families are optimal when the surplus function is convex and there exists a technology type that is simultaneously least productive and least efficient. We also provide succinct expressions to quantify the optimality gap of any surplus function, based on its concave biconjugate. This new framework complements the current literature in two ways: invention of a new toolset; understanding affine contracts' performance in a larger landscape. Our results also shed light on the technical roots of this question: why are there more positive results in the recent literature that show simple contracts' optimality in robust settings rather than stochastic settings? This phenomenon is related to two technical facts: the sum of quasi-concave functions is not quasi-concave, and the maximization and expectation operators do not commute.
Paper Structure (32 sections, 8 theorems, 38 equations, 8 figures, 2 tables)

This paper contains 32 sections, 8 theorems, 38 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Principal's Distributional Robust Perspective
  3. Agent's Decision and Technology Type
  4. On Risk Aversion
  5. Relationship to the Literature
  6. Contributions
  7. A Theoretical Framework to Bound Contract Optimality Gap under General Uncertainty Structure
  8. General Results
  9. Game I:
  10. Game II:
  11. Game III:
  12. Claim 1: Left-hand side column is less than or equal to the right-hand side column.
  13. Claim 2: For any fixed contract family, Game I's principal payoff is less than or equal to Game II's principal payoff.
  14. Claim 3: $z_{\text{III}}(\mathcal{W}) = z_{\text{III}}(\mathcal{W}_0)$.
  15. Claim 4: $z_{\text{II}}(\mathcal{W}_0) = z_{\text{III}}(\mathcal{W}_0)$.
  16. ...and 17 more sections

Key Result

Theorem 2.1

We define Games I, II, and III below. Let the principal's payoffs under these games be $z_{\text{I}}(\mathcal{W})$, $z_{\text{II}}(\mathcal{W})$, and $z_{\text{III}}(\mathcal{W})$. Given any surjective family $\mathcal{W}$, the optimality gap of $\mathcal{W}$ is bounded above by $z_{\text{III}}(\mat

Figures (8)

  • Figure 1: Sequence of the game.
  • Figure 2: Sequence of Game II.
  • Figure 3: Sequence of Game III.
  • Figure 4: Games III can be mapped to this extensive form game (omitting the type discovery step).
  • Figure 5: An example of (non-convex, non-concave) $g(c)$.
  • ...and 3 more figures

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['thm:main']}
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lemma:key']}
  • Proposition 3.1
  • ...and 12 more