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Existence of Equilibrium Mechanisms in Generalized Principal-Agent Problems with Interacting Teams

Brian Roberson

Abstract

We study incentive design when multiple principals simultaneously design mechanisms for their respective teams in environments with strategic spillovers. In this environment, each principal's set of incentive-compatible mechanisms--those that satisfy their own agents' incentive compatibility constraints--depends on the mechanisms offered by the other teams. Following a classic example by Myerson (1982), such games may lack equilibrium due to discontinuities in the correspondence of incentive-compatible mechanisms. We establish general conditions for equilibrium existence by introducing a novel approach that involves tracking both the outcome distributions along the truthful-obedient path and the sets of outcome distributions achievable through unilateral deviations, thereby providing a foundation for analyzing a wide range of multi-principal mechanism design with team production and agency problems.

Existence of Equilibrium Mechanisms in Generalized Principal-Agent Problems with Interacting Teams

Abstract

We study incentive design when multiple principals simultaneously design mechanisms for their respective teams in environments with strategic spillovers. In this environment, each principal's set of incentive-compatible mechanisms--those that satisfy their own agents' incentive compatibility constraints--depends on the mechanisms offered by the other teams. Following a classic example by Myerson (1982), such games may lack equilibrium due to discontinuities in the correspondence of incentive-compatible mechanisms. We establish general conditions for equilibrium existence by introducing a novel approach that involves tracking both the outcome distributions along the truthful-obedient path and the sets of outcome distributions achievable through unilateral deviations, thereby providing a foundation for analyzing a wide range of multi-principal mechanism design with team production and agency problems.
Paper Structure (23 sections, 8 theorems, 53 equations, 1 figure)

This paper contains 23 sections, 8 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Myerson (1982) Example
  3. Model
  4. First (Private-Type) Stage
  5. Second (Type-Reporting) Stage
  6. Third (Action) Stage
  7. Fourth (Team-Winnings) Stage
  8. Principals' Initial Problems
  9. Multi-Principle Interaction with Team Production and Agency Concerns
  10. Assumptions and the Metric Structure
  11. First (Private Type) Stage:
  12. Second (Type Reporting) Stage:
  13. Third (Action Reporting) Stage:
  14. Fourth (Team Winnings) Stage:
  15. Principal's Initial Problems
  16. ...and 8 more sections

Key Result

Theorem 1

Endow $M^N$ with the $d^*_{M^N}$ metric. If $IC(\mathbf{m}) \coloneqq \prod_{j=1}^N IC_j(\mathbf{m}_{-j})$ admits a selection, then under Assumptions 1--4, for each team $j$: Therefore, there exists a BNPE.

Figures (1)

  • Figure 1: Illustration of Team $j$'s Mechanism

Theorems & Definitions (18)

  • Definition 3.1
  • Definition 3.2: Bayesian-Nash Principals' Equilibrium (BNPE)
  • Definition 4.1: Truthful-obedient induced law
  • Definition 4.2: Narrow topology on laws
  • Definition 4.3: Prokhorov Metric
  • Definition 4.4: Truthful-obedient behavior strategies
  • Definition 4.5: Induced law under an arbitrary unilateral deviation
  • Definition 4.6: Hausdorff metric on compact sets of laws
  • Definition 4.7: Robust narrow metric
  • Theorem 1
  • ...and 8 more