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Multi-Agent Contracts

Paul Duetting, Tomer Ezra, Michal Feldman, Thomas Kesselheim

TL;DR

A natural combinatorial single-principal multi-agent contract design problem, in which a principal motivates a team of agents to exert effort toward a given task, and seeks to design computationally efficient algorithms for finding optimal linear contracts for reward functions that belong to the complement-free hierarchy.

Abstract

We study a natural combinatorial single-principal multi-agent contract design problem, in which a principal motivates a team of agents to exert effort toward a given task. At the heart of our model is a reward function, which maps the agent efforts to an expected reward of the principal. We seek to design computationally efficient algorithms for finding optimal (or near-optimal) linear contracts for reward functions that belong to the complement-free hierarchy. Our first main result gives constant-factor approximation algorithms for submodular and XOS reward functions, with value oracles for submodular reward functions and value and demand oracles for XOS reward functions. It relies on an unconventional use of ``prices'' and (approximate) demand queries for selecting the set of agents that the principal should contract with, and exploits a novel scaling property of XOS functions and their marginals, which may be of independent interest. As our second main result, we show that constant approximation is the best we can get for submodular reward functions, even with both value and demand oracles. For the larger class of subadditive reward functions, we establish an $Ω(\sqrt{n})$ impossibility for settings with $n$ agents. A striking feature of this impossibility is that it applies to subadditive functions that are constant-factor close to submodular. This rapid degradation presents a surprising departure from previous literature, e.g., on combinatorial auctions, where approximation guarantees tend to deteriorate more

Multi-Agent Contracts

TL;DR

A natural combinatorial single-principal multi-agent contract design problem, in which a principal motivates a team of agents to exert effort toward a given task, and seeks to design computationally efficient algorithms for finding optimal linear contracts for reward functions that belong to the complement-free hierarchy.

Abstract

We study a natural combinatorial single-principal multi-agent contract design problem, in which a principal motivates a team of agents to exert effort toward a given task. At the heart of our model is a reward function, which maps the agent efforts to an expected reward of the principal. We seek to design computationally efficient algorithms for finding optimal (or near-optimal) linear contracts for reward functions that belong to the complement-free hierarchy. Our first main result gives constant-factor approximation algorithms for submodular and XOS reward functions, with value oracles for submodular reward functions and value and demand oracles for XOS reward functions. It relies on an unconventional use of ``prices'' and (approximate) demand queries for selecting the set of agents that the principal should contract with, and exploits a novel scaling property of XOS functions and their marginals, which may be of independent interest. As our second main result, we show that constant approximation is the best we can get for submodular reward functions, even with both value and demand oracles. For the larger class of subadditive reward functions, we establish an impossibility for settings with agents. A striking feature of this impossibility is that it applies to subadditive functions that are constant-factor close to submodular. This rapid degradation presents a surprising departure from previous literature, e.g., on combinatorial auctions, where approximation guarantees tend to deteriorate more
Paper Structure (46 sections, 20 theorems, 87 equations, 1 figure, 1 table, 4 algorithms)

This paper contains 46 sections, 20 theorems, 87 equations, 1 figure, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Model -- Key ingredients
  3. Our Contribution
  4. Our Techniques
  5. Constant-factor approximations for submodular and XOS
  6. Inapproximability result for submodular
  7. Inapproximability result for subadditive
  8. Related Work
  9. Optimizing the effort of others
  10. Combinatorial optimization and auctions
  11. Computational approaches to contracts
  12. Follow-Up Work
  13. Organization
  14. Preliminaries
  15. The multi-agent hidden-action principal-agent setting
  16. ...and 31 more sections

Key Result

Proposition 2.1

(a) A set of agents $S \subseteq A$ can be incentivized by some contract if and only if no agent $i \in S$ simultaneously has $f(i \mid S \setminus \{i\}) = 0$ and $c_i > 0$. If a set of agents $S \subseteq A$ can be incentivized by some contract then it can also be incentivized by a linear contract

Figures (1)

  • Figure 1: Illustration of Algorithm \ref{['alg:scaling']}. The blue line corresponds to the value $f(T_t)$ for $t = 0, \ldots, |T_0|$. The algorithm first sets a threshold of $\Psi$, and sets $j^\star$ as the last time step with a value larger than $\Psi$. Then it sets a threshold of $(1-\delta)f(T_{j^\star})$, and sets $k^\star$ as the first time step with a value at most this threshold. Between these time steps (excluding $k^\star$), the $f$ value of the set must satisfy Inequality \ref{['eq:prop_val']}, while the proof shows that there must exists a time step within this range with high marginals (in comparison to the marginals of $T_0$) satisfying Inequality \ref{['eq:prop_marg']}.

Theorems & Definitions (58)

  • Example 1.1: Multiple agents with submodular $f$
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3: Extension to more general equilibrium notions
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 48 more