# Multi-Agent Contract Design: How to Commission Multiple Agents with Individual Outcome

arXiv: https://arxiv.org/abs/2301.13654 · HTML: https://mlsanity.com/paper/2301.13654/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jan 2023.

## Built on by (most cited first, 6 of 6)

- [Contracting with a Learning Agent](https://mlsanity.com/paper/2401.16198/) (2024)
- [Learning Optimal Contracts: How to Exploit Small Action Spaces](https://mlsanity.com/paper/2309.09801/) (2023)
- [The Pseudo-Dimension of Contracts](https://mlsanity.com/paper/2501.14474/) (2025)
- [Deep Incentive Design with Differentiable Equilibrium Blocks](https://mlsanity.com/paper/2603.07705/) (2026)
- [Steering No-Regret Agents in MFGs under Model Uncertainty](https://mlsanity.com/paper/2503.09309/) (2025)
- [Fair Contracts in Principal-Agent Games with Heterogeneous Types](https://mlsanity.com/paper/2506.15887/) (2025)

## Abstract (arXiv)

We study hidden-action principal-agent problems with multiple agents. These are problems in which a principal commits to an outcome-dependent payment scheme in order to incentivize some agents to take costly, unobservable actions that lead to favorable outcomes. Previous works on multi-agent problems study models where the principal observes a single outcome determined by the actions of all the agents. Such models considerably limit the contracting power of the principal, since payments can only depend on the joint result of all the agents' actions, and there is no way of paying each agent for their individual result. In this paper, we consider a model in which each agent determines their own individual outcome as an effect of their action only, the principal observes all the individual outcomes separately, and they perceive a reward that jointly depends on all these outcomes. This considerably enhances the principal's contracting capabilities, by allowing them to pay each agent on the basis of their individual result. We analyze the computational complexity of finding principal-optimal contracts, revolving around two newly-introduced properties of principal's rewards, which we call IR-supermodularity and DR-submodularity. Intuitively, the former captures settings with increasing returns, where the rewards grow faster as the agents' effort increases, while the latter models the case of diminishing returns, in which rewards grow slower instead. These two properties naturally model two common real-world phenomena, namely diseconomies and economies of scale. In this paper, we first address basic instances in which the principal knows everything about the agents, and, then, more general Bayesian instances where each agent has their own private type determining their features, such as action costs and how actions stochastically determine individual outcomes.