Multi-Agent Combinatorial Contracts
Paul Duetting, Tomer Ezra, Michal Feldman, Thomas Kesselheim
TL;DR
This work studies algorithmic contracts for a principal coordinating a team of agents who may perform multiple actions, with outcomes observed only via a binary success event. It introduces subset stability and the Doubling Lemma to design contracts that guarantee strong principal utility even under strategic equilibria, leveraging value and demand oracles for submodular rewards. The authors prove a constant-factor approximation in the multi-agent, multi-action setting, show no PTAS is possible, and provide an FPTAS for the single-agent subproblem, along with robustness results to handle multi-agent equilibria. They further bound the welfare-utility gap (second-best vs. first-best) and present a reduction framework that reduces general instances to no-large-agent or single-agent cases. Collectively, the results advance understanding of combinatorial contracts and yield practical, approximation-guaranteed mechanisms for complex team-based outcomes with hidden actions.
Abstract
Combinatorial contracts are emerging as a key paradigm in algorithmic contract design, paralleling the role of combinatorial auctions in algorithmic mechanism design. In this paper we study natural combinatorial contract settings involving teams of agents, each capable of performing multiple actions. This scenario extends two fundamental special cases previously examined in the literature, namely the single-agent combinatorial action model of [Duetting et al., 2021] and the multi-agent binary-action model of [Babaioff et al., 2012, Duetting et al., 2023]. We study the algorithmic and computational aspects of these settings, highlighting the unique challenges posed by the absence of certain monotonicity properties essential for analyzing the previous special cases. To navigate these complexities, we introduce a broad set of novel tools that deepen our understanding of combinatorial contracts environments and yield good approximation guarantees. Our main result is a constant-factor approximation for submodular multi-agent multi-action problems with value and demand oracles access. This result is tight: we show that this problem admits no PTAS (even under binary actions). As a side product of our main result, we devise an FPTAS, with value and demand oracles, for single-agent combinatorial action scenarios with general reward functions, which is of independent interest. We also provide bounds on the gap between the optimal welfare and the principal's utility. We show that, for subadditive rewards, perhaps surprisingly, this gap scales only logarithmically (rather than linearly) in the size of the action space.