Multi-Agent Combinatorial Contracts

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: the single-agent combinatorial action model of [18], and the multi-agent binary-action model of [4, 19]. This setting presents fundamentally different challenges compared to the previous special cases, as it lacks key properties that have been crucial for resolving these scenarios. To navigate these challenges, we develop a broad set of novel tools that allow us to establish approximation guarantees for this setting. Our main result is a constant-factor approximation for multi-agent multi-action problems with submodular rewards, given access to value and demand oracles. This result is tight: we show that this problem admits no PTAS (even under binary actions). As a byproduct of our main result, we devise an FPTAS, given value and demand oracles, for single-agent combinatorial action scenarios with general reward functions, which is of independent interest. Finally, we show that for subadditive rewards, perhaps surprisingly, the gap between the optimal welfare and the principal’s utility scales logarithmically (rather than linearly) with the size of the action space.