Black-Box Lifting and Robustness Theorems for Multi-Agent Contracts
Paul Dütting, Tomer Ezra, Michal Feldman, Thomas Kesselheim
TL;DR
The paper addresses how a principal can design contracts in multi-agent, combinatorial settings when agents may play under mixed, correlated, or coarse-correlated equilibria. It introduces two core tools—Scaling-for-Existence and Scaling-for-Robustness—to convert broad equilibrium insights into robust, computable guarantees for submodular and XOS rewards, achieving constant-factor approximations between CCE outcomes and PNE outcomes via black-box lifting. It extends the landscape to subadditive, supermodular, and general rewards, deriving precise gap bounds (e.g., MNE-PNE lower bounds, CCE-PNE upper bounds) and identifying fundamental limits (unbounded gaps for XOS/general rewards). The approach relies on dropout stability and potential-game structure to enable tractable transformations and poly-time algorithms using value and demand oracles, thereby enabling robust contract design that remains effective under learning dynamics. Overall, the work broadens the design space for multi-agent contracts, clarifies when black-box guarantees are possible, and provides concrete algorithms and bounds that apply across learning and equilibrium regimes.
Abstract
Multi-agent contract design has largely evaluated contracts through the lens of pure Nash equilibria (PNE). This focus, however, is not without loss: In general, the principal can strictly gain by recommending a complex, possibly correlated, distribution over actions, while preserving incentive compatibility. In this work, we extend the analysis of multi-agent contracts beyond pure Nash equilibria to encompass more general equilibrium notions, including mixed Nash equilibria as well as (coarse-)correlated equilibria (CCE). The latter, in particular, captures the limiting outcome of agents engaged in learning dynamics. Our main result shows that for submodular and, more generally, XOS rewards, such complex recommendations yield at most a constant-factor gain: there exists a contract and a PNE whose utility is within a constant factor of the best CCE achievable by any contract. This provides a black-box lifting: results established against the best PNE automatically apply with respect to the best CCE, with only a constant factor loss. For submodular rewards, we further show how to transform a contract and a PNE of that contract into a new contract such that any of its CCEs gives a constant approximation to the PNE. This yields black-box robustness: up to constant factors, guarantees established for a specific contract and PNE automatically extend to the modified contract and any of its CCEs. We thus expand prior guarantees for multi-agent contracts and lower the barrier to new ones. As an important corollary, we obtain poly-time algorithms for submodular rewards that achieve constant approximations in any CCE, against the best CCE under the best contract. Such worst-case guarantees are provably unattainable for XOS rewards. Finally, we bound the gap between different equilibrium notions for subadditive, supermodular, and general rewards.