Black-Box Lifting and Robustness Theorems for Multi-Agent Contracts

2025-11-24

AI Summary

Traditional multi-agent contract design focuses solely on pure Nash equilibria (PNE), yet principals can achieve higher utility by recommending mixed or coarse correlated action distributions while preserving incentive compatibility. This paper extends the analysis for the first time to mixed Nash equilibria and coarse correlated equilibria (CCE), establishing black-box “boosting” and “robustness” theorems for submodular and XOS reward functions. Methodologically, it integrates game-theoretic analysis, structural properties of submodular/XOS functions, and convergence theory of learning dynamics. Key contributions are: (1) For submodular rewards, it proves a constant-factor upper bound on the utility gain achievable via complex recommendation strategies; (2) It designs a polynomial-time algorithm that achieves a constant-factor approximation to the optimal contract’s equilibrium utility over any CCE; (3) It shows this result does not extend to XOS functions, precisely characterizing the limits of equilibrium expansion.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

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.

Problem

• Extends multi-agent contract analysis beyond pure Nash equilibria to broader equilibrium concepts.
• Shows complex recommendations yield only constant-factor gains for submodular and XOS rewards.
• Provides black-box lifting and robustness theorems connecting different equilibrium solution concepts.

Innovation

• Extends contract analysis to mixed and correlated equilibria
• Provides black-box lifting from pure Nash to correlated equilibria
• Transforms contracts for robustness across equilibrium concepts