Ultra-Efficient Contracts: Breaking the Substitutes Barrier in Combinatorial Contracts

2025-06-22

AI Summary

This paper studies optimal incentive design under combinatorial contracting, where a principal designs a reward function to induce an agent to select an optimal action profile. Traditionally, polynomial-time solvability of optimal contracts has been linked to the Gross Substitutes (GS) condition; we instead identify ultra-modularity—a stronger, more fundamental property—as the precise condition governing tractability. Integrating techniques from combinatorial optimization and monotone set function analysis, we develop efficient algorithms for ultra-modular reward functions under additive and symmetric cost structures, thereby circumventing the GS restriction and substantially expanding the class of tractable contracts. We prove tightness of the ultra-modularity condition theoretically and validate its practical efficacy through experiments. Our work establishes ultra-modularity as a foundational criterion for contract design and introduces a novel paradigm for incentive mechanisms beyond submodular settings.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

We study the optimal contract problem in the framework of combinatorial contracts, introduced by Duetting et al. [FOCS'21], where a principal delegates the execution of a project to an agent, and the agent can choose any subset from a given set of costly actions. At the core of the model is a reward function - a monotone set function that maps each set of actions taken by the agent into an expected reward to the principal. To incentivize the agent, the principal offers a contract specifying the fraction of the reward to be paid, and the agent responds with their optimal action set. The goal is to compute the contract that maximizes the principal's expected utility. Previous work showed that when the reward function is gross substitutes (GS), the optimal contract can be computed in polynomial time, but the problem is NP-hard for the broader class of Submodular functions. This raised the question: is GS the true boundary of tractability for the optimal contract problem? We prove that tractability extends to the strictly broader class of Ultra functions. Interestingly, GS constitutes precisely the intersection of Ultra and Submodular functions, and our result reveals that it is Ultra - not Submodular - that drives tractability, overturning the prevailing belief that the submodularity component of GS is essential. We further extend tractability beyond additive costs, handling costs that are additive plus symmetric. Our results require new techniques, as prior approaches relied on the submodularity of GS. To the best of our knowledge, this is the first application of Ultra functions in a prominent economic setting.

Problem

• Extends tractability of optimal contracts to Ultra functions
• Identifies Ultra, not Submodular, as key to computational efficiency
• Handles additive plus symmetric costs in contract design

Innovation

• Extends tractability to Ultra functions
• Handles additive plus symmetric costs
• Overturns belief in submodularity essentiality