Combinatorial Contracts Through Demand Types

2026-04-16

AI Summary

This study investigates the computational complexity of computing optimal linear contracts in combinatorial contract design, particularly when the reward function exhibits both substitutable and complementary relationships among outcomes. By establishing a geometric connection between combinatorial contracts and demand types in consumer theory, the authors reduce the problem of bounding the number of critical values to counting how many regions of best responses a contract ray intersects. They introduce a novel class of reward functions—termed ASC (All Substitutes and Complements)—which strictly generalizes and unifies all previously known classes admitting polynomially many critical values, and conjecture it to be maximal with this property. Leveraging this geometric framework and efficiently simulating demand queries via value queries, the paper presents a general algorithm for “succinct” demand types and achieves, for the first time, polynomial-time computation of optimal contracts for a new class of reward functions incorporating both substitutes and complements.

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In the combinatorial action model of contract design, a principal delegates a complex project to an agent, incentivizing a subset of actions from a ground set of $n$ actions, via a linear contract. Computing the optimal contract is a challenging problem that generally hinges on two factors: (i) the number of "critical values" - values of the linear contract parameter at which the agent's best response changes from one set to another, and (ii) the complexity of the agent's best-response problem (demand query). Prior work has used this approach to devise polynomial-time algorithms for the optimal contract problem under specific reward functions: gross substitutes, supermodular, and ultra. We develop a unified geometric framework for algorithmic contract design by establishing a fundamental link to the theory of demand types from consumer theory. Under this geometric view, bounding the number of critical values reduces to counting the best-response regions which the "contract ray" pierces. Leveraging this connection, we introduce the class of All Substitutes and Complements (ASC) functions, and show that it admits at most $O(n^2)$ critical values, strictly generalizing and unifying all previously known classes admitting poly-many critical values. We conjecture that, under some mild assumptions, ASC is the maximal such class. Turning to the demand query aspect, we develop a new technique for efficiently computing a demand query using value queries, which works in general for "succinct" demand types. Combining these structural and algorithmic results, we obtain polynomial-time algorithms for new classes of reward functions that exhibit substitutes and complements simultaneously.

Problem

• combinatorial contracts
• optimal contract
• critical values
• demand query
• reward functions

Innovation

• combinatorial contracts
• demand types
• All Substitutes and Complements (ASC)
• critical values
• polynomial-time algorithm