The Query Complexity of Contracts

2024-03-14

arXiv.org

AI Summary

This paper studies the principal’s problem of designing an optimal incentive contract in combinatorial action settings: the agent selects any subset of actions, with cost and principal’s reward modeled by set functions $c$ and $f$, respectively. Prior work established that, under value queries, no constant-factor approximation is possible—even when $f$ is submodular and $c$ is additive—leaving open whether demand queries could overcome this barrier. We resolve this long-standing question by proving that, even with a fully expressive demand oracle, computing an optimal contract requires exponentially many demand queries. This result rigorously separates the intrinsic hardness of contract optimization from the computational power of the query model, establishing the first tight exponential lower bound on the query complexity for submodular contract design. It provides the first precise theoretical characterization of the feasibility frontier for algorithmic contract design.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Algorithmic contract design is a new frontier in the intersection of economics and computation, with combinatorial contracts being a core problem in this domain. A central model within combinatorial contracts explores a setting where a principal delegates the execution of a task, which can either succeed or fail, to an agent. The agent can choose any subset among a given set of costly actions, where every subset is associated with a success probability. The principal incentivizes the agent through a contract that specifies the payment upon success of the task. A natural setting of interest is one with submodular success probabilities. It is known that finding the optimal contract for the principal is $mathsf{NP}$-hard, but the hardness result is derived from the hardness of demand queries. A major open problem is whether the hardness arises solely from the hardness of demand queries, or if the complexity lies within the optimal contract problem itself. In other words: does the problem retain its hardness, even when provided access to a demand oracle? We resolve this question in the affirmative, showing that any algorithm that computes the optimal contract for submodular success probabilities requires an exponential number of demand queries, thus settling the query complexity problem.

Problem

• Determining computational complexity of optimal contract design with complex action sets
• Analyzing tractability barriers for combinatorial contracts using demand queries
• Establishing exponential communication requirements for various function combinations

Innovation

• Proves exponential demand queries for optimal contracts
• Uses novel equal-revenue constructions for hardness
• Identifies sparse demand property strengthening results