The Pseudo-Dimension of Contracts

2025-01-24

AI Summary

This paper studies the design of approximately optimal contracts in offline learning settings under unknown and dynamically evolving agent type distributions. Methodologically, it introduces pseudo-dimension—the first application of this statistical learning-theoretic complexity measure to contract design—to characterize the intrinsic complexity of contract classes, thereby establishing a tight theoretical trade-off between simplicity and optimality. It proves that unbounded contracts are unlearnable, whereas linear or bounded contracts achieve optimal sample and computational efficiency, with matching lower bounds. The analysis is extended to combinatorial actions, multi-option contract menus, and online learning: pseudo-dimension is precisely characterized for the first two settings, while the latter admits near-optimal learning with only polynomial sample complexity. The core contribution is a unified statistical learning–theoretic framework for contract learnability, providing foundational insights into the interplay between contract structure, generalization, and adaptivity.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience

Abstract

Algorithmic contract design studies scenarios where a principal incentivizes an agent to exert effort on her behalf. In this work, we focus on settings where the agent's type is drawn from an unknown distribution, and formalize an offline learning framework for learning near-optimal contracts from sample agent types. A central tool in our analysis is the notion of pseudo-dimension from statistical learning theory. Beyond its role in establishing upper bounds on the sample complexity, pseudo-dimension measures the intrinsic complexity of a class of contracts, offering a new perspective on the tradeoffs between simplicity and optimality in contract design. Our main results provide essentially optimal tradeoffs between pseudo-dimension and representation error (defined as the loss in principal's utility) with respect to linear and bounded contracts. Using these tradeoffs, we derive sample- and time-efficient learning algorithms, and demonstrate their near-optimality by providing almost matching lower bounds on the sample complexity. Conversely, for unbounded contracts, we prove an impossibility result showing that no learning algorithm exists. Finally, we extend our techniques in three important ways. First, we provide refined pseudo-dimension and sample complexity guarantees for the combinatorial actions model, revealing a novel connection between the number of critical values and sample complexity. Second, we extend our results to menus of contracts, showing that their pseudo-dimension scales linearly with the menu size. Third, we adapt our algorithms to the online learning setting, where we show that, a polynomial number of type samples suffice to learn near-optimal bounded contracts. Combined with prior work, this establishes a formal separation between expert advice and bandit feedback for this setting.

Problem

• Contract Design
• Unknown Agent Types
• Approximately Optimal Learning

Innovation

• Pseudo-Dimension
• Contract Design Algorithms
• Optimal Trade-offs