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The Pseudo-Dimension of Contracts

Author

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  • Paul Duetting
  • Michal Feldman
  • Tomasz Ponitka
  • Ermis Soumalias

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.

Suggested Citation

  • Paul Duetting & Michal Feldman & Tomasz Ponitka & Ermis Soumalias, 2025. "The Pseudo-Dimension of Contracts," Papers 2501.14474, arXiv.org.
  • Handle: RePEc:arx:papers:2501.14474
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    References listed on IDEAS

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    1. Schorfheide, Frank & Wolpin, Kenneth I., 2016. "To hold out or not to hold out," Research in Economics, Elsevier, vol. 70(2), pages 332-345.
    2. Grossman, Sanford J & Hart, Oliver D, 1983. "An Analysis of the Principal-Agent Problem," Econometrica, Econometric Society, vol. 51(1), pages 7-45, January.
    3. Gabriel Carroll, 2015. "Robustness and Linear Contracts," American Economic Review, American Economic Association, vol. 105(2), pages 536-563, February.
    4. Frank Schorfheide & Kenneth I. Wolpin, 2012. "On the Use of Holdout Samples for Model Selection," American Economic Review, American Economic Association, vol. 102(3), pages 477-481, May.
    5. Burkett, Justin & Rosenthal, Maxwell, 2024. "Data-driven contract design," Journal of Economic Theory, Elsevier, vol. 221(C).
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