IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2211.05732.html
   My bibliography  Save this paper

The Sample Complexity of Online Contract Design

Author

Listed:
  • Banghua Zhu
  • Stephen Bates
  • Zhuoran Yang
  • Yixin Wang
  • Jiantao Jiao
  • Michael I. Jordan

Abstract

We study the hidden-action principal-agent problem in an online setting. In each round, the principal posts a contract that specifies the payment to the agent based on each outcome. The agent then makes a strategic choice of action that maximizes her own utility, but the action is not directly observable by the principal. The principal observes the outcome and receives utility from the agent's choice of action. Based on past observations, the principal dynamically adjusts the contracts with the goal of maximizing her utility. We introduce an online learning algorithm and provide an upper bound on its Stackelberg regret. We show that when the contract space is $[0,1]^m$, the Stackelberg regret is upper bounded by $\widetilde O(\sqrt{m} \cdot T^{1-1/(2m+1)})$, and lower bounded by $\Omega(T^{1-1/(m+2)})$, where $\widetilde O$ omits logarithmic factors. This result shows that exponential-in-$m$ samples are sufficient and necessary to learn a near-optimal contract, resolving an open problem on the hardness of online contract design. Moreover, when contracts are restricted to some subset $\mathcal{F} \subset [0,1]^m$, we define an intrinsic dimension of $\mathcal{F}$ that depends on the covering number of the spherical code in the space and bound the regret in terms of this intrinsic dimension. When $\mathcal{F}$ is the family of linear contracts, we show that the Stackelberg regret grows exactly as $\Theta(T^{2/3})$. The contract design problem is challenging because the utility function is discontinuous. Bounding the discretization error in this setting has been an open problem. In this paper, we identify a limited set of directions in which the utility function is continuous, allowing us to design a new discretization method and bound its error. This approach enables the first upper bound with no restrictions on the contract and action space.

Suggested Citation

  • Banghua Zhu & Stephen Bates & Zhuoran Yang & Yixin Wang & Jiantao Jiao & Michael I. Jordan, 2022. "The Sample Complexity of Online Contract Design," Papers 2211.05732, arXiv.org, revised May 2023.
  • Handle: RePEc:arx:papers:2211.05732
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2211.05732
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Sergiu Hart & Noam Nisan, 2013. "The Menu-Size Complexity of Auctions," Discussion Paper Series dp637, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    2. Schorfheide, Frank & Wolpin, Kenneth I., 2016. "To hold out or not to hold out," Research in Economics, Elsevier, vol. 70(2), pages 332-345.
    3. Hurwicz,Leonid & Reiter,Stanley, 2008. "Designing Economic Mechanisms," Cambridge Books, Cambridge University Press, number 9780521724104.
    4. Bernard Salanié, 2005. "The Economics of Contracts: A Primer, 2nd Edition," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262195259, April.
    5. 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.
    6. Roger B. Myerson, 1981. "Optimal Auction Design," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 58-73, February.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Banghua Zhu & Sai Praneeth Karimireddy & Jiantao Jiao & Michael I. Jordan, 2023. "Online Learning in a Creator Economy," Papers 2305.11381, arXiv.org.
    2. Mehran Garmehi & Morteza Analoui & Mukaddim Pathan & Rajkumar Buyya, 2015. "An economic mechanism for request routing and resource allocation in hybrid CDN–P2P networks," International Journal of Network Management, John Wiley & Sons, vol. 25(6), pages 375-393, November.
    3. de Bresser, Jochem, 2021. "Evaluating the Accuracy of Counterfactuals The Role of Heterogeneous Expectations in Life Cycle Models," Other publications TiSEM a7e2b4d8-fed0-4e86-926f-d, Tilburg University, School of Economics and Management.
    4. Zhang, Hanzhe, 2021. "The optimal sequence of prices and auctions," European Economic Review, Elsevier, vol. 133(C).
    5. de Bresser, Jochem, 2021. "Evaluating the Accuracy of Counterfactuals The Role of Heterogeneous Expectations in Life Cycle Models," Discussion Paper 2021-034, Tilburg University, Center for Economic Research.
    6. Tomoya Kazumura & Debasis Mishra & Shigehiro Serizawa, 2017. "Strategy-proof multi-object auction design: Ex-post revenue maximization with no wastage," ISER Discussion Paper 1001, Institute of Social and Economic Research, Osaka University.
    7. Chen, Xi & Diakonikolas, Ilias & Paparas, Dimitris & Sun, Xiaorui & Yannakakis, Mihalis, 2018. "The complexity of optimal multidimensional pricing for a unit-demand buyer," Games and Economic Behavior, Elsevier, vol. 110(C), pages 139-164.
    8. Yeon-Koo Che & Weijie Zhong, 2021. "Robustly Optimal Mechanisms for Selling Multiple Goods," Papers 2105.02828, arXiv.org, revised Aug 2024.
    9. Hart, Sergiu & Nisan, Noam, 2017. "Approximate revenue maximization with multiple items," Journal of Economic Theory, Elsevier, vol. 172(C), pages 313-347.
    10. Babaioff, Moshe & Gonczarowski, Yannai A. & Nisan, Noam, 2022. "The menu-size complexity of revenue approximation," Games and Economic Behavior, Elsevier, vol. 134(C), pages 281-307.
    11. Alon Eden & Michal Feldman & Ophir Friedler & Inbal Talgam-Cohen & S. Matthew Weinberg, 2021. "A Simple and Approximately Optimal Mechanism for a Buyer with Complements," Operations Research, INFORMS, vol. 69(1), pages 188-206, January.
    12. Shih-Tang Su & Vijay G. Subramanian, 2022. "Order of Commitments in Bayesian Persuasion with Partial-informed Senders," Papers 2202.06479, arXiv.org.
    13. Hartline, Jason D. & Kleinberg, Robert & Malekian, Azarakhsh, 2015. "Bayesian incentive compatibility via matchings," Games and Economic Behavior, Elsevier, vol. 92(C), pages 401-429.
    14. Plehn-Dujowich, Jose M., 2009. "Endogenous growth and adverse selection in entrepreneurship," Journal of Economic Dynamics and Control, Elsevier, vol. 33(7), pages 1419-1436, July.
    15. Babaioff, Moshe & Blumrosen, Liad & Schapira, Michael, 2013. "The communication burden of payment determination," Games and Economic Behavior, Elsevier, vol. 77(1), pages 153-167.
    16. Tomer Siedner, 2019. "Optimal pricing by a risk-averse seller," Discussion Paper Series dp725, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    17. Azar, Pablo D. & Kleinberg, Robert & Weinberg, S. Matthew, 2019. "Prior independent mechanisms via prophet inequalities with limited information," Games and Economic Behavior, Elsevier, vol. 118(C), pages 511-532.
    18. Devanur, Nikhil R. & Haghpanah, Nima & Psomas, Alexandros, 2020. "Optimal multi-unit mechanisms with private demands," Games and Economic Behavior, Elsevier, vol. 121(C), pages 482-505.
    19. Çağıl Koçyiğit & Halil I. Bayrak & Mustafa Ç. Pınar, 2018. "Robust auction design under multiple priors by linear and integer programming," Annals of Operations Research, Springer, vol. 260(1), pages 233-253, January.
    20. Tang, Pingzhong & Wang, Zihe, 2017. "Optimal mechanisms with simple menus," Journal of Mathematical Economics, Elsevier, vol. 69(C), pages 54-70.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2211.05732. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.