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A Tale of a Principal and Many, Many Agents

Author

Listed:
  • Romuald Elie

    (Université Paris–Est Marne–la–Vallée, Champs–sur–Marne 77454 Marne–la–Vallée CEDEX 2, France)

  • Thibaut Mastrolia

    (CMAP, École Polytechnique, Université Paris Saclay, 91128 Palaiseau, France)

  • Dylan Possamaï

    (Columbia University, New York, New York 10027)

Abstract

In this paper, we investigate a moral hazard problem in finite time with lump-sum and continuous payments, involving infinitely many agents with mean-field type interactions, hired by one principal. By reinterpreting the mean-field game faced by each agent in terms of a mean-field forward-backward stochastic differential equation (FBSDE), we are able to rewrite the principal’s problem as a control problem of the McKean-Vlasov stochastic differential equations. We review one general approach to tackling it, introduced recently using dynamic programming and Hamilton-Jacobi-Bellman (HJB for short) equations, and mention a second one based on the stochastic Pontryagin maximum principle. We solve completely and explicitly the problem in special cases, going beyond the usual linear-quadratic framework. We finally show in our examples that the optimal contract in the N -players’ model converges to the mean-field optimal contract when the number of agents goes to +∞.

Suggested Citation

  • Romuald Elie & Thibaut Mastrolia & Dylan Possamaï, 2019. "A Tale of a Principal and Many, Many Agents," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 440-467, May.
    • Handle: RePEc:inm:ormoor:v:44:y:2019:i:2:p:440-467
    DOI: 10.1287/moor.2018.0931
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    References listed on IDEAS

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    Cited by:

    1. Emma Hubert & Thibaut Mastrolia & Dylan Possamai & Xavier Warin, 2020. "Incentives, lockdown, and testing: from Thucydides's analysis to the COVID-19 pandemic," Papers 2009.00484, arXiv.org, revised Apr 2022.
    2. Herty, Michael & Steffensen, Sonja & Thünen, Anna, 2022. "Multiscale control of Stackelberg games," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 468-488.
    3. Camilo Hern'andez & Dylan Possamai, 2023. "Time-inconsistent contract theory," Papers 2303.01601, arXiv.org.
    4. Xie, Yimei & Ding, Chuan & Li, Yang & Wang, Kaihong, 2023. "Optimal incentive contract in continuous time with different behavior relationships between agents," International Review of Financial Analysis, Elsevier, vol. 86(C).
    5. Marcel Nutz & Yuchong Zhang, 2021. "Mean Field Contest with Singularity," Papers 2103.04219, arXiv.org.
    6. Daniel Krv{s}ek & Dylan Possamai, 2023. "Randomisation with moral hazard: a path to existence of optimal contracts," Papers 2311.13278, arXiv.org.
    7. Steven Campbell & Yichao Chen & Arvind Shrivats & Sebastian Jaimungal, 2021. "Deep Learning for Principal-Agent Mean Field Games," Papers 2110.01127, arXiv.org.
    8. Bastien Baldacci & Philippe Bergault, 2021. "Optimal incentives in a limit order book: a SPDE control approach," Papers 2112.00375, arXiv.org, revised Oct 2022.
    9. René Carmona & Gökçe Dayanıklı & Mathieu Laurière, 2022. "Mean Field Models to Regulate Carbon Emissions in Electricity Production," Dynamic Games and Applications, Springer, vol. 12(3), pages 897-928, September.
    10. Arvind V. Shrivats & Dena Firoozi & Sebastian Jaimungal, 2022. "A mean‐field game approach to equilibrium pricing in solar renewable energy certificate markets," Mathematical Finance, Wiley Blackwell, vol. 32(3), pages 779-824, July.
    11. Rene Carmona, 2020. "Applications of Mean Field Games in Financial Engineering and Economic Theory," Papers 2012.05237, arXiv.org.
    12. Xiang Yu & Yuchong Zhang & Zhou Zhou, 2020. "Teamwise Mean Field Competitions," Papers 2006.14472, arXiv.org, revised May 2021.
    13. Dena Firoozi & Arvind V Shrivats & Sebastian Jaimungal, 2021. "Principal agent mean field games in REC markets," Papers 2112.11963, arXiv.org, revised Jun 2022.
    14. Dylan Possamai & Nizar Touzi, 2020. "Is there a Golden Parachute in Sannikov's principal-agent problem?," Papers 2007.05529, arXiv.org, revised Oct 2022.
    15. Robert Denkert & Ulrich Horst, 2024. "Extended mean-field games with multi-dimensional singular controls and non-linear jump impact," Papers 2402.09317, arXiv.org, revised Nov 2024.
    16. René Carmona, 2022. "The influence of economic research on financial mathematics: Evidence from the last 25 years," Finance and Stochastics, Springer, vol. 26(1), pages 85-101, January.
    17. René Carmona & Peiqi Wang, 2021. "A Probabilistic Approach to Extended Finite State Mean Field Games," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 471-502, May.

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