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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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    1. 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.
    2. 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.
    3. Robert Denkert & Ulrich Horst, 2024. "Extended mean-field games with multi-dimensional singular controls and non-linear jump impact," Papers 2402.09317, arXiv.org.
    4. Bastien Baldacci & Philippe Bergault, 2021. "Optimal incentives in a limit order book: a SPDE control approach," Papers 2112.00375, arXiv.org, revised Oct 2022.
    5. Xiang Yu & Yuchong Zhang & Zhou Zhou, 2020. "Teamwise Mean Field Competitions," Papers 2006.14472, arXiv.org, revised May 2021.
    6. Dena Firoozi & Arvind V Shrivats & Sebastian Jaimungal, 2021. "Principal agent mean field games in REC markets," Papers 2112.11963, arXiv.org, revised Jun 2022.
    7. 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.
    8. 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).
    9. Rene Carmona, 2020. "Applications of Mean Field Games in Financial Engineering and Economic Theory," Papers 2012.05237, arXiv.org.
    10. 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.
    11. Camilo Hern'andez & Dylan Possamai, 2023. "Time-inconsistent contract theory," Papers 2303.01601, arXiv.org.
    12. Marcel Nutz & Yuchong Zhang, 2021. "Mean Field Contest with Singularity," Papers 2103.04219, arXiv.org.
    13. 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.
    14. 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.
    15. Daniel Krv{s}ek & Dylan Possamai, 2023. "Randomisation with moral hazard: a path to existence of optimal contracts," Papers 2311.13278, arXiv.org.
    16. Steven Campbell & Yichao Chen & Arvind Shrivats & Sebastian Jaimungal, 2021. "Deep Learning for Principal-Agent Mean Field Games," Papers 2110.01127, arXiv.org.
    17. Dylan Possamai & Nizar Touzi, 2020. "Is there a Golden Parachute in Sannikov's principal-agent problem?," Papers 2007.05529, arXiv.org, revised Oct 2022.

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