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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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