A Tale of a Principal and Many, Many Agents

• 成果类型: Article
• 署名作者: Elie, Romuald; Mastrolia, Thibaut; Possamai, Dylan
• 署名单位: Universite Gustave-Eiffel; Institut Polytechnique de Paris; Ecole Polytechnique; Universite Paris Saclay; Columbia University
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2018.0931
• 发表日期: 2019
• 页码: 440-467
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; mean-field games; nonlinear 2nd-order equations; infinite dimensions; VISCOSITY SOLUTIONS; DISCRETE-TIME; moral hazard; linearity; systems

摘要

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 +infinity.