[Paper Review] Principal-agent problem with multiple principals

Kaitong Hu, Zhenjie Ren+1 more|arXiv (Cornell University)|Mar 30, 2019

Stochastic processes and financial applications9 citations

TL;DR

This paper studies a continuous-time moral hazard model with one agent facing multiple principals, where the agent optimally switches between principals over time. Using backward SDEs and a randomized formulation, the authors derive a representation of the agent’s value function and optimal effort, and establish convergence to a mean field game equilibrium as the number of principals grows large.

ABSTRACT

We consider a moral hazard problem with multiple principals in a continuous-time model. The agent can only work exclusively for one principal at a given time, so faces an optimal switching problem. Using a randomized formulation, we manage to represent the agent's value function and his optimal effort by an Itô process. This representation further helps to solve the principals' problem in case we have infinite number of principals in the sense of mean field game. Finally the mean field formulation is justified by an argument of propagation of chaos.

Motivation & Objective

• To model a moral hazard setting where a single agent can work exclusively for one principal at a time, facing an optimal switching problem.
• To extend the classical principal-agent framework to multiple principals by introducing a randomized formulation and backward SDE techniques.
• To solve the principals' optimal contracting problem in the limit of infinitely many principals using mean field game theory.
• To justify the mean field approximation via a novel backward propagation of chaos argument.
• To provide a tractable representation of the agent’s value function and optimal effort as an Itô process.

Proposed method

• Formulate the agent’s problem using a randomized optimal switching framework, where the agent chooses among principals stochastically.
• Represent the agent’s value function and optimal effort via a backward stochastic differential equation (BSDE), enabling dynamic characterization.
• Use the dual representation of the agent’s optimization to link the contract payoff and effort to the solution of a BSDE with driver $ c^*(Z_t) $.
• Introduce a mean field game formulation in the limit of infinitely many principals, where each principal faces a representative agent with a common law of motion.
• Establish convergence of the finite-$ n $ principal problem to the mean field solution using a backward propagation of chaos argument.
• Leverage Grönwall’s inequality and law of large numbers to prove convergence of empirical measures of principal types to the limiting mean field distribution.

Experimental results

Research questions

• RQ1 How can the principal-agent problem be extended to a setting with multiple principals and a single agent who can only serve one principal at a time?
• RQ2 Can the agent’s optimal switching behavior be characterized using backward SDEs and a randomized formulation?
• RQ3 What is the limiting behavior of the contracting problem when the number of principals tends to infinity?
• RQ4 How can the mean field game formulation be justified rigorously in this context?
• RQ5 Does the finite-$ n $ principal problem converge to the mean field solution, and under what conditions?

Key findings

• The agent’s value function and optimal effort are represented as an Itô process through a backward SDE, enabling a dynamic characterization of the optimal contract.
• The optimal effort $ \alpha^*_t $ is given by $ \arg\max_a \{ aZ_t - c(a) \} $, where $ Z_t $ is the martingale part of the BSDE solution.
• The finite-$ n $ principal problem converges to the mean field game equilibrium as $ n \to \infty $, with convergence rate controlled by $ \mathbb{E}[d^2_0(p^n, p^*)] \leq C(\mathbb{E}[d^2_0(\nu^n, p^*)] + n^{-2}) $.
• The backward propagation of chaos is established, showing that the empirical measure of principal types converges to the limiting mean field distribution.
• The value function of each principal in the finite-$ n $ case converges to the mean field game value, with the limit given by $ \mathbb{E}\left[\int_0^T \beta^*_u(\alpha^*_u W^i_u - U(\theta^i_u) + 1)du + \beta^*_T(W^i_T - U(Y^{*,i}_T))\right] $.
• The convergence is proven under boundedness and Lipschitz continuity assumptions on the utility and cost functions, ensuring stability of the mean field limit.