Gradient-Based Bilevel Optimization for Principal–Agent Contract Design

We study a bilevel \emph{max–max} optimization framework for principal–agent contract design, where a principal selects incentives to maximize utility while anticipating the agent’s best response. This setting, central in moral hazard and contract theory, arises in applications such as market design, delegated portfolio management, hedge fund fee design, and executive compensation. While some special cases admit closed-form solutions, realistic contracts involve nonlinear utilities, stochastic dynamics, and high-dimensional actions for which analytical solutions do not exist. We remove this dependence on closed forms by introducing a scalable framework that solves general principal–agent problems without restrictive assumptions. Our approach adapts \emph{machine learning} techniques for bilevel optimization—specifically, implicit differentiation with conjugate gradient (CG)—to compute hypergradients using only Hessian–vector products, avoiding explicit Hessian inversion and scaling to high-dimensional contracts. Applied to the classic Holmström–Milgrom model, the method recovers the exact analytical optimum and converges reliably from random initialization. Because it is matrix-free and problem-agnostic, it extends directly to complex, nonlinear principal–agent models, providing a new computational tool for contract design, specifically in financial markets.