A primal-dual price-optimization method for computing equilibrium prices in mean-field games models

Abstract: We develop a simple yet efficient Lagrangian method for computing equilibrium prices in a mean-field game price-formation model. We prove that equilibrium prices are optimal in terms of a suitable criterion and derive a primal-dual gradient-based algorithm for computing them. One of the highlights of our computational framework is the efficient, simple, and flexible implementation of the algorithm using modern automatic differentiation techniques. Our implementation is modular and admits a seamless extension to high-dimensional settings with more complex dynamics, costs, and equilibrium conditions. Additionally, automatic differentiation enables a versatile algorithm that requires only coding the cost functions of agents. It automatically handles the gradients of the costs, thereby eliminating the need to manually form the adjoint equations.