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

TL;DR

This paper introduces a primal-dual gradient-based method utilizing automatic differentiation to efficiently compute equilibrium prices in mean-field game models, adaptable to high-dimensional and complex scenarios.

Contribution

The authors develop a novel, flexible Lagrangian approach with automatic differentiation for equilibrium price computation in mean-field games, enhancing scalability and ease of implementation.

Findings

Efficient algorithm for equilibrium prices in mean-field games.

Modular implementation suitable for high-dimensional models.

Automatic differentiation simplifies gradient calculations.

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.