Mean-Field Games of Finite-Fuel Capacity Expansion with Singular Controls

TL;DR

This paper analyzes mean-field games of finite-fuel capacity expansion with singular controls, providing a probabilistic approach to approximate Nash equilibria in large symmetric stochastic games.

Contribution

It introduces a probabilistic method to construct solutions for mean-field games with singular controls and demonstrates their use in approximating Nash equilibria for finite-player games.

Findings

Constructed MFG solutions via iterative Skorokhod reflection.

Established convergence of approximate Nash equilibria as number of players increases.

Extended the link between singular control and optimal stopping to mean-field settings.

Abstract

We study Nash equilibria for a sequence of symmetric $N$-player stochastic games of finite-fuel capacity expansion with singular controls and their mean-field game (MFG) counterpart. We construct a solution of the MFG via a simple iterative scheme that produces an optimal control in terms of a Skorokhod reflection at a (state-dependent) surface that splits the state space into action and inaction regions. We then show that a solution of the MFG of capacity expansion induces approximate Nash equilibria for the $N$-player games with approximation error $\varepsilon$ going to zero as $N$ tends to infinity. Our analysis relies entirely on probabilistic methods and extends the well-known connection between singular stochastic control and optimal stopping to a mean-field framework.