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 in action and inaction region. We then show that a solution of the MFG of capacity expansion induces approximate Nash equilibria for the N-player games with approximation error 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.
(Joint work with L. Campi, M. Ghio and G. Livieri)