  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.
