We consider a class of N-player stochastic games of multi-dimensional singular

control, in which each player faces a minimization problem of monotone-follower type with

submodular costs. We call these games monotone-follower games. In a not necessarily

Markovian setting, we establish the existence of Nash equilibria. Moreover, we introduce

a sequence of approximating games by restricting, for each n ∈ ℕ, the players' admissible

strategies to the set of Lipschitz processes with Lipschitz constant bounded by n. We prove

that, for each n ∈ ℕ, there exists a Nash equilibrium of the approximating game and that the

sequence of Nash equilibria converges, in the Meyer-Zheng sense, to a weak (distributional)

Nash equilibrium of the original game of singular control. As a byproduct, such a convergence

also provides approximation results of the equilibrium values across the two classes of games.

We finally show how our results can be employed to prove existence of open-loop Nash

equilibria in an N-player stochastic differential game with singular controls, and we propose

an algorithm to determine a Nash equilibrium for the monotone-follower game.