  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 natural number n, the players'
admissible strategies to the set of Lipschitz processes with Lipschitz constant
bounded by n. We prove that, for each 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 findings 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.
