  We consider a class of zero-sum stopper vs. singular-controller games in
which the controller can only act on a subset $d_0<d$ of the $d$ coordinates of
a controlled diffusion. Due to the constraint on the control directions these
games fall outside the framework of recently studied variational methods. In
this paper we develop an approximation procedure, based on $L^1$-stability
estimates for the controlled diffusion process and almost sure convergence of
suitable stopping times. That allows us to prove existence of the game's value
and to obtain an optimal strategy for the stopper, under continuity and growth
conditions on the payoff functions. This class of games is a natural extension
of (single-agent) singular control problems, studied in the literature, with
similar constraints on the admissible controls.
