  In this paper we consider a local service-requirement assignment problem
named exact capacitated domination from an algorithmic point of view. This
problem aims to find a solution (a Nash equilibrium) to a game-theoretic model
of public good provision. In the problem we are given a capacitated graph, a
graph with a parameter defined on each vertex that is interpreted as the
capacity of that vertex. The objective is to find a DP-Nash subgraph: a
spanning bipartite subgraph with partite sets D and P, called the D-set and
P-set respectively, such that no vertex in P is isolated and that each vertex
in D is adjacent to a number of vertices equal to its capacity. We show that
whether a capacitated graph has a unique DP-Nash subgraph can be decided in
polynomial time. However, we also show that the nearby problem of deciding
whether a capacitated graph has a unique D-set is co-NP-complete.
