  We study the computational complexity of an important property of simple,
regular and weighted games, which is decisiveness. We show that this concept
can naturally be represented in the context of hypergraph theory, and that
decisiveness can be decided for simple games in quasi-polynomial time, and for
regular and weighted games in polynomial time. The strongness condition poses
the main difficulties, while properness reduces the complexity of the problem,
especially if it is amplified by regularity. On the other hand, regularity also
allows to specify the problem instances much more economically, implying a
reconsideration of the corresponding complexity measure that, as we prove, has
important structural as well as algorithmic consequences.
