  Gvozdeva, Hemaspaandra, and Slinko (2011) have introduced three hierarchies
for simple games in order to measure the distance of a given simple game to the
class of (roughly) weighted voting games. Their third class
$\mathcal{C}_\alpha$ consists of all simple games permitting a weighted
representation such that each winning coalition has a weight of at least 1 and
each losing coalition a weight of at most $\alpha$. For a given game the
minimal possible value of $\alpha$ is called its critical threshold value. We
continue the work on the critical threshold value, initiated by Gvozdeva et
al., and contribute some new results on the possible values for a given number
of voters as well as some general bounds for restricted subclasses of games. A
strong relation beween this concept and the cost of stability, i.e. the minimum
amount of external payment to ensure stability in a coalitional game, is
uncovered.
