On $${\alpha }$$ α -roughly weighted games

Gvozdeva et al. (Int J Game Theory, doi: 10.1007/s00182-011-0308-4 , 2013 ) 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 $${\mathcal {C}}_\alpha $$ C α consists of all simple games permitting a weighted representation such that each winning coalition has a weight of at least $$1$$ 1 and each losing coalition a weight of at most $$\alpha $$ α . For a given game the minimal possible value of $$\alpha $$ α 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 between this concept and the cost of stability, i.e. the minimum amount of external payment to ensure stability in a coalitional game, is uncovered. Copyright Springer-Verlag Berlin Heidelberg 2014