On <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${\alpha }$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">α</mi> </math> </EquationSource> </InlineEquation>-roughly weighted games
Gvozdeva et al. (Int J Game Theory, doi:<ExternalRef> <RefSource>10.1007/s00182-011-0308-4</RefSource> <RefTarget Address="10.1007/s00182-011-0308-4" TargetType="DOI"/> </ExternalRef>, <CitationRef CitationID="CR17">2013</CitationRef>) 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$${\mathcal {C}}_\alpha $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi mathvariant="script">C</mi> <mi mathvariant="italic">α</mi> </msub> </math> </EquationSource> </InlineEquation> consists of all simple games permitting a weighted representation such that each winning coalition has a weight of at least <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>1</mn> </mrow> </math> </EquationSource> </InlineEquation> and each losing coalition a weight of at most <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$\alpha $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">α</mi> </math> </EquationSource> </InlineEquation>. For a given game the minimal possible value of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$\alpha $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">α</mi> </math> </EquationSource> </InlineEquation> 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
