The Nakamura number is an appropriate invariant of a simple game in order to study the existence of social equilibria and the possibility of cycles. For symmetric quota games its number can be obtained by an easy formula. For some subclasses of simple games the corresponding Nakamura number has...

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>...</equationsource></equationsource></inlineequation></citationref></refsource></externalref>

Some distinguished types of voters, as vetoes, passers or nulls, as well as some others, play a significant role in voting systems because they are either the most powerful or the least powerful voters in the game independently of the measure used to evaluate power. In this paper we are...