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>

Tomiyama [Tomiyama, Y., 1987. Simple game, voting representation and ordinal power equivalence. International Journal on Policy and Information 11, 67-75] proved that, for every weighted majority game, the preorderings induced by the classical Shapley-Shubik and Penrose-Banzhaf-Coleman indices...

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...

It is well known that he influence relation orders the voters the same way as the classical Banzhaf and Shapley–Shubik indices do when they are extended to the voting games with abstention (VGA) in the class of complete games. Moreover, all hierarchies for the influence relation are achievable...

In this paper, we characterize the games in which Johnston, Shapley–Shubik and Penrose–Banzhaf–Coleman indices are ordinally equivalent, meaning that they rank players in the same way. We prove that these three indices are ordinally equivalent in semicomplete simple games, which is a newly...