This work is concerned about obtaining insight into two main goals: the structure of important subclasses of simple games (or monotonic Boolean functions) and into the qualitative behavior of power indices when applied to simple games. The notion of power can be viewed here as a price or as an influence. Power indices are important mathematical tools broadly used in Social Science, Political Science and Economics, as well as in other technological sciences.
The first part, devoted to the structure of simple games, presents a novel concept of dimension and codimension for the class of simple games. It introduces a dual concept of dimension which is obtained by considering the union instead of the intersection as the basic operation, and several other extensions of the notion of dimension. It also proves the existence and uniqueness of a minimum subclass of games, with the property that every simple game can be expressed as an intersection, or respectively the union, of its elements. We show the importance of these subclasses in the description of a simple game, and give a practical interpretation of them. Some results of this part are already published in the journal: TOP, vol. 17, núm. 2, p. 407-414. The results in this former part are extensions of some results on dimension by Taylor and Zwicker and published in 1993.
In the second part we introduce and examine the egalitarian property for the most established power indices. This property means that after intersecting a game with a symmetric game or k-out-of-n game, the difference between values of two comparable players does not increase. We prove that the Shapley¿Shubik index, the Banzhaf and Johnston scores satisfy this property. We also give counterexamples for the Holler, Deegan¿Packel, normalised Banzhaf and Johnston indices. We prove that Egalitarian property is a stronger condition for efficient power indices than the Lorentz domination, which extends Peleg¿s 1992 result.
In the third part we study ordinal equivalence of the Shapley¿Shubik and Penrose¿Banzhaf¿Coleman indices with the Johnston index. We prove that these three indices are ordinally equivalent for the semicomplete simple games, a newly defined class that contains complete games, so that the three power indices are ordinally equivalent for the most real-world examples of binary voting systems. The ordinal equivalence arises for the three power indices and it holds for a larger class of complete games. This result is an extension of the Diffo Lambo and Moulen¿s result 2002.