Hierarchies achievable in simple games

Abstract

A previous work by Friedman et al. (Theory and Decision, 61:305–318, 2006) introduces the concept of a hierarchy of a simple voting game and characterizes which hierarchies, induced by the desirability relation, are achievable in linear games. In this paper, we consider the problem of determining all hierarchies, conserving the ordinal equivalence between the Shapley–Shubik and the Penrose–Banzhaf–Coleman\ud power indices, achievable in simple games. It is proved that only four hierarchies are\ud non-achievable in simple games.Moreover, it is also proved that all achievable hierarchies\ud are already obtainable in the class of weakly linear games. Our results prove that\ud given an arbitrary complete pre-ordering defined on a finite set with more than five elements, it is possible to construct a simple game such that the pre-ordering induced by the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices coincides with the given pre-ordering.Postprint (published version

The influence relation for ternary voting games

“…The number of players in the ordered list of equivalence classes is called a hierarchy. Many papers in the literature deal with the concept of hierarchies in the context simple games (Carreras and Freixas, 1996;Friedman et al, 2006;Bean et al, 2008;Freixas and Pons, 2008). In Section 7 this concept is extended to TVGs.…”

Achievable hierarchies in voting games with abstention

“…Indeed, in Friedman et al [20] it is proved that complete simple games and, particularly, weighted simple games show many di¤erent hierarchies, although two sequences of hierarchies are never achievable. Freixas and Pons [17] proved that all hierarchies are achievable in the class of weakly complete games as long as the number of voters is greater than 5. For less than 6 voters, only four hierarchies are not achieved in this class of games, and they are : (1; 1; 1); (1; 1; 1; 1), (2; 1; 1) and (2; 1; 1; 1).…”

“…For example, a …ve-player game G has hierarchy (1; 3; 1) means that one player has less in ‡uence than all the others, one player has more, and the other three players have the same in ‡uence as each other, they are equivalent. From works by Friedman et al [20] and Freixas and Pons [17] on hierarchies, it can be stated that given any complete pre-ordering de…ned on a …nite set of more than 5 voters, it is possible to construct a simple game such that the pre-orderings induced by SS [27] and the BColeman ( [2] and [6]) power indices coincide with the given pre-ordering (when the number of voters is 5 or less, there are four non achievable hierarchies). These results hold under the condition that the game be modeled by a simple game.…”

Political influence in multi-choice institutions: cyclicity, anonymity, and transitivity

“…fw 1 ; w 2 ; :::; w v g where fw 1 ; w 2 ; :::; w k g is the value set of G whose elements are any v objects equipped with a 1 0 The works of Tomiyama (1987) and Di¤o and Moulen (2002) have inspired several other studies on the topic of ordinal equivalence and the subject of achievable hierarchies (Carreras and Freixas 2008;Freixas and Pons 2008;Bean et al 2008;Friedman et al 2006), but all have been conducted in the basic framework of binary voting games. 1 1 Tchantcho et al (2008) compare the preorderings (SS) and (BC) with the combined in ‡uence relation; so our contribution in this regard is in providing a basis for the ordinal equivalence of six relations, which is a richer result.…”

“…It consists of a binary relation called "replacement relation" or "in ‡uence relation", which ranks players according to their a priori in ‡uence in a vote. 1 The two approaches have been essentially developed in the basic framework of binary voting games, where a voter may only vote "yes" or "no" (see, e.g., Laruelle and Valenciano 2001;Carreras and Freixas 2005;Freixas and Pons 2008).…”