Voting games with abstention: Linking completeness and weightedness

Abstract

Weighted games for several levels of approval in input and output were introduced in [9]. An extension of the desirability relation for simple games, called the influence relation, was introduced for games with several levels of approval in input in [24] (see also [18]). However, there are weighted games not being complete for the influence relation, something different to what occurs for simple games. In this paper we introduce several extensions of the desirability relation for simple games and from the completeness of them it follows the consistent link with weighted games, which solves the existing gap. Moreover, we prove that the influence relation is consistent with a known subclass of weighted games: strongly weighted games.Keywords: Decision making process, Voting systems in democratic organizations, Multiple levels of approval, Weightedness and completeness, Desirability relations 2000 MSC: 91A12, 90B50, 91A35, 05C65, 94C10

Achievable hierarchies in voting games with abstention

“…In this paper, we point out some achievable and non achievable hierachies in the class of H-complete (3; 2) games, a class of game recently introduced by Freixas et al [18] in order to handle some shortcoming of the I-in ‡uence, namely the fact that there exist weighted (3; 2) games that are not I-complete. When the number of voters n, is two or three, the strict hierarchy is never achieved.…”

“…However, there exist weighted (3; 2) games that are not complete under the I-in ‡uence relation. Although I-completeness is not consistent for the notion of weighted (3; 2) games, it was shown in [18] that it is for the notion of strongly weighted (3; 2) games. The I-in ‡uence is indeed too demanding.…”

“…The purpose of this section is to prove that all hierarchies are achievable in the subclass of zero-centered strongly weighted games. For that, we will need the following result that was proved in [18], that establishes some links between weights and desirability relations for (3; 2) games. Proposition 4.2 Given two arbitrary players p and r in a weighted (3; 2) game, for any weight function representing it, we have : From each of these weighted representations we de…ne a zero-sum strongly weighted (3; 2) voting game as follows : the quota is Q and the weights for voters are w(i) = (w i ; 0; w i ) for all i 2 N .…”

“…The I-in ‡uence is indeed too demanding. In [18], the three separate condition of the I-in ‡uence has been studied thus introducing a new class of complete games. such that both p and r belong to S 2 , V (S 1 [ fpg; S 2 n fpg;…”

“…Freixas et al [18] noticed some shortcoming in the I-in ‡uence. There are weighted games not being complete for the in ‡uence relation, something di¤erent to what occurs for simple games.…”