Power in voting rules with abstention: an axiomatization of a two components power index

Abstract

Abstract:In order to study voting situations when voters can also abstain and the output can be only binary, i.e., either approval or rejection, a new extended model of voting rule was defined. Accordingly, indices of power, in particular Banzhaf's index, were considered. In this paper we argue that in this context a power index should be a pair of real numbers, since this better highlights the power of a voter in two different cases, i.e., her being crucial when switching from being in favor to abstain, and from abstain to be contrary. We also provide an axiomatization for both indices, and from this a characterization as well of the standard Banzhaf index (the sum of the former two) is obtained. Some examples are provided to show how the indices behave. Powered by Editorial Manager® and ProduXion Manager® from Aries Systems CorporationPower in voting rules with abstention: an axiomatization of a two components power index Abstract: In order to study voting situations when voters can also abstain and the output can be only binary, i.e., either approval or rejection, a new extended model of voting rule was defined. Accordingly, indices of power, in particular Banzhaf's index, were considered. In this paper we argue that in this context a power index should be a pair of real numbers, since this better highlights the power of a voter in two different cases, i.e., her being crucial when switching from being in favor to abstain, and from abstain to be contrary. We also provide an axiomatization for both indices, and from this a characterization as well of the standard Banzhaf index (the sum of the former two) is obtained. Some examples are provided to show how the indices behave.

An Axiomatization for Two Power Indices for (3,2)-Simple Games

“…Moreover, it can be of future interest to study other axiomatizations of power indices for (3,2)-simple games, not related to the behavior on unanimity games, as done in [Freixas and Lucchetti [2016]] and in [Bernardi [2017]] only for the Banzhaf index for (3,2)-simple games.…”

An Axiomatization for Two Power Indices for (3,2)-Simple Games

“…Moreover, it can be of future interest to study other axiomatizations of power indices for (3,2)-simple games, not related to the behavior on unanimity games, as An Axiomatization of the Shapley-Shubik and the Banzhaf Indices done, only for the Banzhaf index for (3,2)-simple games, in Freixas and Lucchetti [2016] who use the axiom of individual block effect and in Bernardi [2018], who uses the average gain-loss balance. Following this line of investigation, it could be interesting also to use different properties extending to (3,2)-simple games other axiomatizations for cooperative games such as Casajus [2012].…”

Does everyone have equal voting power?

“…Let n i = ∑ m i j=1 n j i . n i is the size of the 7 Also see: Freixas and Lucchetti (2016), Leech (2013), Kurz (2012), Lindner (2008), Lindner and Owen (2008), Lindner and Machover (2004), Tolle (2003) and Fischer and Schotter (1978). 8 Also see: Deemen and Rusinowska (2003) and Gelmen (2004).…”