Using the multilinear extension to study some probabilistic power indices

The final publication is available at Springer via http://dx.doi.org/10.1007/s10726-016-9514-6We consider binary voting systems modeled by a simple game, in which voters vote independently of each other, and the probability distribution over coalitions is known. The Owen’s multilinear extension of the simple game is used to improve the use and the computation of three indices defined in this model: the decisiveness index, which is an extension of the Banzhaf index, the success index, which is an extension of the Rae index, and the luckiness index. This approach leads us to prove new properties and inter-relations between these indices. In particular it is proved that the ordinal equivalence between success and decisiveness indices is achieved in any game if and only if the probability distribution is anonymous. In the anonymous case, the egalitarianism of the three indices is compared, and it is also proved that, for these distributions, decisiveness and success indices respect the strength of the seats, whereas luckiness reverses this order.Peer ReviewedPostprint (author's final draft

A Three-Dimensional Voting System in Hong Kong

The voting system in the Legislative Council of Hong Kong (Legco) is sometimes unicameral and sometimes bicameral, depending on whether the bill is proposed by the Hong Kong government. Therefore, although without any representative within Legco, the Hong Kong government has certain degree of legislative power --- as if there is a virtual representative of the Hong Kong government within the Legco. By introducing such a virtual representative of the Hong Kong government, we show that Legco is a three-dimensional voting system. We also calculate two power indices of the Hong Kong government through this virtual representative and consider the $C$-dimension and the $W$-dimension of Legco. Finally, some implications of this Legco model to the current constitutional reform in Hong Kong will be given

Power theories for multi-choice organizations and political rules: Rank-order equivalence

AbstractVoting power theories measure the ability of voters to influence the outcome of an election under a given voting rule. In general, each theory gives a different evaluation of power, raising the question of their appropriateness, and calling for the need to identify classes of rules for which different theories agree. We study the ordinal equivalence of the generalizations of the classical power concepts–the influence relation, the Banzhaf power index, and the Shapley–Shubik power index–to multi-choice organizations and political rules. Under such rules, each voter chooses a level of support for a social goal from a finite list of options, and these individual choices are aggregated to determine the collective level of support for this goal. We show that the power theories analyzed do not always yield the same power relationships among voters. Thanks to necessary and/or sufficient conditions, we identify a large class of rules for which ordinal equivalence obtains. Furthermore, we prove that ordinal equivalence obtains for all linear rules allowing a fixed number of individual approval levels if and only if that number does not exceed three. Our findings generalize all the previous results on the ordinal equivalence of the classical power theories, and show that the condition of linearity found to be necessary and sufficient for ordinal equivalence to obtain when voters have at most three options to choose from is no longer sufficient when they can choose from a list of four or more options

Achievable hierarchies in voting games with abstention

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 the class of complete VGA. The aim of this paper is twofold. Firstly, we show that all hierarchies are achievable in a subclass of weighted VGA, the class of weighted games for which a single weight is assigned to voters. Secondly, we conduct a partial study of achievable hierarchies within the subclass of H-complete games, that is, complete games under stronger versions of influence relation. (C) 2013 Elsevier B.V. All rights reserved.Peer ReviewedPostprint (author’s final draft

A probabilistic unified approach for power indices in simple games

The final publication is available at Springer via https://doi.org/10.1007/978-3-662-60555-4_11Many power indices on simple games have been defined trying to measure, under different points of view, the “a priori” importance of a voter in a collective binary voting scenario. A unified probabilistic way to define some of these power indices is considered in this paper. We show that six well-known power indices are obtained under such a probabilistic approach. Moreover, some new power indices can naturally be obtained in this way.Peer ReviewedPostprint (author's final draft

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

The final publication is available at Springer via http://dx.doi.org/10.1007/s10479-016-2124-5In order to study voting situations when voters can also abstain and the output is 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.Peer ReviewedPostprint (author's final draft