Search

In this paper, we extend the Banzhaf–Coleman–Dubey–Shapley sensitivity index to the class of dichotomous voting games with several levels of approval in input, also known as (j, 2)-simple games. For previous works, on classical simple games ((2, 2)-simple games), a sensitivity index reflects the volatility or degree of suspense in the voting body. Using a set of independent axioms, we provide an axiomatic characterization of that extension on the class of (j, 2)-simple games. [ABSTRACT FROM AUTHOR]

In this paper, we propose an axiomatization of the totally critical raw Banzhaf index (TCRBI). In the literature, the TCRBI was introduced as a generalization of the Banzhaf power index on the class of yes–no voting games with several ordered levels of approval in input ((j, 2)-simple games). Its principal strength, compared to earlier generalizations of the Banzhaf power index, is the fact that it preserves the desirability relation regardless of the number of input levels of approval. Our main result is that, up to a multiplicative positive real number, the TCRBI is the only power index on the class of (j, 2)-simple games, satisfying the following set of axioms : Equal treatment, Positivity, Null Player, and Linear Transfer. [ABSTRACT FROM AUTHOR]

We introduce the axiom of composition independence for power indices and value maps. In the context of compound (two-tier) voting, the axiom requires the power attributed to a voter to be independent of the second-tier voting games played in all constituencies other than that of the voter. We show that the Banzhaf power index is uniquely characterized by the combination of composition independence, four semivalue axioms (transfer, positivity, symmetry, and dummy), and a mild efficiency-related requirement. A similar characterization is obtained as a corollary for the Banzhaf value on the space of all finite games (with transfer replaced by additivity). [ABSTRACT FROM AUTHOR]

We introduce a new generating function based method to compute the Banzhaf, Deegan–Packel, Public Good (a.k.a. the Holler power index) and Shapley–Shubik power indices in the presence of incompatibility among players. More precisely, given a graph G = V , E with V the set of players and E the edge set, our extension involves multiple weighted voting games (MWVG's) and incompatible players, i.e., pairs of players belonging to E are not allowed to cooperate. The route to obtain the aforementioned generating functions comprises the use of a key lemma characterizing the set of minimal winning coalitions of the game with incompatibility due to Alonso-Meijide et al. (Appl Math Comput 252(1):377–387, 2015), a tool from combinatorial analysis, namely, the Omega calculus in partition analysis, and basic tools borrowed from commutative algebra involving the computation of certain quotients of polynomial rings module polynomial ideals. Using partition analysis, we obtain new generating functions to compute the Deegan–Packel and Public Good power indices with incompatibility leading to lower time complexity than previous results of Chessa (TOP 22(2):658–673, 2014) and some results of Alonso-Meijide et al. (Appl Math Comput 219(8):3395–3402, 2012). Using a conjunction of partition analysis and commutative algebra, we extend to MWVG's the generating function approach to compute the Banzhaf and Shapley–Shubik power indices in the presence of incompatibility. Finally, an example taken from the real-world, i.e., the European Union under the Lisbon Treaty, is used to illustrate the usefulness of the Omega package, a symbolic computational package that implements the Omega calculus in Mathematica, due to Andrews et al. (Eur J Comb 22(7):887–904, 2001) in the context of MWVG's by computing the PG power index of the associated voting game. [ABSTRACT FROM AUTHOR]

Participants in parliamentary voting-usually political parties-are evaluated with a value that assigns them predicted power with respect to their strength by so called power indices. However, in real world, political parties' representatives act not strictly as predicted in theory. One possibility how the representatives differ from theory is the way they form coalitions; the coalitions can be announced (for example official governmental coalition parties in multiparty parliaments) or hidden. To incorporate the coalition forming influence, Bilal et al. (, ) proposed to consider additional weights to possible coalitions into power indices. This article applies the concept of additional weights to calculate an ex post power distribution using Shapley-Shubik power index together with Banzhaf power index on real voting data, namely the data from the Chamber of Deputies of the Czech Parliament with the emphasis on the State Budget voting issues during 2006-2010 parliamentary period. [ABSTRACT FROM AUTHOR]