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

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Abstract

In 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.

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Notes

  1. If we modelize this example as a weighted game the weights for yes are the same, the details are exposed in Example 1.3-(iii).

  2. In the Sect. 6 we shall say something about axiomatization of \(I^{YN}_p\).

  3. This shows that the null-axiom and the reduced axiom at the highest level imply YA-equal treatment on unanimity games, which by transfer can be extended to all games. YA-equal treatment for the (3, 2) game V and players \(p,r \in N\) means that \(P_p^{YA}[V] = P_r^{YA}[V]\) whenever \(V(S)-V(S_{p\downarrow })=V(S)-V(S_{r\downarrow })\) for all tripartitions \(S \in 3^N\) with \(p,r \in S_1\). Analogously, one may consider YN- and AN-equal treatment with the corresponding implications.

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Acknowledgments

The ideas of the paper were discussed, and the paper itself was prepared mostly during some exchange visits of the two authors. Both are grateful to the hosting departments for their warm hospitality. They also acknowledge a grant from GNAMPA, CNR, supporting the visit in Italy of the first author and Grant MTM2012-34426/FEDER of the Ministry of Economy and Competitiveness. The research of the second author was partially supported by Ministero dell’Istruzione, dell’Universitá e della Ricerca Scientifica (COFIN 2009). We also thank the three referees of the paper and Giulia Bernardi for their helpful comments, which allowed to improve the final version of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Technical University of Catalonia (Campus of Manresa), Manresa, Spain

    Josep Freixas

  2. Politecnico di Milano, Milano, Italy

    Roberto Lucchetti

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  1. Josep Freixas
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  2. Roberto Lucchetti
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Correspondence to Roberto Lucchetti.

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Freixas, J., Lucchetti, R. Power in voting rules with abstention: an axiomatization of a two components power index. Ann Oper Res 244, 455–474 (2016). https://doi.org/10.1007/s10479-016-2124-5

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