Abstract
We exhibit the hidden beauty of weighted voting and voting power by applying a generalization of the Penrose-Banzhaf index to social choice rules. Three players who have multiple votes in a committee decide between three options by plurality rule, Borda’s rule, and antiplurality rule, or one of the many scoring rules in between. A priori influence on outcomes is quantified in terms of how players’ probabilities are pivotal for the committee decision compared to a dictator. The resulting numbers are represented in triangles that map out structurally equivalent voting weights. Their geometry and color variation reflect fundamental differences between voting rules, such as their inclusiveness and transparency.
We thank an anonymous referee for his or her thoughtful reading and several helpful suggestions.
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Notes
- 1.
We recommend the contributions in Holler and Nurmi (2013) for a good overview of typical applications of power indices. Somewhat atypical applications are discussed by Kovacic and Zoli (2021) and Napel and Welter (2021). Napel (2019) provides a short introduction to power measurement with many further references.
- 2.
See Kurz et al. (2020) for details and related literature: the article defines weighted committee games, characterizes and counts equivalence classes for selected voting rules, and provides lists of structurally distinct committees. Mayer and Napel (2021) does similarly for the special case of scoring rules. Kurz et al. (2021) generalizes the Penrose-Banzhaf and Shapely-Shubik indices to committee games. For a practical application of the framework, see Mayer and Napel (2020).
- 3.
See Fig. 5 in Mayer and Napel (2021). It provides a map of the 51 Borda equivalence classes for \(n=m=3\) and \(w_1\ge w_2\ge w_3\) with a reference distribution of weights for each class. Maps could be constructed for more than three alternatives, too, but the higher number of preference profiles and perturbations has considerable computational costs. Equivalence classes for scores \(0<s<1\) change fast: the number of Borda classes rises from 51 to 505 and \(\ge \! 2251\) for \(m=3\), 4, and 5 (Kurz et al. 2020). Corresponding analogues of Fig. 3 exhibit smoother transitions with even more shades of color.
- 4.
The Shapley-Shubik index (1954) belongs to the same family of indices but supposes a positive correlation of yes-or-no preferences across voters. In technical terms, it assumes an impartial anonymous culture (IAC), while the Penrose-Banzhaf index reflects an impartial culture (IC). The Holler-Packel index (1983) does not consider all coalitions of yes-supporters but only minimal winning coalitions \(S\subseteq N\) in which every yes-vote is pivotal for the outcome.
- 5.
Replacing the IC assumption that underlies Eq. (1) by the IAC assumption (cf. fn. 4) naturally generalizes the Shapley-Shubik index (see Kurz et al. 2021). By contrast, generalization of the Holler-Packel index would first require the definition of a suitable analogue of minimal winning coalitions in weighted committee games. One possibility would be to study each winning alternative \(a\in A\) separately and to consider a-minimal preference profiles \(\textbf{P}\) where \(\rho (\textbf{P})=a\) such that \(\rho (\textbf{P}')\ne a\) for any profile \(\textbf{P}'\) in which a is ranked lower by some voter with constant preferences on subset \(A\smallsetminus a\).
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Kurz, S., Mayer, A., Napel, S. (2023). The Art and Beauty of Voting Power. In: Leroch, M.A., Rupp, F. (eds) Power and Responsibility. Springer, Cham. https://doi.org/10.1007/978-3-031-23015-8_7
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