Abstract
Proportionality plays a role in many principles of fairness or justice. In particular, it is often invoked in electoral systems aiming at a similarity of opinion distributions in the parliament and in the electorate at large. The proportional systems of representation (PR systems, for short) strive for maximal similarity between these two distributions. Unfortunately, the concept of proportionality is imprecise in two senses: it is vague and ambiguous. In other words, once a clear criterion of what we mean by proportionality is given, different PR systems may differ in the degree of proportionality achieved in any given election. This is what will be referred to as the vagueness of proportionality (of election outcomes). Proportionality is, however, also ambiguous in that it refers to different things depending on how the voters are expected to signal their opinions and on what is it that one wishes to distribute proportionally.
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Notes
Candidate C receives x Borda points from a voter if this voter ranks x candidates lower than C. Summing the Borda points given to C by all voters, we get C’s Borda score. The Borda Count is a system that ranks candidates in the order of their Borda scores.
A reviewer accurately points out that the concept of proportionality per se is neither vague nor ambiguous in the purely technical sense. Proportionality refers to a relation between two distributions, e.g. of votes and seats. The smaller the difference—measured with some agreed-upon metric—the more proportional is the system. However, and this where ambiguity enters the picture, votes or ballots can be of different kinds. Hence, a seat distribution that is reasonably proportional in the sense of plurality ballots, may be rather disproportional when approval or STV ballots are used. Vagueness, in turn, is a result of using different PR formulae, like in the example at the beginning of this section.
A brief analysis of results and campaign is given by Nurmi and Nurmi (2012).
The island province of Åland is a single-member constituency. Its party system also differs from the mainland. Hence it is excluded from the present discussion.
The new system—supported by a majority in the 2007–2011 parliament—is unlikely to be adopted anytime soon, since it was not accepted by the majority government following the 2011 election. So it remains a theoretical benchmark for comparing election results.
In the table the strict preference orders are written from the most preferred (top-most) to the last preferred one. We shall also use ≻ to express strict preference so that A≻B denotes a strict preference of A over B.
A reviewer points out that the Martin index discussed by Riker (1986) is a precursor of the Penrose one. Overall, the literature on power indices in vast. For a thorough historical and theoretical account, see Felsenthal and Machover (1998). A more recent treatment is Laruelle and Valenciano (2008). For applications to the European Union, see Cichocki and Zyczkowski (2010).
For a recent and relatively extensive discussion on fair distribution of voting power among representatives, the reader is referred to Cichocki and Zyczkowski (2010).
The argument is a slight modification of Baigent’s (1987, 163) illustration.
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Acknowledgements
The author wishes to thank two anonymous reviewers for constructive criticism and suggestions. The support of Academy of Finland and University of Turku through their funding of the Public Choice Research Centre of University of Turku is gratefully acknowledged.
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Nurmi, H. Some remarks on the concept of proportionality. Ann Oper Res 215, 231–244 (2014). https://doi.org/10.1007/s10479-012-1252-9
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DOI: https://doi.org/10.1007/s10479-012-1252-9