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Aitchison and angular distances for votes and seats


The following is a notebook in the cloud (1 MB), that one can best download and run on the desktop (see the browser in the lower right). It has initialisation cells for two small packages.

Readers who recently looked at the Brexit discussion will find the topic familiar, see

The summary of the notebook is:

Votes and seats satisfy only two of seven criteria for application of the Aitchison distance. Vectors of votes and seats, say for elections for political parties the House of Representatives, can be normalised to 1 or 100%, and then have the outward appearance of compositional data. The Aitchison geometry and distance for compositional data then might be considered for votes and seats too. However, there is an essential zero when a party gets votes but doesn't gain a seat, and a zero gives an undefined logratio. In geology, changing from weights to volumes affects the percentages but not the Aitchison distance. For votes and seats there are no different scales or densities per party component however, and thus reportioning (perturbation) would be improper. Another key issue is subcompositional dominance. For votes {10, 20, 70} and seats {20, 10, 70} it is essential that we consider three parties. For a disproportionality measure we would value it positively that there is a match on 70. The Aitchison distance looks at the ratio {10, 20, 70} / {20, 10, 70} = {1/2, 2, 1} and then neglects a ratio equal to 1. In this case it essentially compares the subcompositions, i.e. votes {10, 20} and seats {20, 10}, rescales to {1/3, 2/3} and {2/3, 1/3}, and finds high disproportionality. This means that it essentially looks at a two party outcome instead of a three party outcome. It follows that votes and seats are better served by another distance measure. Suggested is the angular distance and the Sine-Diagonal Inequality / Disproportionality (SDID) measure based upon this. Users may of course apply both the angular and the Aitchison measures while being aware of the crucial differences in properties.

NB. This discussion forms part of a larger framework given most recently by my other paper: "One woman, one vote. Though not in the USA, UK and France". My diagnosis is that "political science on electoral systems" is still in the Humanities and pre-science, notably by relying more upon common language instead of sharp definitions that are relevant for empirics. On the other hand, there are also mathematicians who deal with their definitions abstractly, without a proper grounding in empirical research. My invitation to empirical researchers is to help make a difference, notably in re-engineering the theory on electoral systems.

POSTED BY: Thomas Colignatus
1 month ago

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