W.r.t. the referendum on PR

May 27 2010
Thomas Colignatus
Econometrician and teacher of mathematics in The Netherlands

Published in Mathematics Teaching 222, January 5 2011

The UK coalition government plans a referendum on more proportional representation (PR) in national elections. The UK already has PR for the EU elections but these draw less attention. The current district representation (DR) for Westminster elections is seen by many as vital for democracy however and the planned referendum may meet with contempt or heated passions. For a mathematics course the issue can be interesting both for its intrinsic complexity and the voting paradoxes that tickle the imagination. I have used voting in class to collect opinions on solutions and solution strategies and this tends to work well if used wisely. A discussion on voting procedures is a next level. It might be a volunteer project alongside the official curriculum, possibly combined with a debating event. Currently I am not teaching math in highschool but I can imagine that these issues can provoke some interest amongst pupils and a sense of how mathematics can contribute to understanding. Some points in voting theory are not well-known and kids can derive some pleasure in explaining those to their parents and grandparents.

Elsewhere I have elaborated on the didactics of discussing PR versus DR. That paper is written in the environment of Mathematica and uses programs in that computer algebra. By consequence that paper is less suited for publication in journals that have other formats. The paper may however still be useful for fellow math teachers who might be interested in considering the subject. The purpose of this note thus is to provide a summary of that longer paper and provide a link to it.

The UK situation can be compared to Holland where PR exists for a century and where a culture of coalition government has grown. In the paper I discuss the UK 2010 Westminster elections and the Dutch 2006 elections and it appears that discussing a real PR system has some advantages above merely simulating some hypothetical PR system for the UK.

Curiously in Holland there are Liberal Democrats who want a greater role of districts and they currently opt for the German model where a voter has two votes, one for the district and one for the national proportions. Hence an obvious question is whether that indeed is a good compromise. Two votes may send mixed messages and this seems less transparent. It seems more like a trick in Germany to somewhat undo its high electoral threshold.

With a single vote (the preferred approach) there is crucial link between number of voters (n), number of seats (s), average district size (d = n / s) and the criteria for electoral quota (q = n / s as well) and majority rule. In PR seats are allocated to parties so that (almost) each MP has the electoral quota. In DR that would mean that the MP must have 100% of the vote in the district. A majority condition that a MP is elected with 50% plus 1 of the district means that 49% of the country may be unrepresented. This causes a discussion what "representation" actually means, and also causes a discussion of majority versus plurality (though the crucial factor remains the dispersion over districts). Pure DR neglects the national condition while pure PR neglects local information. A compromise of proportionality and districts is to allow free (non-district) seats for the overflow. E.g. if half of the seats in Parliament are for single seat districts then the district size can be twice the electoral quota and a district candidate is (ideally) elected when gaining a majority of at least one quota.

For apportionment, mathematics developed threshold methods versus the mechanisms of highest average, greatest remainder and the principle of Sainte-Laguë & Webster. The latter can be optimal for apportionment of states or districts that will get at least one seat. That kind of optimality can be dubious for political parties. Firstly because a party with a majority in the turnout may miss out on majority in Parliament and secondly since voters for some party A may not want that their vote, if wasted, goes to some party B. A proportional representation of the wasted vote w in total n is also possible by leaving seats empty or by filling the seats and taking a qualified majority f = 1/2 * n / (n - w). We thus should distinguish the mirroring of the proportions in the vote and the mirroring of a majority (and it is not quite true that the first takes care of the latter). For a coalition formed after the elections there is the more complex threshold of a "coalition qualified majority" since the coalition may not always be a solid block.

A multiple seats election is not quite the same as a series of single seat elections. Direct single seat elections such as for the chief executive (President) are riddled with voting paradoxes. Superior to a single vote are some methods with preference orderings like the Borda Fixed Point but these are somewhat complex. Optimal seems the indirect method where the electorate chooses Parliament in a single vote multiple seats election and that Parliament then applies the complexer preference methods for the single seat election of the Premier. For example, David Cameron could be the Borda Fixed Point winner, second to Nick Clegg in a Borda count but still winning in a pairwise vote.

Details are in Colignatus (2010), "Single vote multiple seats elections. Didactics of district versus proportional representation, using the examples of the United Kingdom and The Netherlands", http://mpra.ub.uni-muenchen.de/22782/