Single vote multiple seats elections. Didactics of district versus proportional representation, using the examples of the United Kingdom and The Netherlands

No new issues are discussed but we try to improve on the didactics of some well-known elementary features of multiple seats elections that rely on a single vote such as common elections for Parliament or the U.S. Congress. The didactics concentrate on proportionality versus districts. Since some people in the UK want more proportionality and some people in Holland want more districts, the examples of the UK 2010 and Dutch 2006 general elections are developed in some detail. Subordinate issues are (1) majority versus plurality, and (2) 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 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. An algorithm is given that includes such rules and some simulations are shown. 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, though voters only gave a single vote, David Cameron would be the Borda Fixed Point winner, second to Nick Clegg in a Borda count but still winning in a pairwise vote. It is also explained how to use some new routines in Mathematica.