Voting Qutopianism (Warning: High Wonk Content Contained Within)

People who favour democracy should wish for votes to translate into MPs, but alas, as things stand the constituency boundaries currently favour Labour and disadvantage the Conservatives, because they are not of equal size, and because Tory votes are often heavily concentrated in safe seat areas.

The proposal to reform the boundaries was blocked by the Lib-Dems in coalition a couple of years ago - a reform that would have seen the number of MPs reduced to 600, and the constituency boundaries being of roughly equal size, making the political outcome much more representative as the Conservatives would have gained between 20 and 25 extra MPs. Until those boundary changes occur, the people of Great Britain will not be democratically represented in a way that reflects their votes.

Not only does percentage of votes not translate consistently into number of seats - number of seats doesn't translate into voting power either. At the low end there is no direct link between a party's influence and its number of votes.

Imagine a reduced parliament in which there are only 120 MPs.  The Conservatives have 50, Labour has 40, The Lib Dems have 20 and UKIP has 10. Despite Labour having twice the number of MPs as The Lib Dems, a coalition between The Conservatives and The Lib Dems would give them a majority, and give Labour less power than a party that obtained half their MPs. There are a vast number of inter-party permutations for coalitions, meaning that a party with relatively few votes can be nearly as powerful as a major party in a coalition, or equally pretty much powerless, depending on how the land of the coalitions lies.

While we're in the mood for oddities - some of you may have heard of Kenneth Arrow's 'impossibility theorem', which proves that an aggregation of society's individual preferences doesn't translate into a comprehensive aggregate societal preference. Let me try and break it down this way. Imagine the three big leaders are running for a national popularity contest, and observe the strange goings on that occur here. Let us say that a third of the electorate prefers Ed Miliband( M ) to Nick CKlegg ( K ) to David Cameron ( C ), another third of the electorate prefers K to C to M; and the remaining third prefers C to M to K. There is nothing particularly strange about this until we consider what happens in two person contests given the above preferences.. M can boast that two-thirds of the electorate prefers him to K. C responds that two-thirds of the electorate prefer him to M. Finally, K counters by noting that two-thirds of the electorate prefers him to C.

In mathematics, a binary relation over a set is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c. If the societal preferences in what I’ve just said are determined by majority vote, we have an irrational ordering of preferences; that is, ‘society’ prefers M over K, K over C, and C over M. Thus even if the preferences of all the individual voters are transitive (by that I mean that transitivity holds if, wherever a voter prefers Cameron to Miliband and Miliband to Clegg, he or she prefers Cameron to Clegg), the societal preferences determined by the majority vote are not necessarily transitive and thus not necessarily rational either.

Kenneth Arrow's theorem demonstrates that given the foregoing all reasonable voting systems (or equivalently, economic market systems) are subject to such irrationalities.

Let me try to posit some further clarity with a different illustration. Think of our three leaders M, K, and C as cars rather than people, and then think of a woman deciding which of the three cars to buy. Let’s give her three criteria (interchangeable and commensurate with one another) for making this decision; looks, affordability and performance. Car M looked better than car K, which looked better than car C. On the other hand, car K was more affordable than car C, which in turn was more affordable than car G. Finally, car C performed better than car M, which performed better than car K.

Since the woman placed equal and commensurate measure on each of these criteria, she would be in a bit of quandary here. She clearly preferred car G to car K (M outscored K on two criteria). She also preferred car K to car C (for the same reason) - yet she preferred car C to car M. And if you are following here you will see that the same problem of non-transitivity holds for individuals, and that when broadened out to an election it leaves a bit of a detritus. In the case above one only need induce the woman to declare one of the criteria more important than the others. This is easier than convincing one third of the electorate to change its mind.

Given the foregoing; there are four conditions under which consistency will show that we cannot derive societal preferences from individual preferences…

1) The societal preferences must be transitive (if society prefers x to y and y to z then it must prefer x to z)

2) The societal preferences must satisfy the principle - if alternative x is preferred to alternative y by a majority in the society, then society must prefer x to y.

3) The societal preferences must satisfy the independence of irrelevant alternatives (the societal preference depends only on the orderings of the individuals with respect to alternatives in that environment).

4) The societal preferences must not be susceptible to autocracy - there is no individual whose preferences automatically determine all of society’s preferences.

As for the realities of the electoral situation, of course we know that the political portrait of lucidity has been gravely disfigured from the bottom up as much as the top down, so the absolute best that one can hope for is that through the media-manipulating smokescreen the impressionability and cognitive indigence does not wholly impair the view of those gazing in, and that in the absence of a good rationale people’s gut instincts amount to enough in seeing who is very evidently the least bad party for the job - at least in the next few years.