Voting Qutopianism

So what’s been happening in the world of politics then? Isn’t it great having Ken Clarke back in the Shadow Cabinet? He is one of the few Tory politicians in the past 20 years that I have liked and respected - and anyone that makes Dozy Dave look even more soulless gets my vote. I was sitting at home recently with my sprained knee resting on a table watching Prime Minister’s questions, and I was getting so bored with Dozy Dave’s ramblings that I started writing down a little experiment. I was imagining a poll conducted entirely for the purposes of ‘popularity’ amongst voters - one that had nothing to do with the system of election - but was based exclusively on who was good enough to be considered ‘most popular’. Picking one great leading man from each of the main parties, I realised that I had to go right the way back to Charles Kennedy to find a half decent prominent member of the Lib Dems. As for the other two, I’m quite happy with Gordon Brown and Ken Clarke.

Now imagine all three are running for a national popularity contest, and observe the oddities that occur here. Let us say that a third of the electorate prefers Gordon Brown ( G ) to Ken Clarke ( K ) to Charles Kennedy ( C ), another third of the electorate prefers K to C to G; and the remaining third prefers C to G to K. There is nothing particularly strange about this until we consider what happens in two person contests given the above preferences. Think about this. G can boast that two-thirds of the electorate prefers him to K. C responds that two-thirds of the electorate prefer him to G. Finally, K counters by noting that two-thirds of the electorate prefers him 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 G over K, K over C, and C over G. Thus even if the preferences of all the individual voters are transitive (by that I mean that transitivity holds if, wherever a voter prefers x to y and y to z, he or she prefers x to z), the societal preferences determined by the majority vote are not necessarily transitive and thus not necessarily rational either.

So then that got me thinking, could a general theorem be proved showing that given the foregoing all reasonable voting systems (or equivalently, economic market systems) are subject to such irrationalities?

In one sense, yes. Let me offer you further clarity with a different illustration. Think of our three leaders G, 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 G 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 G, 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 (G outscored K on two criteria). She also preferred car K to car C (for the same reason) - yet she preferred car C to car G. And if you are following here you will see that the same problem of non-transitivity holds for individuals, yet it seems more tractable. 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.

The conclusion is this; 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 everyone 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).

And finally..

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 best party for the job - at least in the next few years.