Friday, 31 August 2012

Fair Elections?

Elections for Dutch parliament are only a couple of weeks away. The papers, television, internet and social media are swamped with election rhetoric, opinions and polls. Not much different from the election circus in the States I guess. For the Dutch this election is of high importance. The national economy is under pressure because of rising national debt, sluggish domestic demand and the housing market slump. The next government needs to come up with measures to control and reduce the national debt without hurting economic growth. Which government that will be is up to the voters. But will an election result in a government that represents the preferences of voters? There are many examples of voting systems in which that is not the case, and I am not talking about elections in a banana republic.

To give an example; in 2005 the Labour party got 57% of the seats in the British House of Commons with only 36% of the votes. Is that fair? This outcome is a consequence of the first-past-the-post electoral system, or winner-takes-all system, in multiple districts. In this case, the candidate that gets the most votes, whether he/she reaches a majority of votes or not ("first past the post") wins the seat in parliament. It was designed to support for a two-party electoral contest. It worked well when there were only Whigs and Tories it is however ill-suited to a multi-party political landscape. It thwarts the will of the voters, leaving millions without political representation in parliament.

In the Netherlands the House of Commons is elected using a system of open party lists and only one district (the whole nation), resulting in proportional representation. Question is if this results in a fair representation of the voters preferences. Unfortunately this is also not the case. In 1994 the majority of the voters preferred the Democrats 66 party (D66) to the Dutch Labour Party (PvdA), Christian Democratic Appeal (CDA) and People's Party for Freedom and Democracy (VVD) so you would expect that the D66 would also have the majority of the seats in the House of Commons. However PvdA, CDA, VVD all ended up with more seats that D66. Is this fair?

Take a look at the vote and seat distribution of the 1994 elections. As you can see, CDA and PvdA suffered great losses, D66 and VVD won a lot of votes. Now consider the election matrix as retrieved prior to the elections (source Dutch Parliament Elections Studies). The cell (D66, PvdA) in this matrix contains the number of 655 while the cell (PvdA, D66) contains the number of 580. This means that 655 respondents strictly prefer D66 to PvdA and that 580 strictly prefer the PvdA.

Using this matrix a majority ranking can be constructed in such a way that the party higher in rank holds a majority over all parties at lower ranks. From the majority ranking it is immediately clear that although D66 has the majority vote it is only ranked fourth based on number of seats. This is an example of what is called The More-Preferred-Less-Seats Paradox.

Can’t we improve voting systems to overcome these kinds of paradoxes? Unfortunately not. Like Kurt Gödel's proof that there will always be facts which cannot be proved or disproved in any mathematical system, the Impossibility Theorem on social choice of Arrow precludes the ideal of a perfect democracy. So abolish democracy? Of course not! The perfect democracy doesn’t exist, the challenge is to find the best possible. With Arrow’s theorem available a trade-off can be made on the properties a voting system should have and whether the above paradox is acceptable or not. But how to decide on that? Take a vote?