The best way to elect

Some of you may have noticed the small matter of an impending general election in the UK. As a bunch of statistics geeks we could try to guestimate the outcome of the closest run election in living memory, but there are thousands of 'experts' paraded through our news bulletins trying to do that already. So we thought we'd look at something a little different, but still relevant.

So the election looks set to deliver another un-satisfying coalition government. Last time this happened, many chose to level criticism at the voting system itself. Not at the concept of democracy itself, just the system that translates individual votes into an election result. The UK went as far as to have a referendum to change it. We didn't. It is still first past the post.

So what is a voting system and does it matter which one we choose? Which one is the best?

Voting systems fall into a larger category of algorithms called social choice functions. These essentially represent different ways you can translate the preferences of a group of individuals into a single choice for the whole group. These need not be voting systems, although these are certainly the most familiar. So the task is to develop the ideal social choice function. Unfortunately even under some pretty common sense characteristics (such as it shouldn't be a dictatorship) a clever chap called Kenneth Arrow proved that no such voting system exists.

So we are left to try and choose the best of an imperfect bunch. Rather than trying to create a theoretically perfect voting system, we can consider the practical aspects of voting systems, and their practical pros and cons. 

One way to evaluate a voting system is through the concept of power. In a voting situation, the concept of power can be interpreted as the proportion of scenarios where a voter, or group of voters can influence the outcome of the vote, assuming no one else changes their vote. Some voting systems can actually benefit minority groups, in terms of their power. Consider the Alternative Vote system. In this system if there is no outright winner by considering the first choices of the voter, the candidate with the fewest votes is eliminated from the election, and the votes of the people that voted for them are redistributed to their second choice. If there's still no clear winner, the second smallest party is eliminated, and so on, until a clear winner emerges.

The issue with the Alternative vote is that, when considering the people that voted for the smallest parties as a block, the number of situations where they can influence the vote increases. The probability that they have the final say in the outcome actually increases. More simply, in the case of an unclear outcome, the first people that are asked to decide one way or another are those who voted for the most extreme, minority parties. In the UK this could be people who voted for the BNP, or the Communist Party of Britain.

Another approach is one analogous to a game theory approach, from the perspective of the people to be elected.

So far in this article (and in much research), the assumption has been that the options in the vote are static, clearly defined options. For much of th 20th century in the UK this was the case. However in the 1997 election in the UK, the Labour party departed from its traditional left wing roots and adopted a centre ground position, or even right wing. Indeed some of its policies were later criticised for being almost Thatcherite. 

The point is that Labour, under Tony Blair, departed from the usual process of coming up with policies they thought were right and then presenting them to the public. In effect, they considered the question, which policies should we choose in order to to get elected?

They were attempting to solve a game theory problem, where the options are the different policy choices, and the outcome of the game is win or not win.

This approach has come in for some criticism since, and even in the current election parties have accused each other of developing polocies that are purely "populist". This accusation always generates a smile. Surely the whole point of an election is to have policies that are popular!

The best solution to this game theoretic approach are the policy choices that generate the most votes, and this is why this approach has some merit. The winner is hopefully the one who best represents the views of the most people, and therefore the most people should be satisfied with the outcome. In some ways this is the most democratic outcome.

However the choice of the phrase 'generates the most votes' in the previous paragraph was intentional. Some voting systems distort the way popular policies translate into votes. It is important to choose the voting system which doesn't distort votes in this way. Whilst we're not aware of any research into this, it is probable that some voting systems do distort votes, meaning that more votes can be gained by choosing less popular policies and again the Alternative Vote is a prime candidate, since it encourages candidates to select policies which appeal to the more radical parties. On the other hand it is probable that Proportional Representation and the plain ordinary first past the post system are the best.

The final aspect that will be considered in this article is a purely practical one, that of local representation. With the proportional representation, seats in a government are allocated according to the proportion of votes received. In order to this a reasonable number of seats must be allocated to each constituency area. Typically there are several seats for each area, and the area sizes are increased accordingly. The trade off is a reduction in local reprerepresentation. The more you increase the constituency size to improve proportional representation, the more you lose local representation. 

Based in the Medway towns, the importance of the Medway towns is acutely obvious. Thousands of Medway residents and businesses use the Medway Tunnel each day. For the engineering geeks out there it was the first Immersed Tube tunnel to be built in England. That means they dig a tube under the river, but first built the tunnel tube on dry land, floated it into the river Medway, and then dropped into the river.

It is relevant because it would probably not have been built without 3 local MPs driving it through parliament. Certainly the Wainscott bypass, the road that connects the tunnel to any other road of significance would not have been built.

Yet thousands of people rely on these major transport arteries every day. So in this instance, local representation was critical to getting this vital project done.

So in summary the selection of a voting system is more complex than it first seems. We shouldn't be so quick to dismiss the basic first past the post system, especially given its most appealing characteristic, its simplicity. There is no definite 'right' voting system and there are trade-offs to be made. We should choose whichever had the characteristics our particular situation demands, but that those characteristics can be measured and compared.

Whatever you do, make sure you vote!