*I got sent this wonderful, tongue in cheek but nonetheless provocative, calculation, based on several conflict, mediation and choice theories, written by Gordon Burt of the Conflict Research Society. *

David Cameron and Nick Clegg are having constructive talks and Alex Salmond is saying there is an alternative to David and Nick. Here is a conflict! And one response to thinking about conflict is to think of voting systems and social choice theory. So I thought I myself would have a go! NOTE: I should stress that this is a purely academic exercise which seems to me to follow from applying the ideas in Arrow et al. (2002).

People disagree. That seems to me to be an important starting point. The key question is then: what should be done when people disagree? Fortunately there is a variety of criteria which we can use to decide what to do – unfortunately the criteria can sometimes disagree about what to do! Three criteria have certain attractions. Let us apply these criteria to the question of whether David Cameron, Nick Clegg or Gordon Brown should be prime minister. Each gets D, N and G % first preference votes, respectively. D, N and G are each less than 50%.

The Condorcet majority winner is the candidate who defeats all the other candidates in a pairwise contests. Nick Clegg would defeat David Cameron if as seems plausible Gordon Brown’s supporters voted for Nick Clegg. Likewise Nick Clegg would defeat Gordon Brown if as seems plausible David Cameron’s supporters voted for Nick Clegg. So Nick Clegg is the winner by this first criterion.

The De Borda winner is the candidate who gets the highest rankings. We suppose that Nick Clegg gets the second preference votes of the others. The calculations below show that Nick Clegg beats both the others. So Nick Clegg is the winner by this second criterion.

[David Cameron scores 3D+2pN+qN+G; Nick Clegg scores 2D+3N+2G; and Gordon Brown scores D+2qN+pN+3G, where p+q=1. Nick defeats David because

(2D+3N+2G)-( 3D+2pN+qN+G)=(-D+N(3-2p-q)+G)≥(-D+N+G)>0.]

The welfare winner is the candidate who gets the highest satisfaction scores. If satisfaction scores have much the same character as the rankings then it seems plausible that a similar result will ensue. In other words Nick Clegg will be the winner by this third criterion.

The findings accord with the most celebrated result in the literature: the median voter theorem. Judged by all three attractive criteria, Nick Clegg should be the winner. Indeed in almost none of the previous elections has either of the other parties gained more than 50% of the vote. So because it is the middle party the leader of the Liberal Democrats should always be the prime minister!

Discuss!

*Arrow, K. J., Sen, A. K. and Suzumura, K. (Eds.) (2002) Handbook of social choice and welfare. Volume 1. Amsterdam: Elsevier*.