Statistical Modeling, Causal Inference, and Social Science: Why the square-root rule for vote allocation is a bad idea

Why the square-root rule for vote allocation is a bad idea

Commentators and experts have taken two positions on the allocation of votes in a two-stage voting system, such as block voting in the European Union or the Electoral College in the United States. From one side (for example, this article by Richard Baldwin and Mika Widgren), there is the claim that mathematical considerations of fairness demand that countries (or, more generally, blocks) get votes in proportion to the square root of their populations. From the other side (for example, this article by Gideon Rachman), there is the claim that such mathematical rules are irrelevant to the real world of politics. This debate has real-world importance, in particular because of Poland's recent lobbying for square-root allocation in the European Union, in opposition to Germany's support of something closer to proportionality.

I make a different claim, which is that mathematical rules are relevant to the real world, but that when the mathematics and statistics are done correctly, we find that proportional allocation is much more fair than square-root allocation, in the sense of giving more equal voting power--probability of decisiveness--to individual voters. This sense of voting power is the criterion used by the square-root-rule proponents. Thus, I am taking them at their own word and saying that, under their own rules, the square-root rule is not fair.

LemmusLemmus

You should send the German government a letter. There might be money for you in this.

Alex F

"I make a different claim, which is that mathematical rules are relevant to the real world, but that when the mathematics and statistics are done correctly, we find that proportional allocation is much more fair than square-root allocation, in the sense of giving more equal voting power--probability of decisiveness--to individual voters. This sense of voting power is the criterion used by the square-root-rule proponents. Thus, I am taking them at their own word and saying that, under their own rules, the square-root rule is not fair."

Of course, their criterion is silly. The primitive of the model -- and the real-world relevant issue -- is that people have preferences over outcomes, not over being decisive. A fair system will be one that chooses an outcome which gives equal weight to preferences of people everywhere, not one which makes each person equally likely to be decisive. An optimal system will be one that chooses "the best" outcome according to some criterion like maximizing sums of normalized utilities. But there's no reason in the world -- not in mathematics, not in politics, not in social intuition -- why we should design a system to give everyone an equal chance of being decisive.

We can argue about whether a system which weights country sizes by the 1/2, 9/10, or 1st power is the one which best equalizes probability of being decisive. But I just don't see that there can be any argument that it's an answer to the wrong question.

Marian

In my opinion it might be true that this abstract model hasn't got much to do with the real world, and might not be appropriate to the requirements of daily politics.

But there are some serious errors in reasoning in the 2nd comment ("Posted by: LemmusLemmus at October 11, 2007 8:55 AM."):

"The primitive of the model -- and the real-world relevant issue -- is that people have preferences over outcomes, not over being decisive."
I think that for nearly everybody, knowing to be decisive (or at least as much decisive as every other person) is very important. Voter participation is decreasing in most countries, not just in the EU, but all over the world. One reason could be that voters don't feel powerfull, and as a consequence useless, in the face of globalisation and the cession of more and more rigths of the national goverments to the institutions of the EU. A decisive voter is a characteristic of an efficient democracy!!

"A fair system will be one that chooses an outcome which gives equal weight to preferences of people everywhere, not one which makes each person equally likely to be decisive."
A system which makes each person equally likely to be decisive is the only conversant / established way of giving equal weight to preferences of people "everywhere". --> If everyone's vote is euqally likely to be crucial, everybodys preferences are represent equally strong!

And what are "normalized utilities"?! I would be glad if you could define this, LemmusLemmus...

I'm looking forward to your re-comments or e-mails ( themaryman@gmail.com )!