Fair voting weights

Andrew Gelman weighs in the issue of voting allocation fairness in a two-stage voting system, such as block voting in the EU (an issue I covered, albeit from a different perspective, in an older post) and the electoral college in the US. Here's an edited down version of his argument:

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. [...]

My claim (and that of Jonathan Katz and Joe Bafumi, my coauthors), thus, is that even if one accepts the voting power criterion, the square-root rule is inappropriate. Could we be right? Is it possible that the consensus of experts in voting power in Europe are wrong, and three political science professors from the United States got it right?

A quick summary of our argument: The square-root-rule is derived from a game-theoretic argument that also implies that elections in large countries will be much much closer (on average) than elections in small countries. This implication is in fact crucial to the reasoning justifying the square-root rule. But it's not empirically correct. For example, if a country is 9 times larger, its elections should be approximately 3 times closer to 50/50. This doesn't happen. Larger elections are slightly closer than small elections, but by very little, enough that perhaps a 0.9 power rule would be appropriate, not a square-root (0.5 power) rule.

[...]I think it's really time for the voting-power subfield of political science, economics, and mathematics to move beyond this silly model (i.e. the square-root).