Serguei Kaniovski sent me this paper. Here’s the abstract:

I [Kaniovski] discuss a numerical scheme for computing the Banzhaf swing probability when votes are not equiprobable and independent. Examples indicate a substantial bias in the Banzhaf measure of voting power if either assumption is not met. The analytical part derives the exact magnitude of the bias due to the common probability of an affirmative vote deviating from one half and due to common correlation in unweighted simple-majority games. The former bias is polynomial, whereas the latter is linear. I derive a modified square-root rule for two-tier voting systems which takes into account both the homogeneity and the size of constituencies. The numerical scheme can be used to calibrate an accurate empirical model of a heterogeneous voting body, or to estimate such a model from ballot data.

The model in the paper is related to my paper with Tuerlinckx and my paper with Katz and Bafumi. It has a slightly different from our work in focusing on settings with small numbers of voters (e.g., 2, 3, or 4), whereas we are interested in elections with thousands or millions of voters. That stated, I completely disagree with the focus on the square-root rule (section 4.1 of the paper), since the underlying assumption here–that elections with large numbers of voters will be much closer (in vote proportion) than elections with small numbers of voters–does not occur in reality; see here for a discussion in the U.S. context. (Kaniovski notes this in a footnote on page 20 of his paper, but I don’t think he’s fully internalized our results–if he had, I don’t think he’d focus on the square root rule.) The mathematics of the paper seem reasonable so it might point to a useful future direction of this research.

Finally, I like the last paragraph of the paper:

The main conclusion of this paper is that, despite the Banzhaf measure being a valid measure of a priori voting power and thus useful for evaluating the rules at the constitutional stage of a voting body, it is a poor measure of the actual probability of being decisive at any time past that stage. The Banzhaf measure cannot be used to forecast how frequent a voter will be decisive.

Except that, given that the Banzhaf measure is a poor measure etc., I don’t see why it is appropriate to claim that it is “useful for evaluating the rules at the constitutional stage.” If it’s wrong in all the empirical examples, I don’t see how it can be useful at the constitutional stage. I mean, if you lose on red and you lose on black, and you lose on 0 and you lose on 00, then you don’t need to actually spin the wheel to know you’re in trouble!