Wow—this was even better than being mentioned in Private Eye. The subject line of the email was, “Voting power in three-way election,” and I was like, OK, this guy is not just about comedy and poker . . .

. . . OK, it wasn’t that Gabe Kaplan. Here’s what he wrote:

I’m a senior in high school from Minneapolis, Minnesota. I’m working on a math paper on voting power in local elections and found your 2004 paper — “Standard Voting Power Indexes Do Not Work: An Empirical Analysis”. I’m attempting to modify a few of the equations to test voting power in a 3-way race (assuming random, plurality voting).

It is totally ok if you do not have time, but I was wondering how you divided the continuous probability density and produced a result only in terms of ‘n_j’ in the paper’s second equation? It is in section 2.2.2 on page 662: Pr(a given vote is decisive in jurisdiction j)=p_v_j÷(2n_j). Did you use the normal distribution formula and somehow define ‘x’ in terms of ‘n_j’?

My reply: I used the binomial(n_j, 0.5) distribution, which can be approximated as a normal distribution with mean 0.5*n_j and sd 0.5*sqrt(n_j). From there, you approximate the probability of a tied election as the probability that this continuous random variable is in the range [0.5*n_j +/- 0.5] (or shifted by 0.5 if n_j is even).

Also, please don’t try to apply the random voting model to real elections. That’s the whole point of our paper, not to do this!