Stephen Pettigrew at FiveThirtyEight writes about the 2014 elections that ended in a tie. These are generally resolved by games of chance. The elections that are mentioned there are small ones (two city councilors and a county commissioner); is this because most elections are small or because small elections are more likely to end in ties? Some of both, I think.

Empirically, according to this paper by Casey Mulligan and Charles Hunter, in the period 1898-1992, six out of 16,577 elections to the US House of Representatives were decided by ten votes or less (one was exactly tied), while nine out of 40,036 elections to state legislatures were decided by one vote or less (and two were exactly tied). Looking at this suggests that state elections are more likely to be tied – perhaps because they’re smaller. And in fact the probability of having a “pivotal” election (an election that could be changed with a single vote, according to Mulligan and Hunter, appears to vary like 1 over the number of votes.

This is easy to explain. Very roughly speaking, if we look at two-party elections (I’ll follow Brian Hayes and call the two parties Red and Blue), the proportion of the vote that Blue receives has some distribution; for the sake of argument let’s say that it’s uniform on [0.25, 0.75]. (I don’t actually expect that it’s uniform, but the value at 0.5 is what matters.) Then for Blue’s proportion of the vote received to be in [0.5 – 0.5/n, 0.5 + 0.5/n] in an n-vote election (if n is even, which is the only case when a tie can actually happen) has probability 2/n. The constant 2 depends on my assumption of a prior distribution (although it turns out to match up well with the empirical data) but the 1/n dependence does not. Mulligan and Hunter go a bit deeper and suggest a two-tiered model (there’s a prior on the true proportion of Blue voters, and then voters vote at random according to this) but the broad conclusion is the same.

This suggests that models like the Banzhaf power index don’t make sense in measuring voting power, since votes are not coin flips. The paper The Mathematics and Statistics of

Voting Power by Gelman, Katz, and Tuerlinckx suggets some more realistic models than the coin-flip model.

Incidentally, before looking at the Mulligan and Hunter paper, I was actually thinking that the probability of a tied election would go like the -3/4 power of the number of voters. My thinking was as follows: I had an argument like the 1/n one in my head, but the probability of getting n/2 heads in n coin flips goes like n^{-1/2}, so why not pick something in between?

And once we’ve found ties, how do we break them? In the three ties mentioned by Pettigrew, one was broken by picking a name from a hat, one by picking blocks from a bag, and one (for a seat on the Neptune Beach, Florida city council) by the following baroque procedure:

Rory Diamond and Richard Arthur had each received 1,448 votes for Seat 4 on the Neptune Beach City Council. To break the tie, Diamond’s name was drawn from a bag by a third party. This allowed Diamond the chance to call the coin toss. He won the toss by calling heads. Because of this, he could decide whether to draw first or second from a bag of ping-pong balls, numbered one through 20. He deferred to Arthur, who drew No. 12. The ball was replaced, and Diamond then drew No. 4. Arthur won the seat.

Why three rounds? Does this provide more randomness than just one round? This Florida Times-Union article suggests that Supervisor of Elections Jerry Holland “wanted the results to seem as random as possible.”