Vote counts are posted Nov. 7 at the Citywide Senior Center in Cambridge’s Central Square.

It is a truism of Cambridge elections that the fine details of how to count the vote don’t matter.

If you look deeply enough into anything, you can find something complicated. The question is whether that complexity actually matters.

This year’s School Committee election is balanced on a knife’s edge, so details matter as they never have before.

It is basically a coin flip between Andrew King and Richard Harding. If King requests a recount, a change in a single ballot could swing the election in his favor. Or maybe not.

Transfer votes in proportional representation

In Cambridge’s proportional representation system of ranked choice voting, a voter’s ballots end up going to one candidate and one candidate only. For voters who rank a single candidate as their No. 1 choice, the vote always stays with that candidate. But for voters who rank more than one candidate, the exact mechanism is a little tricky. It also almost never matters, until this year.

In this year’s School Committee election, with 23,092 ballots cast, a quota of 3,037 votes was required to elect each of the six committee members. The top two vote-getters, incumbent Rachel B. Weinstein and newcomer Elizabeth Hudson, each surpassed quota from their No. 1 votes alone: 3,651 votes for Weinstein and 3,516 votes for Hudson. So for each of those candidates, their surplus votes – the difference between quota and their No. 1 votes – get distributed to the No. 2 choices on each respective ballot.

That’s a surplus of 614 ballots for Weinstein and a surplus of 479 for Hudson. But which 614 of Weinstein’s 3,651? Which 479 of Hudson’s 3,516?

The “Cincinnati method”

According to state law, the method of surplus transfer the Election Commission uses can be any method in use “in any city of the United States” on Jan. 1, 1938.

In part because vote tabulation in the 1930s did not use computers, methods of tabulation in use favored those accomplished easily by hand counting. Cambridge uses the “Cincinnati method,” in use in the Ohio city in 1938.

In the Cincinnati method, the surplus ballots are selected by sampling ballots from a candidate’s stack of ballots. For Weinstein, the 614 ballots are chosen by taking every fifth ballot in order from her stack (because 3,037 divided by 614 is almost five), and for Hudson, the 479 ballots are chosen taking every sixth ballot from her stack (3,037 divided by 479 is just over six).

That means order matters. To keep the system as fair as possible, a random draw is conducted before the election to provide an ordering of precincts. In this year’s random draw, precinct 9-3 (the Russell Youth Center on Huron Avenue) was first, and precinct 2-3 (the O’Connell Library in East Cambridge) was last.

If, during a recount, an additional ballot is found (say, stuck in a machine or ballot box) or removed (say, ruled invalid on closer inspection) from a precinct early in the draw such as 9-3, it can change which ballots are transferred.

For instance, in the election tabulated on Friday, the fifth, 10th, 15th, etc. ballots from Weinstein’s stack were transferred (assuming they had a No. 2 vote selected); if a ballot were added to the top of Weinstein’s stack during a recount, the fifth, 10th and 15th ballots drawn during the recount would be the fourth, ninth and 14th ballots from Friday’s count.

Because not all of Weinstein’s ballots have the same No. 2 votes marked, that can change the results.

Analysis of ballot data

We don’t have to guess about whether transfers can make a difference, because the Election Commission makes the actual ballot data available as required under law. The software to run the election, Choice Plus Pro, is also freely available.

If you rerun the election by removing the first ballot for Weinstein, King beats Harding. But not if you remove two. If you rerun the election by removing the first ballot for Hudson, King beats Harding. The same if you remove two.

If you rerun the election by adding four ballots for Hudson or five for Weinstein, King and Harding tie: 2,619 to 2,619 in the Hudson case, or 2,615 to 2,615 in the Weinstein case. In either case, according to the rules, King would prevail. In the words of Choice Plus Pro, “Richard Harding Jr. is declared DEFEATED because s/he was tied for last place; but in the previous count(s), s/he had the fewest votes.”

Overall, I reran the election 36 different ways, adding and subtracting up to nine votes each for Weinstein and Hudson, all votes that voted only for Weinstein or Hudson.

In 51.4 percent of those cases, Harding prevailed, as he did on election night and at the count Friday. In 48.6 percent of those cases, King prevailed, as he did in the count on Nov. 9.

Avoiding the randomness

Most proportional representation elections that are not subject to the limits of Massachusetts Chapter 54A §16(b) (repealed, but still in effect for Cambridge) do not use the Cincinnati method. Most popular is “fractional transfer,” in which the No. 2 votes of all of a candidate’s ballots are transferred in proportion to the candidate’s surplus size.

If the ballot data is tabulated using the fractional transfer method, King would beat Harding 2,606.8 votes to 2,603.6 votes. In that tabulation, instead of transferring 614 votes, Weinstein would transfer 617.99791, of which 51.02182 would go to King and 37.53734 would go to Harding. Then Hudson would transfer 469.09513, of which 31.36725 would go to King and 24.04569 to Harding. After transferring the fractional votes from elected and defeated candidates in the next four rounds, King would end up with 2,606.76301 votes and Harding with 2,603.56601.

The problems with the Cincinnati method have been known for a long time, and this year’s Charter Review Committee has discussed them. Although it has not finalized language, it is expected to recommend the Election Commission have the flexibility to choose fractional transfer over Cincinnati. Just a little bit late for this year’s municipal election, and it would depend on the state Legislature and Cambridge voters to enact the change in time for the 2025 municipal election.

This post was updated Nov. 21, 2023, to correct a figure to “sixth” from “fifth” when taking ballots from Elizabeth Hudson’s stack in a vote count.