Cambridge Civic Journal Forum

It has been suggested at various times that Cambridge should consider modifications to its current proportional representation election method – especially in regard to its method of distribution of surplus #1 votes using the “Cincinnati Method” which is dependent on the order in which ballots are initially tabulated. An alternate method that is often suggested (but rarely explained) is known as “Fractional Transfer” and various other names. Indeed, the tabulation software currently used by Cambridge (ChoicePlusPro) has Fractional Transfer as its default method, and our own “Cambridge Rules” must be invoked for our local Cambridge elections. The Election Quota is calculated in the same manner, i.e. the total number of valid ballots divided by one more than the number to be elected, rounded up to the nearest integer (or add 1 if a whole number).

There are several key differences between the Cambridge Rules and Fractional Transfer:
(1) Under the Cambridge Rules, any overvote where the same rank is given to more than one candidate is ignored. Under Fractional Transfer, overvotes may be counted (for example if 4 candidates are given a #1 vote they would each get 0.25 votes) or they may be ignored. This is a choice that would have to be made.

(2) Under the Cambridge Rules, surplus #1 votes are redistributed to the next highest ranked continuing candidates as whole ballots where the whole ballots are chosen via the Cincinnati Method, i.e. every nth ballot where n is the nearest integer to the quotient of the total and the number of surplus ballots. For example, if Quota was 2000 and a candidate had 2600 #1 votes, there would be 600 surplus votes and 2600/600 would be approximately 4.3 and the ballots chosen for redistribution would be (in sequence) #4, #8, #12, etc. Any surplus ballot with no valid next preference would not be transferable and would remain with the #1 choice. Thus there can be no “exhausted” ballots during the surplus distribution. The distribution of surplus ballots continues until the elected candidate’s number of votes is reduced to the Election Quota. If during this surplus distribution another candidate reaches Quota, that candidate would be declared elected and would no longer be eligible to receive additional ballots with any subsequent ballots transferred to the next preference candidate on that ballot still eligible to receive transfers. There are thus two ways in which the initial ballot order can affect the election results – the specific ballots chosen for redistribution and the point at which any other candidate reaches Quota.

Under Fractional Transfer, any elected candidate with surplus votes would have a fraction of ALL ballots transferred to the next preferred continuing candidate with a corresponding weight. For example, if the Quota was 2000 and the candidate had 2500 votes (so the surplus would be 500), then ALL of that candidates ballots would be transferred to the next preferred candidate with a weight of 1/5 or 0.2 with the elected candidate retaining 0.8 of all of all ballots – thus reducing the total to the election quota. In the case where there is no valid next preference, that weight (0.2 in the example) would be exhausted, so there can be ballot exhaustion during the surplus distribution in order for the election to be independent of ballot order. If another candidate reaches Quota during this distribution (or any subsequent surplus distribution), the distribution will continue allowing the newly elected candidate to exceed Quota. A subsequent count will then take place to also reduce that candidates total down to Quota – again transferring a fraction of ALL of that candidate’s ballots in the same manner. Any candidate who has reached Quota at the end of any round is declared elected and becomes ineligible to receive transfers.

(3) Under the Cambridge Rules, after all #1 vote surpluses have been fully distributed, the next Round is the “Under 50” Round where all candidates with fewer than 50 votes at that point are simultaneously defeated and all ballots transferred to next preferred eligible candidates or exhausted if there is no additional valid choice.

Under Fractional Transfer, all candidates who have been “mathematically eliminated” are defeated simultaneously. This means that the sum of all of the votes of those candidates at that point is less than the number of votes for the next lowest candidate. If any continuing candidate reaches Quota during this round, that candidate is declared elected at the end of the round, and any surplus ballots are subsequently redistributed in a subsequent round.

(4) Under the Cambridge Rules, the remainder of the process is a series of runoffs where the candidate with the fewest votes at the end of each round is defeated and all of that candidate’s ballots are transferred to the next highest ranked continuing candidate or exhausted. This continues until the required number of candidates have been elected either by reaching Quota or by having not been defeated at the point where the requisite number of candidates have not been defeated. If any candidate reaches Quota during a round, that candidate is declared elected and is no longer eligible to receive additional ballots. This is another way in which the original ordering of ballots can affect the election outcome. After the initial #1 surplus distributions, no candidate can ever have more than the Election Quota of ballots.

Under Fractional Transfer, the election proceeds in much the same way via a series of runoffs, but whenever a candidate reaches Quota during a round, the count continues until all of the defeated candidate’s ballots have been transferred or exhausted, and any surplus ballots of an elected candidate are transferred in a subsequent surplus distribution round to reduce that elected candidate’s total to Quota. This process continues until the number of candidates is reduced to the number to be elected. In the final round some candidates may go over Quota, but the standard rule is that the election is declared to be complete at that point without any additional surplus distribution.

(5) Under the Cambridge Rules, if a vacancy occurs, the vacancy is filled via a “Vacancy Recount” using only the Quota of ballots that were used to elect that candidate. This is simply a series of runoffs to elect one candidate where all candidates not previously elected are eligible to receive votes (but not including any votes previously received in the original election).

There is no established rule for how a vacancy would be filled under Fractional Transfer. It could be done in the same manner as the Cambridge Rules, but candidates elected in the final round might have a substantial number of surplus ballots compared to any candidates elected during previous rounds all of whom would have exactly a Quota of ballots.

Here is a comparison of three methods for the most recent (2021) Cambridge City Council election: (1) the official results using the Cambridge Rules; (2) Fractional Transfer with overvotes included; and (3) Fractional Transfer with all overvotes ignored. As you can see, the same candidates are elected with the order of election differing slightly and the rounds somewhat different due to differences in the rules – most notably in the introduction of surplus distributions after any candidate reaches Quota during a round.

Official Count:
CouncilFinal2021

Fractional Transfer including overvotes:
Fractional2021

Fractional Transfer – No Overvotes:
Fractional2021NoOvervotes