Simple examples of how to conduct an election by the single transferable vote

To home page.

Links to sections:

The next sections follow the steps to proportional representation, using the Senatorial Rules of transferable voting:

Learning By Doing the democratic voting method: Thomas Wright Hill's intuitive transferable voting.

I realised that scientific method could solve the problem of voting method. But such a formal approach may only appeal to specialists and not the general public. As a student, who hadnt yet voted, I thought there was nothing to know about voting, apart from placing an X. Beginners dont know they are beginners, and are not happy about finding it out. Perhaps, the best way to begin with them is at the beginning of electoral reform.

John Dewey's great educational reform slogan was 'Learning By Doing', because that was much more effective than cramming children with academic abstractions.
In 1821, Thomas Wright Hill suggested the principle of proportional representation as it was informally practised in his boys school. To quote Enid Lakeman, in How Democracies Vote:

...his pupils were asked to elect a committee by standing beside the boy they liked best. This first produced a number of unequal groups, but soon the boys in the largest groups came to the conclusion that not all of them were actually necessary for the election of their favourite and some moved on to help another candidate, while on the other hand the few supporters of an unpopular boy gave him up as hopeless and transferred themselves to the candidate they considered the next best. The final result was that a number of candidates equal to the number required for the committee were each surrounded by the same number of supporters, with only two or three boys left over who were dissatisfied with all those elected. This is an admirable example of the use of STV.

It is useful to know that some candidate is a clear favorite, because he may be made the chairman. Other posts on the committee may go to those next most favored with more votes than they needed to be elected.

But Hill's essential idea is of transfering surpluses from those who have more to those who have less than they need. And it may have been the moral to the Gospel incident of the loaves and the fishes. Jesus had his disciples go round an audience to gather the left-overs for those who had still not been fed.

Thomas Hill's original proportional representation, by transferable voting, is still the best way to grasp the principle. Its virtue is that it is genuinely democratic. No-one tells the boys who to support. No masters or prefects draw up 'party lists' to draft the boys into school 'houses' or teams or whatever.
The beauty of Hill's election is that the voters know intuitively how it works. Strictly speaking, nobody has to do any arithmetic.
No ballot papers have to be filled: A voter doesnt have to note down that first ( or number one on a ballot paper ) he or she stood by the school favorite, who didnt need the support; and that in the second place ( number 2 ) she moved to a candidate, who had helped her with her problems, but who was also exceeding the share of votes needed for a committee seat; the voter in question perhaps finds that by moving on a third time ( which would be number 3 on a ballot paper ) she helps a personal friend to just reach the portion of votes to ensure a seat, and so stays in the queue of her third choice.

So, this informal election does not require voters to be numerate. They dont have to count up to three or five. But by moving on, or transfering themselves, as surplus voters, from one candidate's queue to another, they are just as surely expressing an order of preference, by voting with their feet.

Also, a returning officer, vital to a formal election, is not needed. No-one has to do sums to show a candidate has over half the voters needed to win a single seat. If 16 children line up behind one of their two favorites, and one queue is 9 children long and the other 7, you know that the candidate with two extra kids has won. You dont have to count the two rows.
If both rows are the same length ( 8 kids each ) then lots are drawn or you toss a coin to decide which candidate wins.

If there are two seats, the winners need one-third of the votes each. Suppose there are 24 voters. Provided two of the candidates both collect rows of 8 children behind them, they cannot be beaten by any other candidate. It is just possible that a third candidate may muster the remaining 8 children.

This would be an unusual alignment. Probably, it would be evident, which of the three candidates was a home of last resort, by having his queue of 8 completed last.
If not, one of the three tied candidates, who drew the short staw, would step down.

( If the result was still in dispute, then the arithmetic method of transferable voting, described below, might have to be used. For practical purposes, that has sufficed. Transferable voting doesnt have to be perfect to be very much on the right lines. Those who demand perfection in this method usually want some other voting system very much worse, instead. )

Electing a committee of three, we follow on from the above examples of single-seat and double-seat elections. The three winning candidates, who have one-quarter of the votes each, cannot be beaten. Again, the winning proportion ( or the 'quota' ) is 8 votes each, if there are 32 voters, this time.
A four-way tie is very unlikely. Any tie is more unlikely in an informal election than a formal one. Because, the voters, themselves, seeing a tie, are likely to break ranks just enough to provide the tie-break.

The formal election may gain in precision, but lack the informal election's ability to modify the result, under circumstances that could persuade some voters to revise their decisions.

Nor does it matter if three rows of 9 children form to ensure the three most popular candidates elected. All that means is the winning candidates have surpassed the lowest hurdle of popularity, of just 8 votes each, needed for election - and which ensures the minimal proportional representation of three quarters ( or 3 x 8 = 24 out of 32 ) of the voters.

Remember, I have to describe this informal election to you, in terms of arithmetic, but the children elect three candidates in terms of the three longest queues. This is no doubt a sophisticated process, involving subtle judgements of the characters of their school fellows, but it is not formal arithmetic. Such powers of mental arithmetic, that individuals may bring to their aid, avoid the tedium of working with abstract numbers. The children are the numbers. They perform a collective calculation by the operations of moving about between the candidates.

The American civic organisation Democracy 2000 conducted decision-making workshops both in schools and town halls. They found that citizens dont take to the formal arithmetic of transferable voting that the returning officer has to do. For beginners, I recommend Thomas Hill's method to show a small group, in practise, the principle of democratic elections.

'Irish' style elections.

To top.

But on a large scale, these elections have to be organised on formal lines, with ballot papers and a results sheet, recording the stages of the count - even if the whole process is automated. With this in mind, Ive also recommended the ( traditional ) Irish version of single transferable vote.

It allowed the formal electoral procedure of transferable voting to be seen in terms of individual voters ( according to their orders of choice ) allocated to individual representatives making community decisions.

Democracy has been so abstracted and removed from everyday concerns that any number of bad systems are used to the discredit of its name. And few people seem any the wiser.

Modellers of group decision-making, like Democracy 2000, require the voter to be represented by a legislator of his or her choice. They found the Senatorial rules of surplus vote transfer ( a more advanced form of STV, explained below in succeeding sections ) hard to explain to American audiences.
( Many people dont have a head for figures or have a phobia of them. )

The senatorial rules are consistent with one person, one vote ( and one vote one value ). But demonstration elections may want to steer clear of the ambiguity of voters as more or less represented by their range of preferences.

The old Irish system of STV, adapted for audience interaction, is roughly as follows. This account also serves as an attempt to show that STV can be fairly simply explained.
We suppose the local public invited to an informal meeting, so ballot papers or machines are optional. But for the election of any given committee, the audience need to know how many seats there are, and how many people it takes to elect each representative.

For this purpose, it is explained how the Droop quota follows from the single majority system. Half the voters can elect one candidate. ( If two candidates each got 50 out of 100 votes, that would be a tie break and they'd have to draw straws. )

Following from that, two seats can be won by candidates who have each won one third the votes. ( For example, when two candidates have each got 34 votes out of 100, there are only 32 votes left and no remaining candidate can win a seat with less than the required proportion of one-third the votes. Likewise three committee seats are won when three candidates have each reached the quota of one-quarter the votes. And so on.

Alright, we assume your audience have got to know the candidates and they each gather round their most prefered candidate. Usually, some candidates will be more popular than others. Say, there are half a dozen candidates competing for three seats. With 100 voters, election depends on winning one quarter of the votes or 25 votes. But the most popular candidate may have 40 people around her, who like her best.

She doesnt need 15 of those votes to elect her as the 'best' candidate. Those 15 votes are wasted unless they help elect their next best choice. Irish-style elections would pick 15 of the 40 at random, so they are a representative sample of all the voters who gave their first preferences to the leading lady.

( The original Irish method is a statistical approximation of the Senatorial rules - explained in a later section. There's no need for your leading lady's 40 first preference voters to know that the transfer value of their second preferences are 15/40 or 3/8 of a vote each. )

To cut a long story short, say those 15 votes go to a candidate with 10 voters around him. He also is elected with 25 votes. We'll assume another candidate got just 25 votes and all three seats are taken. However, I want to point out that this method of surplus transfer by representative sample gives the one-to-one relations between voters and representatives, that civic groups, like Democracy 2000, require for their democratic workshops.

Note that the method of transfering votes by representative sample also preserves the basic principle of one person one vote.

Tho not necessary for their voting audience, the organisers, as part of their studies, would record the stages of the count - just like returning officers in full scale Irish general elections - which are keenly followed by the public, who voted in two referenda to keep STV, or 'choice voting', as it's often called in America.

The next sections follow the steps to proportional representation, using the Senatorial Rules of transferable voting.

The Preference Vote.

To top.

The candidates' names are printed in random order on the voting papers. Each candidate may label what they stand for. If the election is political, this may be a party label or Independent. The electorate has a preference vote. Each voter may state their order of choice for the candidates, by placing the number 1 by the candidate's name of their first choice, number 2 by their second choice, number 3 by their third choice, and so on, in order of preference, insofar as a given voter pleases.

With the STV method, latter preferences do not count against former preferences. A 3rd choice will not weaken the chances of a 2nd choice, nor a 2nd choice weaken the chances of a 1st choice.

With STV, the voter does not have to vote only for candidates of the same party. The voter may prefer candidates of one party - or candidates of two or three parties, perhaps to express a wish they form a coalition. The voter may prefer among candidates of one party or of several parties, and among Independents.

In fact, the voter may state any order of preference between all the candidates on the voting paper. But it is not compulsory to number-order all the candidates.

The Proportional Count

To top.

It should be stressed that numbered ordering one's choice of candidates is all the voters need to know about the transferable voting method of election. Every kind of election is in two phases. First, the actual voting and second, the count of the votes. The count is conducted by trained personnel.

The Electoral Reform Society Of Great Britain and Ireland has a rule book, by Frank Britton and Robert Newland, to cover every eventuality in the procedure of an STV count and generally organise a parliamentary election down to the last detail. For a long time, they have known about an extremely thoro program for computing transferable votes.
They, or their veteran sister society, the Proportional Repesentation Society of Australia may help with specialised enquiries.

But the following information should be enough to elect your own interest group's committee by STV, as millions of people already do.
Have a calculator handy.

The count of the vote means that, first, the number of voting papers is added up, and each paper is checked to see it has been correctly filled in, with the preference numbers placed within the spaces provided by the candidates' names.
If, for instance, the voter has made a mistake by missing out the number 3 but has placed numbers 1,2,4,5, then his voting intentions will only be counted valid for the first two preferences, 1 and 2. Any invalid voting papers are subtracted from the total vote, to give the total valid vote.

With STV, candidates each need to win a proportion or quota of the total votes before they are elected. This proportion is called, after its inventor, the Droop quota.
It is a rationalisation of the simple majority count, whereby one candidate needs just over half the votes to be elected in a single member constituency. The Droop quota generalises this to say that in a two-member constituency, two candidates need just over one-third the votes each to be elected. This gives a proportional representation of two times one-third or two-thirds of the total vote.
Three members are returned with one-quarter of the votes each, proportionally representing three-quarters of the voters. And so on. Therefore, in a 7-member constituency, the PR is 7/8 of the voters.

As a simple example, take a club of 32 voters, who wish to elect a committee of 3. (A good tip is always to have an odd number of committee members to avoid possible stalemates.) Advisedly, the elective quota is simply one-fourth of 32 votes, which equals 8 votes.

( Round-up small fractional quotas, e.g. 35/(5+1) = 5.84, rounded-up to two decimal places. )

The stages of counting transferable voting

To top

Suppose there are 5 candidates, called A,B,C,D and E. Having counted 32 valid votes, the returning officer then counts how many first preferences each of the 5 candidates have. It is standard practise to openly tabulate the stages of the STV count. So, what you have is a board raised to public view. On top, the quota, 32/(3+1) = 8, should be noted.
The far left side column randomly lists the candidates' names. The stages of the count are recorded in successive columns.

Firstly, the number of first preferences for each candidate are written in the column beside their names. Say, candidate D has 2 first preferences, A has 5, E has 6, C has 12 and B has 7.

( By way of cross-checking, all these first preferences are totted-up. In fact, each successive stage of the count must cross-check with the total votes as originally counted. So, the number of voters ceasing to express a preference must also be noted at the foot of the relevant column. )

Since C has 12 votes, he or she is declared elected with 12, minus the quota of 8 votes, leaving a surplus of 4 votes. With STV, this surplus vote is not wasted.
All the most prefered candidate's voters have an equal right to decide how that candidate's surplus vote is transfered or re-distributed to second preferences, so all the elected candidate's votes are transfered in proportion to the size of C's surplus vote.

The 12 transferable votes from candidate C only have a surplus value of 4 votes above the elective quota of 8 votes. Therefore, the value of these 12 transferable votes counts for only 4/12 or 1/3 of a vote each.

The rule for weighting the count of the transferable votes is: transferable votes minus quota equals surplus, and the surplus is divided by the transferable votes.
( This is Gregory's method or the Senatorial Rules, named after the use of STV in various Commonwealth senates. )

Say, 4 of the 12 transferable votes give their second preferences to D. As each transferable vote is worth 1/3 of a vote, D gains by 4 times 1/3 equals 4/3 votes. Say, 5 of the second preferences go to A, who gets an extra 5/3 votes.
Neither D nor A have yet reached the quota of 8 votes to elect them. Suppose B gets the remaining 3 out of 12 second preferences, worth 3/3 or 1 vote. Since we've said B got 7 first preferences, that transfer value of 1 vote is just enough to elect B.

B has no surplus vote to transfer, and there are no other elected candidates ( whose surpluses would be transfered to decide a contest for more than the three seats of this example ). So, candidate D, with the least votes, is excluded. D's 3 1/3 votes are transfered to their next preferences.
Say, E, with 6 votes, picks up 2 1/3 of D's votes, to be elected to the third and last seat on the committee. This leaves A with 6 2/3 plus, at most, 1 of D's votes.
Had E not won by 1/3 of a vote above the quota, E and A might have tied for third place, with 8 votes each. Then they would have to draw straws or toss a coin for third place, just as when two candidates for a single seat get exactly half the votes each.

Table of the STV Count

Count of the Single Transferable Vote
Candidates 1st stage 2nd stage ctd. 3rd stage ctd.
1st preferences:
C elected
Transfer of
C's surplus
at 4/12 = 1/3
B elected Elimination
of D
E elected
D 2 4/3 3 1/3
A 5 5/3 6 2/3 1 7 2/3
E 6 6 2 1/3 8 1/3
C 12 8 8
B 7 3/3 8 8
valid vote:
32 32 32

The Droop quota: a foot-note

To top

For many years, the Electoral Reform Society would add one vote to the Droop quota. So, the elective quota for one seat, given 100 voters, would be 50 plus one equals 51 votes. Now they dont bother. Because, if two candidates tie with 50 votes each, which candidate takes the one seat is not an elective decision. They have to draw lots.
Similarly, if the three leading candidates in a contest each won 33 out of a total 99 votes, the candidate of these three, who drew the short straw, would have to stand down.

Altho most textbooks define the Droop quota as total votes divided by one more than the number of seats, followed by a 'plus one vote', you can forget about the 'plus one vote'.
For electorates of more than 100 voters, it doesnt matter which version of the formula you use. But for small numbers of voters, the 'plus one vote' may prevent enough candidates gaining their full quota, to take all the seats. ( Nor need one add a small fraction of a vote, tho one must round-up rather than round-down a quota. )

Appendix: Example to show ordinary Meek's method.

To top

Here is an example of how Meek's method works -- without my PR reform of the exclusion count. A more complete example is given by Stephen Todd. ( Follow the link: committees/2003/pdf/ ) Meek's method is first coming into political use in New Zealand's local government elections.

Suppose the preferences in table 1. In the ordinary Meek STV count table 2 below, the count is to pick out three of the 5 candidates.

It may be of interest to follow how Meek's method arrives at the figures. Transfer values are taken to four decimal places, because that was just enough to recover the whole numbers from which they were derived. Of course, computer counts work to far greater extremes of complexity and accuracy.

Table 1: Preferences.
Preferences Votes. Transfer of A's surplus @ 46/74 & B's @ 8/36. A & B both keep 28: 56 votes to add to the surplus transfers.
BCEDA 36 36 x 8/36 = 8
ACDEB 24 24 x 46/74 = 14.9189
ADEBC 20 20 x 46/74 = 12.4324
ABECD 18 18 x 46/74 = 11.1892
ACDBE 8 8 x 46/74 = 4.9730
ABDCE 4 4 x 46/74 = 2.4865
110 54.0000 + 56 = 110.

In table 2, five, of the six permutations, show A to begin the orders. That means 74 voters prefer A for election if these are most preferences ( or for exclusion, when dealing with least preferences ). The quota for 110 voters electing 3 seats is approximately 28. So, A's surplus is 74 - 28 = 46. And A's transfer value is 46/74. Two of the permutations show that C is the next preference to A. So, C gets a surplus value of 14.9189 + 4.9730 = 19.8919.

Also C gets 8 votes from B. This increases the transfered vote to C to 27.8919. This figure is duly recorded in the Meek STV count table 2, stage 2, below.

Likewise, the other figures for stage 2 may be derived from table 2. Here, E gets no surplus votes, because E isnt second in any of the permutations in table 5.

B, who has already surpassed the 28 votes quota, at 36 votes, gets still more votes transfered from A, amounting to 11.1892 + 2.4865 = 13.6757. Therefore, B's new total vote is 49.6757. B still keeps the quota of 28 votes, so B's new keep value is 28/49.6757. And B's new surplus is 21.6757, for a transfer value of 21.6757/49.6757.

The transfer of B's re-calculated surplus is shown at stage 3. The 11.1892 votes going from B to E are multiplied by the transfer value, to give 4.8823.

The 2.4865 votes, going from B to E, multiplied by the transfer value, give 1.0850. Add this to D's existing 12.4324 to equal 13.5174.

The 36 votes from B to C, multiplied by the transfer value, give 15.7084. Add this to surplus votes from A to C, namely 14.9189 + 4.9730 = 19.8919. This comes to 35.003, more than a quota for C.

Table 2: Meek's method STV election count. Quota = 110/(3+1) = 28.
Candidates. Stage 1: First preferences. A & B elected. Stage 2: Transfer A's surplus @ 46/74 & B's @ 8/36. Stage 3: Further transfer B's surplus, @ 21.6757/49.6757. C elected.
A 74 28 28
B 36 28 + 13.6757 28
C 0 27.8919 35.6003
D 0 12.4324 13.5174
E 0 0 4.8823
Total votes: 110 110 110

This completes the example to show how Meek's method might elect three candidates. In this case, A, B and C are elected.
Table 3 compares Meek's method with a manual count of the same ballot.

Table 3: Manual STV election count ( of same election by Meek's method, in above table. ) Quota = 28.
Candidates. Stage 1: First preferences.
A & B elected.
Stage 2: A's surplus transfered @ 46/74. Stage 3: B's surplus transfered. Stage 4: Totals from A's & B's surpluses.
A 74 28
B 36 28
C 0 32 x 46/74 ~ 19.892 8 27.892
D 0 24 x 46/74 ~ 14.918 14.918
E 0 18 x 46/74 ~ 11.189 11.189
Total votes: 110 110 ( to 2 dec. places )

The manual count differs from Meek's method, notably in that C has not yet quite reached a 28 votes quota, at 27.892 votes.
Also, taking counts to different numbers of decimal places for different degrees of accuracy means they are not strictly comparable.

Richard Lung.
Original page from 1999; Last section, February 2004, put here 1 January 2006.

To top

To home page