Example of a Ballot Count using the Single Transferable Vote (STV) Method
Assume the following situation:
There are 500 voters.
There are 3 positions to be filled.
There are 5 candidates (A, B, C, D, & E) on the ballot paper.
All ballots are formal.
Voters are required to indicate preferences for all the candidates. Therefore there is no ‘exhaustion’ of ballot papers as a result of optional preferences.
Calculating the quota:
The first step is to calculate the quota needed for election. The quota is defined as the total number of formal votes divided by one more than the number of positions being filled, plus one, with any remainder being ignored. This produces a quota equal to the minimum number of votes such that, if each of the successful candidates obtained this number, it would not be arithmetically possible for any other candidate to obtain the quota. To demonstrate:
Let the total number of formal votes cast be T
Let the number of positions to be filled be n
Q, the quota, is such that if each of the n successful candidates obtains Q votes, the remaining votes (namely T-nQ) must be less than Q:
T – nQ < Q
nQ + Q > T rearranging the inequality
Q(n + 1) > T factorising Q
Q > T ÷ (n + 1)
In this example T = 500 and n = 3, so Q = (500 ÷ 4) + 1 = 126 votes
The quota of 126 votes remains unchanged throughout the count.
Counting the first preferences:
The next step is to allocate the ballots to each candidate according to the first preferences expressed. Assume that the first preferences were:
A 230
B 100
C 50
D 80
E 40
Total 500
Candidate A has received more than a quota and is elected. He retains his quota of votes (126). His additional votes above the quota (termed his ‘surplus’ votes) are transferred on to the remaining candidates according to the second preferences marked on the ballots by the voters. The question then becomes: which of the 230 ballot papers that A received are to be deemed the ‘surplus’ ones? The fairest way to do this is to examine all of candidate A’s ballot papers and to transfer his first preference votes at a reduced value (known as the ‘transfer’ value). This is calculated in the following way:
Candidate A’s surplus = Total votes – Quota = 230 – 126 = 104
Transfer value of candidate A’s votes = Surplus ÷ Total votes = 104 ÷ 230 = 0.4522
Assume that the 230 first preference votes for candidate A flowed to second preferences in the following way:
10 for candidate B
20 for candidate C
60 for candidate D
140 for candidate E
Therefore the number of votes transferred would be calculated as:
To candidate B: 10 x 0.4522 = 4
To candidate C: 20 x 0.4522 = 9
To candidate D: 60 x 0.4522 = 27
To candidate E: 140 x 0.4522 = 63
Note that, in transferring votes, fractions of votes are ignored.
After the transfer of candidate A’s surplus votes, the count looks like this:
A 126 elected
B 100 + 4 (transferred from A) = 104
C 50 + 9 (transferred from A) = 59
D 80 + 27 (transferred from A) = 107
E 40 + 63 (transferred from A) = 103
Total 499
At this point we check to see if any other candidate has now reached a quota as a result of the surplus transferred from candidate A. If another candidate has reached a quota, the same process of transfer of surplus votes takes place. This process continues until all the positions have been filled, or there is no other candidate who has reached a quota (as is the case here). If so, we move to the next phase of the count.
Exclusion of candidates:
When no further candidate has reached a quota, the candidate in the count who has the lowest number of votes is excluded. His votes are then transferred on to the remaining candidates according to the preferences on the ballots. In this case it is candidate C who is excluded. Assume that all of candidate C’s 59 ballots are examined and:
The 50 first preference votes flowed to second preferences:
10 for candidate B
40 for candidate D
The 9 transferred second preferences flowed to third preferences:
9 for candidate E
No transfer value calculation is required here because candidate C is being excluded.
After the exclusion of candidate C, the count looks like this:
A 126 elected
B 100 + 4 (transferred from A) +10 (transferred from C) = 114
C 0 excluded
D 80 + 27 (transferred from A) + 40 (transferred from C) = 147
E 40 + 63 (transferred from A) + 9 (transferred from C) = 112
Total 499
Another round of transfers:
Remember that the quota is 126 votes. The exclusion of candidate C and the transfer of his votes has now enabled candidate D to obtain a quota, with a surplus. This surplus now has to be transferred. Again a transfer value is calculated:
Candidate D’s surplus = Total votes – Quota = 147 – 126 = 21
Transfer value of candidate D’s votes = Surplus ÷ Total votes = 21 ÷ 147 = 0.1429
We now examine the preferences of the 147 ballots allocated to candidate D. By this stage of the count we may be looking at voters’ fourth and fifth preferences. Assume that the 147 ballots ultimately flow to preferences:
57 to candidate B
90 to candidate E
Therefore the number of votes transferred would be calculated as:
To candidate B: 57 x 0.1429 = 8
To candidate E: 90 x 0.1429 = 12
After transfer the count looks like this:
A 126 elected
B 100 + 4 (transferred from A) +10 (transferred from C) + 8 (transferred from D) = 122
C 0 excluded
D 126 elected
E 40 + 63 (transferred from A) + 9 (transferred from C) + 12 (transferred from D) = 124
Total 498
Final step:
Although candidate E has not obtained a quota, he is elected, because he has more votes than candidate B and there is only one position left to be filled. Note that that total number of votes in the count has gradually fallen due to the rounding down of votes during the transfer process.
Final Outcome:
The successful candidates, in order of election, are A, D and E.
Candidate B, who had the second highest number of first preference votes, has missed out on gaining a position. This is because of the strong lower preference flows to the minor party candidates. The majority of candidate A’s supporters preferred candidate E over candidate B. And while candidate C’s voters mostly chose candidate D as their second preference, they ultimately also preferred candidate E over candidate B. This example emphasises just how important voters’ lower preferences can be in determining the outcome of the final positions.