Counting single transferable votes

Voting

When using an STV ballot, the voter ranks the candidates on the ballot. For example:

Quota

The quota (sometimes called the threshold) is the number of votes a candidate must receive to be elected. The Hare quota and the Droop quota are commonly used to determine the quota.

Hare quota

When Thomas Hare originally conceived his version of Single Transferable Vote, he envisioned using the quota:

In the unlikely event that each successful candidate receives exactly the same number of votes, not enough candidates can meet the quota and fill the available seats in one count. Thus the last candidate cannot not meet the quota, and it may be fairer to eliminate that candidate.

To avoid this situation, it is common instead to use the Droop quota, which is always lower than the Hare quota.

Droop quota

The most common quota formula is the Droop quota, which given as:

Droop produces a lower quota than Hare. If each ballot has a full list of preferences, Droop guarantees that every winner meets the quota rather than being elected as the last remaining candidate after lower candidates are eliminated. The fractional part of the resulting number, if any, is dropped (the result is rounded down to the next whole number.)

It is only necessary to allocate enough votes to ensure that no other candidate still in contention could win. This leaves nearly one quota's worth of votes unallocated, but counting these would not alter the outcome.

Droop is the only whole-number threshold for which (a) a majority of the voters can be guaranteed to elect a majority of the seats when there is an odd number of seats; (b) for a fixed number of seats.

Each winner's surplus votes transfer to other candidates according to their remaining preferences, using a formula s/t*p, where s is a number of surplus votes to be transferred, t is a total number of transferable votes (that have a second preference) and p is a number of second preferences for the given candidate. Meek's counting method recomputes the quota on each iteration of the count.

Example

Two seats need to be filled among four candidates: Andrea, Brad, Carter, and Delilah. 57 voters cast ballots with the following preference orderings:

The quota is calculated as 57 2 + 1 + 1 = 20 .

In the first round, Andrea receives 40 votes and Delilah 17. Andrea is elected with 20 surplus votes. Ignoring how the votes are valued for this example, 20 votes are reallocated according to their second preferences. 12 of the reallocated votes go to Carter, 8 to Brad.

As none of the hopefuls have reached the quota, Brad, the candidate with the fewest votes, is excluded. All of his votes have Carter as the next-place choice, and are reallocated to Carter. This gives Carter 20 votes and he fills the second seat.

Thus:

Counting rules

Under the single transferable vote system, votes are successively transferred to hopefuls from two sources:

The possible algorithms for doing this differ in detail, e.g., in the order of the steps. There is no general agreement on which is best, and the choice of exact method may affect the outcome.

1. Compute the quota.
2. Assign votes to candidates by first preferences.
3. Declare as winners all candidates who received at least the quota.
4. Transfer the excess votes from winners to hopefuls.
5. Repeat 3–4 until no new candidates are elected. (Under some systems, votes could initially be transferred in this step to prior winners or losers. This might affect the outcome.)

If all seats have winners, the process is complete. Otherwise:

6. Eliminate one or more candidates, typically either the lowest candidate or all candidates whose combined votes are less than the vote of the lowest remaining candidate.
7. Transfer the votes of the losers to remaining hopeful candidates.
8. Repeat 3–7 until all seats are full.

Surplus allocation

To minimize wasted votes, surplus votes are transferred to other candidates. The number of surplus votes is known; but none of the various allocation methods is universally preferred. Alternatives exist for deciding which votes to transfer, how to weight the transfers, who receives the votes and the order in which surpluses from two or more winners are transferred. Reallocation occurs when a candidate receives more votes than necessary to meet the quota. The excess votes are reallocated to still other candidates.

Random subset

Some surplus allocation methods select a random vote sample. Sometimes, ballots of one elected candidate are manually mixed. In Cambridge, Massachusetts, votes are counted one precinct at a time, imposing a spurious ordering on the votes. To prevent all transferred ballots coming from the same precinct, every n th ballot is selected, where 1 n is the fraction to be selected.

Hare

Reallocation ballots are drawn at random from those transferred. In a manual count of paper ballots, this is the easiest method to implement; it is close to Thomas Hare's original 1857 proposal. It is used in all universal suffrage elections in the Republic of Ireland. Exhausted ballots cannot be reallocated, and therefore do not contribute to any candidate.

Cincinnati

Reallocation ballots are drawn at random from all of the candidate's votes. This method is more likely than Hare to be representative, and less likely to suffer from exhausted ballots. The starting point for counting is arbitrary. Under a recount the same sample and starting point is used in the recount (i.e., the recount must only be to check for mistakes in the original count, and not a second selection of votes).

Hare and Cincinnati have the same effect for first-count winners, since all the winners' votes are in the "last batch received" from which the Hare surplus is drawn.

Wright

The Wright system is a reiterative linear counting process where on each candidate's exclusion the count is reset and recounted, distributing votes according to the voters nominated order of preference, excluding candidates removed from the count as if they had not nominated.

For each successful candidate that exceeds the quota threshold, calculate the ratio of that candidate's surplus votes (i.e., the excess over the quota) divided by the total number of votes for that candidate, including the value of previous transfers. Transfer that candidate's votes to each voter's next preferred hopeful. Increase the recipient's vote tally by the product of the ratio and the ballot's value as the previous transfer (1 for the initial count.)

The UK's Electoral Reform Society recommends essentially this method. Every preference continues to count until the choices on that ballot have been exhausted or the election is complete. Its main disadvantage is that given large numbers of votes, candidates and/or seats, counting is administratively burdensome for a manual count due to the number of interactions. This is not the case with the use of computerised distribution of preference votes.

From May 2011 to June 2011, The Proportional Representation Society of Australia reviewed the Wright System noting:

While we believe that the Wright System as advocated by Mr. Anthony van der Craats system is sound and has some technical advantages over the PRSA 1977 rules, nevertheless for the sort of elections that we (the PRSA) conduct, these advantages do not outweigh the considerable difficulties in terms of changing our (The PRSA) rules and associated software and explaining these changes to our clients. Nevertheless, if new software is written that can be used to test the Wright system on our election counts, software that will read a comma separated value file (or OpenSTV blt files), then we are prepared to consider further testing of the Wright system.

Hare-Clark

This is a variation on the original Hare method that used random choices. It is used in some elections in Australia. It allows votes to the same ballots to be repeatedly transferred. The surplus value is calculated based on the allocation of preference of the last bundle transfer. The last bundle transfer method has been criticised as being inherently flawed in that only one segment of votes is used to transfer the value of surplus votes denying voters who contributed to a candidate's surplus a say in the surplus distribution. In the following explanation, Q is the quota required for election.

1. Separate all ballots according to their first preferences.
2. Count the votes.
3. Declare as winners those hopefuls whose total is at least Q.
4. For each winner, compute surplus as total minus Q.
5. For each winner, in order of descending surplus:
   1. Assign that candidate's ballots to hopefuls according to each ballot's preference, setting aside exhausted ballots.
   2. Calculate the ratio of surplus to the number of reassigned ballots or 1 if the number of such ballots is less than surplus.
   3. For each hopeful, multiply ratio * the number of that hopeful's reassigned votes and add the result (rounded down) to the hopeful's tally.
6. Repeat 3–5 until winners fill all seats, or all ballots are exhausted.
7. If more winners are needed, declare a loser the hopeful with the fewest votes, recompute Q and repeat from 1, ignoring all preferences for the loser.

Example: If Q is 200 and a winner has 272 first-choice votes, of which 92 have no other hopeful listed, surplus is 72, ratio is 72/(272−92) or 0.4. If 75 of the reassigned 180 ballots have hopeful X as their second-choice, and if X has 190 votes, then X becomes a winner, with a surplus of 20 for the next round, if needed.

The Australian variant of step 7 treats the loser's votes as though they were surplus votes. But redoing the whole method prevents what is perhaps the only significant way of gaming this system – some voters put first a candidate they are sure will be eliminated early, hoping that their later preferences will then have more influence on the outcome.

Gregory

Another method, known as Senatorial rules (after its use for most seats in Irish Senate elections), or the Gregory method (after its inventor in 1880, J.B. Gregory of Melbourne) eliminates all randomness. Instead of transferring a fraction of votes at full value, transfer all votes at a fractional value.

In the above example, the relevant fraction is 75 272 − 92 = 4 10 . Note that part of the 272 vote result may be from earlier transfers; e.g., perhaps Y had been elected with 250 votes, 150 with X as next preference, so that the previous transfer of 30 votes was actually 150 ballots at a value of 1 5 . In this case, these 150 ballots would now be retransferred with a compounded fractional value of 1 5 × 4 10 = 4 50 .

In the Republic of Ireland, Gregory is used only for the Senate, whose franchise is restricted to approximately 1,500 councillors, members of Parliament and National University of Ireland and University of Dublin graduates for 6 of those seats. However, in Northern Ireland beginning in 1973, Gregory was used for all STV elections, with up to 7 fractional transfers (in 8-seat district council elections), and up to 700,000 votes counted (in 3-seat European Parliament elections).

An alternative means of expressing Gregory in calculating the Surplus Transfer Value applied to each vote is