Counting single transferable votes

When Thomas Hare originally conceived his version of single transferable vote, he envisioned using the quota:

${\displaystyle {\frac {\text{valid votes cast}}{\text{available seats}}}}$
The Hare Quota

The Hare quota is mathematically simple. It is the largest number of votes that allows enough to be elected to fill the open seats, barring the presence of exhausted votes.

If even one vote is exhausted, the last seat will have to be filled by a candidate who does not have Hare. The use of Hare means that all ballots must be filled out fully to prevent any possibility of exhausted votes. Under some STV systems, ballots not filled out fully are declared not valid. Or the rules must allow candidates to be elected at the end through plurality, even if they do not achieve quota.

The Hare quota's large size means that elected members have fewer surplus votes and thus other candidates do not get benefit from vote transfers that they would in other systems. Some candidates may be eliminated in the process who may not have been eliminated under systems that transfer more surplus votes. Their elimination may cause a degree of dis-proportionality that would be less likely with a lower quota, such as the Droop quota.

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

${\displaystyle \left\lfloor {\frac {\text{votes}}{{\text{seats}}+1}}\right\rfloor +1}$
The Droop Quota

Droop quota is a smaller number of votes than Hare.

Because of this difference, under Droop it is more likely that every winner meets the quota rather than being elected as the last remaining candidate after lower candidates are eliminated. (But in real-life elections, if it is allowed for valid ballots to not bear full rankings, it is common even under Droop for one or two candidates to be elected with partial quota at the end, as the field of candidates is thinned to the number of remaining open seats and as the valid votes still in play become scarcer.)

The use of Droop sometimes leaves nearly one quota's worth of votes held by unsuccessful candidates; these ballots are effectively ignored. Unlike a system using the Hare quota and mandatory full marking of the ballot, with optional marking and Droop quota, a certain percentage of ballots are not used to elect anyone.

Under Droop, a majority of the voters can be guaranteed to elect a majority of the seats when there is an odd number of seats.

Each winner's surplus votes transfer to other candidates according to their remaining preferences, using a formula (p/t)*s, 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.

Under some systems, a fraction of the vote is transferred, with a fraction left behind with the winner. In others such as Ireland or Malta national elections, transfers are done using whole votes, with some of the votes that are directed to another candidate left behind with the winner and others of the same sort of votes moved in whole to the indicated candidate.

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 ${\displaystyle {57 \over 2+1}+1=20}$.

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 (quota) and he fills the second seat.

Thus:

If the transfer is of surplus received in the first count, transfers are done in reference to all the votes held by the successful candidate.

If the transfer is of surplus received after the first count through transfer from another candidate, transfers are done in reference to all the votes held by the successful candidate or merely in reference to the most recent transfer received by the successful candidate.

Reallocation ballots are drawn at random from those most recently received. In a manual count of paper ballots, this is the easiest method to implement.

Votes are transferred as whole votes. Fractional votes are not used.

This system is close to Thomas Hare's original 1857 proposal. It is used in elections in the Republic of Ireland to Dáil Éireann (the lower chamber),[1] to local government,[2] to the European Parliament,[3] and to the university constituencies in Seanad Éireann (the upper chamber).[4]

This is sometimes described as "random" because it does not consider later back-up preferences but only the next usable one. Through random drawing of the votes to make up the transfer, statistically the transfers often reflect the make-up of the votes held by the successful candidate.

Sometimes, ballots of the 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 ${\displaystyle n}$th ballot is selected, where ${\displaystyle {\begin{matrix}{\frac {1}{n}}\end{matrix}}}$ is the fraction to be selected.

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

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.

This is a variation on the original Hare method that used "random" choices. It is used in Tasmanian and ACT lower house 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. Count the first preferences votes.
2. Declare as winners those candidates whose total is at least Q.
3. For each winner, compute surplus as total number of votes minus Q.
4. For each winner, in order of descending surplus:
   1. Assign that winner's ballots to candidates according to each ballot's next 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 candidate, multiply ratio * the number of that candidate's reassigned votes and add the result (rounded down) to the candidate's tally.
5. Repeat 3–5 until winners fill all seats, or all ballots are exhausted.
6. If more winners are needed, declare a loser the candidate with the fewest votes, and reassign that candidate's ballots according to each ballot's next preference.

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, then the votes X receives is 0.4*75 or 30. If X had 190 votes, then X becomes a winner, with a surplus of 20 for the next round, if needed.

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 ${\displaystyle \textstyle {\frac {72}{272-92}}={\frac {4}{10}}}$. 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 ${\displaystyle \textstyle {\frac {1}{5}}}$. In this case, these 150 ballots would now be retransferred with a compounded fractional value of ${\displaystyle \textstyle {\frac {1}{5}}\times {\frac {4}{10}}={\frac {4}{50}}}$.

In the Republic of Ireland, the Gregory Method is used for elections to the 43 seats on the vocational panels in Seanad Éireann, whose franchise is restricted to 949 members of local authorities and members of the Oireachtas (the Irish Parliament).[5] In Northern Ireland, the Gregory Method has been used since 1973 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 for the Northern Ireland constituency from 1979 to 2020).

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

$${\displaystyle {\text{surplus transfer value}}=\left({{{\text{total value of candidate's votes}}-{\text{quota}}} \over {\text{total value of candidate's votes}}}\right)\times {\text{value of each vote}}}$$

The Unweighted Inclusive Gregory Method is used for the Australian Senate.[6]

The effect of the Gregory system can be replicated without using fractional values by a party-list proportional allocation method, such as D'Hondt, Webster/Sainte-Laguë or Hare-Niemeyer. A party-list proportional representation electoral system allocates a share of the seats in a legislature to a political party in proportion to its share of the votes, a task which is mathematically equivalent to establishing a share of surplus votes to be transferred to a hopeful candidate based on the overall vote for an eliminated candidate.

Example: If the quota is 200 and a winner has 272 first-choice votes, then the surplus is 72 votes. If 92 of the winner's 272 votes have no other hopeful listed, then the remaining 180 votes have a second-choice selection and can be transferred.

Of the 180 votes which can be transferred, 75 have hopeful X as their second-choice, 43 have hopeful Y as their second-choice, and 62 have hopeful Z as their second-choice. The D'Hondt system is applied to determine how the surplus votes would be transferred - successive quotients are calculated for each hopeful candidate, one surplus vote is transferred to the hopeful candidate with the largest quotient, and the hopeful candidate's quotient is recalculated; this is repeated until all surplus votes have been transferred.

As a result of this process, 30 surplus votes have been transferred to hopeful X, 17 to hopeful Y, and 25 to hopeful Z.

In 1969, B. L. Meek devised a vote counting algorithm based on Senatorial (Gregory) vote counting rules. The Meek algorithm uses an iterative approximation to short-circuit the infinite recursion that results when there are secondary preferences for prior winners. This system is currently used for some local elections in New Zealand,[7][8] and for elections of moderators on some internet websites, e.g. Stack Exchange Network portals.[9]

All candidates are allocated one of three statuses – Hopeful, Elected, or Excluded. Hopeful is the default. Each status has a weighting, or keep value, which is the fraction of the vote a candidate will receive for any preferences allocated to them while holding that status.

The weightings are:

Hopeful	${\displaystyle 1}$
Excluded	${\displaystyle 0}$
Elected	${\displaystyle w_{\text{new}}=w_{\text{old}}\times {\frac {\text{Quota}}{\text{Candidate's votes}}}}$ which is repeated until ${\displaystyle {\text{Candidate's votes}}={\text{Quota}}}$ for all elected candidates

Thus, if a candidate is Hopeful they retain the whole of the remaining preferences allocated to them, and subsequent preferences are worth 0.

If a candidate is Elected they retain the portion of the value of the preferences allocated to them that is the value of their weighting; the remainder is passed fractionally to subsequent preferences depending on their weighting. For example, consider a ballot with top preferences A, B, C, and D in that order, where the weightings of the candidates are ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$, and ${\displaystyle d}$, respectively. From this ballot A will retain ${\displaystyle r_{A}=a}$, B will retain ${\displaystyle r_{B}=(1-a)b}$, C will retain ${\displaystyle r_{C}=(1-a)(1-b)c}$, and D will retain ${\displaystyle r_{D}=(1-a)(1-b)(1-c)d}$.

If no candidate on a ballot has a weight of 1 then the sum total retained by the candidates on the ballot will be strictly less than 1. The amount by which 1 exceeds the ballot's total is called the "excess", and the total excess from all ballots is disposed of by altering the quota. Meek's method is the only method to change quota mid-process. The quota is found by

$${\displaystyle {{{\text{votes}}-{\text{excess}}} \over {\text{seats}}+1},}$$

a variation on Droop. This has the effect of also altering the weighting for each candidate.

This process continues until all the Elected candidates' vote values closely match the quota (plus or minus 0.0001%).[10]

In 1994, C. H. E. Warren proposed another method of passing surplus to previously-elected candidates.[11] Warren is identical to Meek except in the numbers of votes retained by winners. Under Warren, rather than retaining that proportion of each vote's value given by multiplying the weighting by the vote's value, the candidate retains that amount of a whole vote given by the weighting, or else whatever remains of the vote's value if that is less than the weighting.

Consider again a ballot with top preferences A, B, C, and D where the weightings are a, b, c, and d. Under Warren's method, A will retain r_A = a, B will retain r_B = min(b, 1−r_A), C will retain r_C = min(c, 1−r_A−r_B), and D will retain r_D = min(d, 1−r_A−r_B−r_C).

Because candidates receive different values of votes, the weightings determined by Warren are in general different from Meek.

Under Warren, every vote that contributes to a candidate contributes, as far as it is able, the same portion as every other such vote.[12]

Bulk exclusion rules can reduce the number of steps required within a count. Bulk exclusion requires the calculation of breakpoints. Any candidates with a tally less than a breakpoint can be included in a bulk exclusion process provided the value of the associated running sum is not greater than the difference between the total value of the highest hopeful's tally and the quota.

To determine a breakpoint, list in descending order each candidates' tally and calculate the running tally of all candidates' votes that are less than the associated candidates tally. The four types are:

• Quota breakpoint—The highest running total value that is less than half of the Quota
• Running breakpoint—The highest candidate's tally that is less than the associated running total
• Group breakpoint—The highest candidate's tally in a Group that is less than the associated running total of Group candidates whose tally is less than the associated Candidate's tally. (This only applies where there are defined groups of candidates such as in Australian public elections, which use an above-the-line group voting method.)
• Applied breakpoint—The highest running total that is less than the difference between the highest candidate's tally and the quota (i.e. the tally of lower-scoring candidates votes does not affect the outcome). All candidates above an applied breakpoint continue in the next iteration.

Quota breakpoints may not apply with optional preferential ballots or if more than one seat is open. Candidates above the applied breakpoint should not be included in a bulk exclusion process unless it is an adjacent quota or running breakpoint (see 2007 Tasmanian Senate count example below).

Quota breakpoint (based on the 2007 Queensland Senate election results just prior to the first exclusion):

Running breakpoint (based on the 2007 Tasmanian Senate election results just prior to the first exclusion):