Counting single transferable votes

The single transferable vote (STV) is a voting system based on proportional representation and ranked voting. Under STV, an elector's vote is initially allocated to his or her most-preferred candidate. After candidates have been either elected (winners) by reaching quota or eliminated (losers), surplus votes are transferred from winners to remaining candidates (hopefuls) according to the surplus ballots' ordered preferences.

The system minimizes "wasted" votes and allows for approximately proportional representation without the use of party lists. A variety of algorithms (methods) carry out these transfers.

Voting

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

Andrea	2
Carter	1
Brad	4
Delilah	3

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:

$\frac{\text{(votes cast)}}{\text{(available seats)}}$
The Hare 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:

$\left(\frac{\text{votes}}{\text{seats}+1}\right)+1$
The Droop Quota

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:

	16 Votes	24 Votes	17 Votes
1st	Andrea	Andrea	Delilah
2nd	Brad	Carter	Andrea
3rd	Carter	Brad	Brad
4th	Delilah	Delilah	Carter

The quota is calculated as ${57 \over 2+1} +1 = 20$.

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 and he fills the second seat.

Thus:

	Round 1	Round 2	Round 3
Andrea	40	20	20	Elected in round 1
Brad	0	8	0	Excluded in round 2
Carter	0	12	20	Elected in round 3
Delilah	17	17	17	Defeated in round 3

Counting rules

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

• Surplus votes (i.e. those in excess of the quota) of elected candidates.
• All votes of eliminated candidates.

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 $\begin{matrix} \frac {1} {n} \end{matrix}$ 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.^[1] 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.

In May 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 .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 $\textstyle\frac{75}{272 - 92} = \frac{4}{10}$. 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 $\textstyle \frac15$. In this case, these 150 ballots would now be retransferred with a compounded fractional value of $\textstyle \frac15 \times \frac{4}{10} = \frac{4}{50}$.

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

$\text{Surplus Transfer Value} = \left( {{\text{Total value of Candidate's votes} - \text{Quota}} \over \text{Total value of Candidate's votes}} \right)\times \text{Value of each vote}$

Secondary preferences for prior winners

Suppose a ballot is to be transferred and its next preference is for a winner in a prior round. Hare and Cincinnati ignore such preferences and transfer the ballot to the next preference.

Alternatively the vote could be transferred to that winner and the process continued. For example, a prior winner X could receive 20 transfers from second round winner Y. Then select 20 at random from the 220 for transfer from X. However, some of these 20 ballots may then transfer back from X to Y, creating recursion. In the case of the Senatorial rules, since all votes are transferred at all stages, the recursion is infinite, with ever-decreasing fractions.

Meek

In 1969, B.L. Meek devised an algorithm based on Senatorial rules, which uses an iterative approximation to short-circuit this infinite recursion. This system is currently used for some local elections in New Zealand and for elections of moderators on some internet websites, for example Stack Exchange Network portals.^[2]

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	$1$
Excluded	$0$
Elected	$w_\text{new} = w_\text{old} \times \frac{\text{Quota}}{\text{Candidate's votes}}$ which is repeated until $\text{Candidate's votes} = \text{Quota}$ 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, using the formula:

$1 -\text{nth Weighting}$

For example, consider a ballot with top preferences A, B, C, where the weightings of the three candidates are $a$, $b$, $c$ respectively. From this ballot A will retain $a$, B will retain $(1-a) b$, and C will retain $(1-a) (1-b) c$.

This may result in a fractional excess, which is disposed of by altering the quota. Meek's method is the only method to change quota mid-process. The quota is found by

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

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 .0001%).^[3]

Warren

In 1994, C. H. E. Warren proposed another method of passing surplus to previously-elected candidates.^[4] Warren is identical to Meek except in the amounts 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, where the weightings are a, b, and c. Under Warren's method, A will retain a, B will retain b (or (1−a) if (1−a)<b), and C will retain c (or (1−a−b) if (1−a−b)<c — or 0 if (1−a−b) is already less than 0).

Because candidates receive different values of votes, the weightings determined by Warren are in general different than 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.^[5]

Distribution of excluded candidate preferences

The method used in determining the order of exclusion and distribution of a candidates' votes can affect the outcome. Multiple methods are in common use for determining the order polyexclusion and distribution of ballots from a loser. Most systems (with the exception of an iterative count) were designed for manual counting processes and can produce different outcomes.

The general principle that applies to each method is to exclude the candidate that has the lowest tally. Systems must handle ties for the lowest tally. Alternatives include excluding the candidate with the lowest score in the previous round and choosing by lot.

Exclusion methods commonly in use:

• Single transaction—Transfer all votes for a loser in a single transaction without segmentation.
• Segmented distribution—Split distributed ballots into small, segmented transactions. Consider each segment a complete transaction, including checking for candidates who have reached quota. Generally, a smaller number and value of votes per segment reduces the likelihood of affecting the outcome.
  • Value based segmentation—Each segment includes all ballots that have the same value.
  • Aggregated primary vote and value segmentation—Separate the Primary vote (full-value votes) to reduce distortion and limit the subsequent value of a transfer from a candidate elected as result of a segmented transfer.
  • FIFO (First In First Out – Last bundle)—Distribute each parcel in the order in which it was received. This method produces the smallest size and impact of each segment at the cost of requiring more steps to complete a count.^[6]
• Iterative count—After excluding a loser, reallocate the loser's ballots and restart the count. An iterative count treats each ballot as though that loser had not stood. Ballots can be allocated to prior winners using a segmented distribution process. Surplus votes are distributed only within each iteration. Iterative counts are usually automated to reduce costs. The number of iterations can be limited by applying a method of Bulk Exclusion.

Bulk exclusions

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).

Example

Quota Breakpoint (Based on the 2007 Queensland Senate election results just prior to the first exclusion)

Candidate	Ballot position	GroupAb	Group name	Score	Running sum	Breakpoint / Status
MACDONALD, Ian Douglas	J-1	LNP	Liberal	345559		Quota
HOGG, John Joseph	O-1	ALP	Australian Labor Party	345559		Quota
BOYCE, Sue	J-2	LNP	Liberal	345559		Quota
MOORE, Claire	O-2	ALP	Australian Labor Party	345559		Quota
BOSWELL, Ron	J-3	LNP	Liberal	284488	1043927	Contest
WATERS, Larissa	O-3	ALP	The Greens	254971	759439	Contest
FURNER, Mark	M-1	GRN	Australian Labor Party	176511	504468	Contest
HANSON, Pauline	R-1	HAN	Pauline	101592	327957	Contest
BUCHANAN, Jeff	H-1	FFP	Family First	52838	226365	Contest
BARTLETT, Andrew	I-1	DEM	Democrats	45395	173527	Contest
SMITH, Bob	G-1	AFLP	The Fishing Party	20277	128132	Quota Breakpoint
COLLINS, Kevin	P-1	FP	Australian Fishing and Lifestyle Party	19081	107855	Contest
BOUSFIELD, Anne	A-1	WWW	What Women Want (Australia)	17283	88774	Contest
FEENEY, Paul Joseph	L-1	ASP	The Australian Shooters Party	12857	71491	Contest
JOHNSON, Phil	C-1	CCC	Climate Change Coalition	8702	58634	Applied Breakpoint
JACKSON, Noel	V-1	DLP	D.L.P. - Democratic Labor Party	7255	49932
Others				42677	42677

Running Breakpoint (Based on the 2007 Tasmanian Senate election results just prior to the first exclusion)

Candidate	Ballot position	GroupAb	Group name	Score	Running sum	Breakpoint / Status
SHERRY, Nick	D-1	ALP	Australian Labor Party	46693		Quota
COLBECK, Richard M	F-1	LP	Liberal	46693		Quota
BROWN, Bob	B-1	GRN	The Greens	46693		Quota
BROWN, Carol	D-2	ALP	Australian Labor Party	46693		Quota
BUSHBY, David	F-2	LP	Liberal	46693		Quota
BILYK, Catryna	D-3	ALP	Australian Labor Party	37189		Contest
MORRIS, Don	F-3	LP	Liberal	28586		Contest
WILKIE, Andrew	B-2	GRN	The Greens	12193	27607	Running Breakpoint
PETRUSMA, Jacquie	K-1	FFP	Family First	6471	15414	Quota Breakpoint
CASHION, Debra	A-1	WWW	What Women Want (Australia)	2487	8943	Applied Breakpoint
CREA, Pat	E-1	DLP	D.L.P. - Democratic Labor Party	2027	6457
OTTAVI, Dino	G-1	UN3		1347	4430
MARTIN, Steve	C-1	UN1		848	3083
HOUGHTON, Sophie Louise	B-3	GRN	The Greens	353	2236
LARNER, Caroline	J-1	CEC	Citizens Electoral Council	311	1883
IRELAND, Bede	I-1	LDP	LDP	298	1573
DOYLE, Robyn	H-1	UN2		245	1275
BENNETT, Andrew	K-2	FFP	Family First	174	1030
ROBERTS, Betty	K-3	FFP	Family First	158	856
JORDAN, Scott	B-4	GRN	The Greens	139	698
GLEESON, Belinda	A-2	WWW	What Women Want (Australia)	135	558
SHACKCLOTH, Joan	E-2	DLP	D.L.P. - Democratic Labor Party	116	423
SMALLBANE, Chris	G-3	UN3		102	307
COOK, Mick	G-2	UN3		74	205
HAMMOND, David	H-2	UN2		53	132
NELSON, Karley	C-2	UN1		35	79
PHIBBS, Michael	J-2	CEC	Citizens Electoral Council	23	44
HAMILTON, Luke	I-2	LDP	LDP	21	21

See also

• Hagenbach-Bischoff quota
• Imperiali quota
• IRV implementations in United States#Independence Party of Minnesota (2004 Presidential poll)
• Voting matters — To advance the understanding of preferential voting systems, McDougall Trust journal comparing and critiquing actual and theoretical varieties of STV and related voting systems.

References

1. Single Transferable Vote Rules UK Electoral Reform Society
2. "Stack Exchange Meta FAQ question "There's an election going on. What's happening and how does it work?"". StackExchange Meta. 10 June 2012. Retrieved 21 April 2015.
3. Hill, I. David; B. A. Wichmann; D. R. Woodall (1987). "Algorithm 123 — Single Transferable Vote by Meek's Method" (PDF). The Computer Journal 30 (2): 277–281. doi:10.1093/comjnl/30.3.277.
4. Warren, C. H. E., "Counting in STV Elections", Voting matters 1 (1994), paper 4.
5. Hill, I. D. and C. H. E. Warren, "Meek versus Warren", Voting matters 20 (2005), pp. 1–5.
6. "Rules of the Proportional Representation Society of Australia for conducting election by Quota-Preferential".

External links

• Fair vote – US campaign for electoral reform.
• Quota Notes – Proportional Representation Society of Australia.
• Australian Electoral Commission Web site.
• Algorithm 123 — Single Transferable Vote by Meek's Method
• OpenSTV – software for computing the single transferable vote
• Irish Proportional Representation System
• Banbridge Council election results giving detailed breakdowns of fractional transfers, illustrating Senatorial Rules.
• New Zealand's Local Electoral Act 2001 – How to count votes under Meek's method
• Animated presentation of how Meek's method is used to count votes in New Zealand STV
• Movable vote workshop tallies STV and related rules.