Counting single transferable votes

There are many ways to count votes in Single Transferable Vote elections.

Voting

If a class of children were choosing representatives, say, they could line-up behind the candidate of their choice. They would all know that each candidate only needs a certain number of classmates to vote for them to be elected. So some of those standing in line for a candidate who has more than enough votes would choose to not waste their vote and would instead move to another line to help someone else to win. Naturally, they would not move to a line that already has enough to win. Likewise, those children whose candidate obviously could not win, would move to another line, and so on, until all the representatives are chosen.

When using an STV ballot, these preferences are set out in advance, as instructions to the counters.

Each voter ranks all candidates in order of preference. For example:

2	Andrea
1	Carter
4	Brad
3	Delilah

Setting the quota

Choice of quota

The quota (sometimes called the threshold) is the number of votes a candidate must receive to be elected. The Droop quota is preferred because it 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, and (b) we cannot elect more candidates than there are seats.

A candidate's surplus votes are transferred to other candidates according to the later preferences. In Meek's method, the quota must be recalculated throughout the count.

With the Hare quota, even if each voter expresses a preference for every candidate, one candidate is likely to be elected with less than a full quota. If each voter expresses a full list of preferences, the Droop quota guarantees that every candidate elected will meet the quota rather than be elected by being the last remaining candidate after lower candidates are eliminated.

Droop quota

The most common formula for the quota used now is the Droop quota which is most often given as:

$\left({{\rm {votes}} \over {\rm {seats}}+1}\right)+1$
The Droop Quota

The fractional part of the resulting number, if any, is dropped; that is, we round down to the next whole number. Unlike the Hare quota, this does not require that all preferences must reach a final home. It is only necessary that enough votes be allocated to ensure that no other candidate still in contention could win. This leaves nearly a quota's worth of votes unallocated, but it is held that counting these votes would not alter the eventual outcome.

Hare quota

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

${\rm {votes}} \over {\rm {seats}}$
The Hare Quota

This has thus become known as the Hare quota. It would require that all votes cast be divided equally between the eventually successful candidates. The only differences, thus, between the votes received for each candidate would be based on the distribution of voters between constituencies (Hare's original proposal was for a single national constituency) and the number of non-continuing votes, i.e. people who did not express a preference for all candidates, meaning that some candidates would be elected with less than a quota as the last remaining.

Counting the votes

The single-transferable-vote process for counting the votes has the following outline. There are several different systems for filling in this outline. Experts do not agree on which one system is best.

We use the term hopefuls for those candidates not yet elected or eliminated at a given stage of the process. We leave out details of what to do for ties or for fewer candidates than seats.

First, compute the threshold quota and assign ballots to candidates by first preferences only.

PROCESS A: Declare as elected all candidates who receive at least the quota; then transfer the excess votes from the winners to later choices on those ballots. Repeat this process until no new candidates are elected. Note: Some systems transfer only to hopeful candidates; others transfer to all candidates. It can make a difference to the result.

If we now have as many winners as there are seats to win, or if there are only as many candidates left as there are seats, we are done.

PROCESS B: Eliminate one or more candidates (most systems eliminate just the lowest vote-getter). Adjust for those freed-up ballots (usually by transferring each one to another choice on the ballot; it depends on the system used). Then go back to Process A and continue from there.

Surplus re-allocation

The votes an elected candidate receives in excess of the quota constitute a surplus. To minimize wasted votes, these are transferred to other candidates. The complex part is deciding which particular votes for winners are excess and therefore to be transferred, as opposed to those votes left with the winner. It makes a significant difference in the results, because people with the same first preference often have different second and third preferences. There are several competing methods, not agreed on by experts. There are also competing methods for deciding which of the later choices the votes are to be transferred to, and competing methods for deciding the order in which excesses from two or more winners are transferred. These differences in methods will cause a difference in results for some sets of voting data.

The Duggan-Schwartz theorem proves that every choice voting system, regardless of which of the competing methods is used, is subject to gaming the system when you have three or more candidates treated impartially and three or more voters treated impartially. Gaming the system (tactical voting) means that there are conditions under which a voter can get what he wants by lying about his preferences on the ballot, when he would not get what he wants by telling the truth about his preferences.

Randomisation

Some of the surplus allocation methods rely on selecting a random sample of the votes. Ensuring randomness is done in various ways. In many cases, all the relevant ballot-papers are simply manually mixed together. In Cambridge, Massachusetts, votes are counted one precinct at a time, imposing a spurious ordering on the votes. To prevent all ballots for transfer being selected from the same precinct, every $n$th ballot paper is selected, where ${\begin{matrix}{\frac {1}{n}}\end{matrix}}$ is the fraction to be selected.

Initial surplus

Suppose candidate X, at a certain stage of the count, has 190 votes, and the quota is 200. Now X receives 30 votes transferred from candidate Y (after Y was either elected or eliminated). This gives X a total of 220 votes, i.e. a surplus of 20 to be transferred. But which 20 votes will be transferred?

Hare method

20 votes are drawn at random from the 30 received from Y's transfers. These 20 votes are each transferred to the next available preference after X stated on the ballot, skipping those that have already been elected or eliminated. In a manual count of paper ballots, this is the easiest method to implement; it is close to Thomas Hare's original proposal in 1857. It is used in all universal suffrage elections in the Republic of Ireland. This is analogous to what happens in the children-voting example above. Some people consider it fair in that, with 200 required for election, the group of 30 with first-preference Y get to influence other preferences, whereas the group of only 190 with first-preference X should just be satisfied to get their candidate elected. But some other people feel the group of 190 should get more influence on other preferences (as in Meek's method below). Also, exhausted ballots are excluded, so if more than 10 of the 30 votes have no preference stated after X, then it is impossible to select 20 to transfer and so some votes must be wasted.

Cincinnati method

20 votes are drawn at random from all 220 votes. This is used in Cambridge, Massachusetts (where every 11th vote ( $\textstyle {\frac {220-200}{220}}={\frac {1}{11}}$ surplus) would be selected for transfer). It is more likely than the Hare method to be representative, and less likely to suffer from too many exhausted ballots. However, the random element is still present. This may tip the balance in close elections; also, if a recount of the votes is required, it must be ensured that the same sample is used in the recount (i.e. the recount must only be to check for mistakes in the original count, not to try another random lottery selection of votes).

If a candidate exceeds the quota on the first count (i.e. purely with first preference votes), the Hare method and the Cincinnati method have the same effect for that candidate, since all the candidate's votes are in the "last batch received" from which the Hare surplus is drawn.

Hare-Clark method

The original Hare-Clark method used random choices. This non-random modern version is based on the way it is now done in public elections in Australia. It presumes that, for instance, voters who are surplus to their first choice and not needed to elect their second choice can give their votes to their third choice. Here Q is the quota required for election (normally the Droop quota). We have one pile of bags of ballots for each candidate, initially empty.

First, separate all ballots into bags according to their first choices and put each bag on its first choice's pile. Write on each bag its voteValue (the number of ballots in the bag). Then Process A is as follows:

1. Every candidate whose total voteValue is at least Q is declared elected. Record each such candidate's surplusVote as that total minus Q.
2. If no candidate has a positive surplusVote, Process A is finished.
3. Otherwise pick the elected candidate with the largest surplusVote to process next. We will move extra vote values to hopefuls only, as follows:
   1. Separate the ballots from the top bag of that candidate's pile into new bags according to the first hopeful listed, and put each new bag on top of its first hopeful's pile.
   2. Put the ballots that have no hopefuls listed (if any) back in the original bag on the elected candidate's pile. Subtract surplusVote from that bag's voteValue.
   3. Calculate a fraction $\textstyle {\text{transferValue}}={\frac {\text{surplusVote}}{\text{number of ballots in all new bags}}}$, but just use 1 if the number of such ballots is less than the surplusVote.
   4. Write on each new bag its ${\text{voteValue}}={\text{transferValue}}\cdot {\text{number of ballots it contains}}$, dropping the fractional part of the result.
4. Repeat this Process A from Step 1.

Example: If the quota Q is 200 and someone has 272 first-choice votes, of which 92 have no other hopeful listed, the surplusVote is 72 and so the transferValue is 72/180 = 40%. If 75 of the remaining 180 ballots have candidate X as the second-choice, and if X has only 190 votes, then the bag of 75 ballots is transferred to X's pile with a voteValue of 30 (40% of 75). So X is declared elected with a surplusVote of 20, and we will next separate those 75 ballots into smaller bags with their voteValues totalling 20 (or less, depending on what fractional parts are dropped).

Process B (if not enough have been elected by Process A): Eliminate the hopeful candidate with the lowest total of voteValues and redo the entire Hare-Clark method from the beginning, except ignore all mention of the eliminated candidate on any ballot.

Actually, the Australian method acts on the eliminated candidate's votes similar to what is described in Step 3 for 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. Note: Step 3 imposes an arbitrary order on simultaneous winners, to simplify hand-counting; if done by computer, it would be better to handle those surpluses simultaneously.

Gregory method

Another method is known variously as the 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). This eliminates all randomness. Instead of transferring a fraction of the votes at full value, one transfers all the votes at a fractional value. If we assume there are 40 ballots expressing no preference after X, the relevant fraction $\textstyle {\frac {20}{220-40}}={\frac {1}{9}}$ in the example. Note that part of X's 220 vote total may already be composed of fractions 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 $\textstyle {\frac {1}{5}}$ of a vote each. In this case, these 150 ballots would now be retransferred with a compounded fractional value of $\textstyle {\frac {1}{5}}\times {\frac {1}{9}}={\frac {1}{45}}$. In practice, the transferred value of a ballot is usually expressed not as a fraction but as a decimal, rounded to 2 or 3 places. To simplify tallying, the initial votes may be given a nominal value of 100 or 1000 to remove the decimal point.

Calculating compound fractions is labour-intensive, so in the Republic of Ireland the method is used only for the Senate whose franchise is restricted to c. 1500 councillors and members of Parliament. However, in Northern Ireland the method has been used for all STV public elections since 1973, 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 the application of the Gregory method in calculating the Surplus Transfer Value applied to each vote is

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

Subsequent surplus

All the above methods apply only to the transfer of an initial surplus, when a previously unelected candidate exceeds the quota of votes for the first time during the count. Suppose a ballot is to be transferred and the next stated preference is for an already-elected candidate. The common practice has been simply to skip over this preference and transfer the ballot instead to the next unelected (and uneliminated) candidate. This is what the Hare method and the Cincinnati method do.

It is possible to apply another method. Suppose we choose to have a previously-elected candidate X receive 20 transfers from a newly-elected candidate Y in addition to the quota of 200 previously retained: we could mix all of X's 220 votes and select 20 at random for transfer from X. However, the problem with this is that some of these 20 ballots may transfer back from X to Y, creating recursion. This is messy; in the case of the Senatorial rules, since all votes are transferred at all stages, it will be an infinite recursion, with ever-decreasing fractions applicable.

Meek's method

In 1969, B.L. Meek devised an algorithm based on Senatorial rules, which uses an iterative approximation to short-circuit this infinite recursion. It requires computer counting. This system is currently used for some local elections in New Zealand.

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	$w_{\text{new}}=w_{\text{old}}\times {\frac {\mathrm {Quota} }{\mathrm {Candidate's\ votes} }}$ which is repeated until $\mathrm {Candidate's\ votes} =\mathrm {Quota}$ 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 a proportion of the value of the preferences allocated to them, which proportion of is the value of their weighting; the remainder of the value of the vote is passed along fractionally to subsequent preferences depending on their weighting, with the formula

$1-{\text{nth Weighting}}$

being carried out at each preference.

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, hence 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 the Droop quota. This has the effect of also altering the weighting for each candidate.

This process continues until all the Elected candidates' vote values almost equal the quota (within a very close range, i.e. between 0.99999 and 1.00001 of a quota).

Warren's method

In 1994, C.H.E. Warren proposed another method of passing on subsequent surplus to previously-elected candidates. Warren's method is essentially identical to Meek's except in the amounts of votes retained by previously-elected candidates. Under Warren's method, 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).

It is important to note that, because the candidates receive different values of votes, the weightings determined by Warren's method will in general be different than the weightings determined by Meek's method.

Under Warren's method, every voter contributing to the election of a candidate contributes, as far as he or she is able, the same portion of his or her vote as every other such voter.

The Wright System

The Wright System - Count Process Flow Chart

In 2008, concerned about the distortion and lack of proportionality in the current Australian proportional counting systems, Systems Analyst and programmer Anthony van der Craats proposed to the Victorian and Australian Parliaments the adoption of the Wright system (named after Jack Wright, author of the book Mirror of a Nation's Mind and past president of the Proportional Representation Society of Australia) as an alternative method of counting the vote.

The Wright system is a refinement of the Australian Senate system replacing the method of distribution and segmentation of preferences with a reiterative linear counting system were the count is reset and restarted on every exclusion.

The Wright System fulfills the two principles identified by Brian Meek

• Principle 1. If a candidate is eliminated, all ballots are treated as if that candidate had never stood.

• Principle 2. If a candidate has achieved the quota, he retains a fixed proportion of every vote received, and transfers the remainder to the next non-eliminated candidate, the retained total equalling the quota.

The system uses the Droop Quota (the integer value of the total number of votes divided by one more than the number of vacant positions plus one) and the Gregory method of weighted surplus transfer value of the vote in calculating a candidate's surplus transfer value which is then multiplied by the value of the each vote received by the candidates whose votes are to be redistributed, as is the case in the Western Australian upper-house elections.

Unlike the Western Australian upper-house electoral system the Wright System proposes a reiterative counting process that differs from the Meek's method as an alternative to the method of segmentation and distribution of excluded candidates' votes.

On every exclusion of a candidate from the count, the counting of the ballot is reset and all valid votes are redistributed to candidates remaining in the count.

In each iteration of the count, votes are first distributed according to the voter's nominated first available preference, with each vote assigned a value of one and the total number of votes tabulated for each candidate and the quota calculated on the value of the total number of valid votes using the Droop quota method.

Any candidate that has total value equal or greater than the quota is provisionally declared elected and their surplus value distributed according to the voter's nominated subsequent preference. If the number of vacancies are filled on the first distribution, the results of the election are declared with all provisionally declared candidates being declared the winner of the election.

If the number of candidates provisionally declared elected is less than the number of vacancies and all candidates' surplus votes have been distributed then the candidate with the lowest value of votes is excluded from the count. The ballot is reset and the process of redistribution restarted with ballot papers being redistributed again according to the voters next available preference allocated to any continuing candidate. This process repeats itself until all vacancies are filled in a single count without the need for any further exclusions.

The Wright System takes into account optional preferential voting in that any votes that do not express a valid preference for a continuing candidate are set aside without-value and the quota is recalculated on each iteration of the count following the distribution of the first available preference. Votes that exhaust as a result of a candidate's surplus transfer are set aside with the value associated with the transfer in which they exhausted.

The main advantage of the Wright System is that is does away with the distortion and bias in the vote that arises from the adopted methods of segmentation and distribution of preferences of excluded candidates. Each vote has proportionally equal weight and is treated in the same manner as every other vote.

Under the current system used in the Australian Senate a voter whose first preference is for a minor candidate and their subsequent second preference for a major candidate that has been declared elected earlier in the count, is denied the opportunity to have their second preference vote allocated to the candidate of their choice. With the reiterative counting system the voter's second preference forms part of the voter's alternative chosen candidate's surplus and is redistributed according to the voter's nominated preference allocation.

Distribution of excluded candidate preferences

The choice of method used in determining the order of exclusion and distribution of a candidates' votes is crucial to the out come of a STV count.

There are a number of methods commonly used in determining the order polyexclusion and distribution of preferences from an excluded candidate. Most of the systems in use (with the exception of a reiterative count) were designed to facilitate a manual counting process and each one in turn affects the outcome of the election.

The general rule and principle that applies to each method is to exclude the candidate that has the lowest vote (Score). If more than one candidate has the same value or number of votes then a means of resolving which candidates are to be excluded needs to be determined. In a tie vote situation the decision can be made by examining the previous score and excluding the candidate that had the lowest previous value, if this is not possible to decide then the exclusion is generally chosen by random lot.

Types of Exclusion methods commonly in use:

• Single transaction
• Segmented distribution
  • Value based segmentation
  • Aggregated primary vote and value segmentation
  • FIFO (First In First Out - Last bundle)
• Reiterative count

Single transaction

A single transaction is when all votes allocated to a candidate to be excluded from the count are transferred in a single transaction without any segmentation.

Segmentation

A segmented distribution is broken down into smaller segmented transactions with each segment being considered a complete transaction at the conclusion of which assessment is made to determine if a candidate has been elected. The choice of segmentation can have a significant impact in the calculation of a candidate's Surplus Transfer Value and as such the outcome of the election. The general rule being the smaller the parcel of and value of votes the less likely that the distortion in the count will effect the over all result.

Aggregated primary vote and value based segmentation

Segmentation is generally based around the value of the vote with all votes that have the same value being transferred collectively as one single transaction. Some methods of segmentation separate the Primary vote (Full-value votes) in order to limited the distortion that occurs in the process of a segmented distribution and as means of trying to limit the value of a Surplus Transfer Value of any candidate elected as result of a segmented transfer.

FIFO (First In First Out)

FIFO is method of segmentation where each parcel of votes is distributed in the order in which they were received. This method produces the smallest number and size of each segmentation but in the process increases the number of steps required to complete a count.

Reiterative count

An alternative to the segmented distribution methods is to undertake a reiterative count where on the exclusion of any candidate, the count is reset and restarted from the beginning with the vote reallocated to the next available candidate in order of the voters nominated preference. A reiterative count treats each ballot paper in the same way as though the candidate excluded from the count did not stand. This allows for votes to be allocated to candidates that may have been declared elected using a segmented distribution process. In a reiterative count, votes that form part of a candidates surplus are distributed only within each iteration of a count. A reiterative count is best suited for a computerised counting system as the potential number of distributions and time required for each iteration can be considerable. The number of iterations of the count can be limited by applying a method of Bulk Exclusion.

Bulk exclusions

In order to reduce the number of steps required within a count it is possible to apply Bulk Exclusion rules to speed up the counting process. Bulk exclusion requires the calculation of Breakpoints.

There are four types of Breakpoints in an STV count.

• Quota Breakpoint
• Running Breakpoint
• Group Breakpoint
• Applied Breakpoint

Any candidates with a total vote (Score) 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 contest candidate's score and the nominated quota.

To determine a Breakpoint:

List in descending order candidates' total value of votes (Score) and calculate the Running Sum value of all candidates' votes that are less than the associated candidates total vote (Score).

Quota Breakpoint

A Quota Breakpoint is the highest Running Sum value that is less than half of the Quota

Running Breakpoint

A Running Breakpoint is the highest Candidate's total vote (Score) that is less than the associated Running Sum value.

Group Breakpoint

A Group Break Point is the highest Candidate's total vote (Score) in a Group that is less than the associated Running Sum of Group candidates whose total vote (Score) is less than the associated Candidate's total vote. (This only applies where there is defined groups of candidates such as in Australian public elections which use an Above-the-line group voting method.)

Applied Breakpoint

An Applied Breakpoint is the highest Running Sum that is less than the difference between the highest Candidate's total vote (score) and the quota. (i.e. the total value of all candidates votes below can not effect the result of the election) All candidates above an Applied Breakpoint continue to contest the election.

Careful consideration is required when applying Quota Breakpoints or Group Breakpoints. Quota Breakpoints may not apply with Optional Preferential ballots or if there is more than one position not yet filled in the count. At no time should a candidate above the Applied Breakpoint 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	539459	Contest
WATERS, Larissa	O-3	ALP	The Greens	254971	431482	Contest
FURNER, Mark	M-1	GRN	Australian Labor Party	176511	278103	Contest
HANSON, Pauline	R-1	HAN	Pauline	101592	154430	Contest
BUCHANAN, Jeff	H-1	FFP	Family First	52838	98233	Contest
BARTLETT, Andrew	I-1	DEM	Democrats	45395	65672	Contest
SMITH, Bob	G-1	AFLP	The Fishing Party	20277	39358	Quota Breakpoint
COLLINS, Kevin	P-1	FP	Australian Fishing and Lifestyle Party	19081	36364	Contest
BOUSFIELD, Anne	A-1	WWW	What Women Want (Australia)	17283	30140	Contest
FEENEY, Paul Joseph	L-1	ASP	The Australian Shooters Party	12857	21559	Contest
JOHNSON, Phil	C-1	CCC	Climate Change Coalition	8702	15957	Contest
JACKSON, Noel	V-1	DLP	D.L.P. - Democratic Labor Party	7255	49932	Applied Breakpoint
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

An example of a STV count

Suppose we conduct an STV election using the Droop quota where there are two seats to be filled and four candidates: Andrea, Brad, Carter, and Delilah. Also suppose that there are 57 voters who cast their 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 threshold 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 excess votes. Her 20 excess votes are reallocated to their second preferences. For example, 12 of the reallocated votes go to Carter, 8 to Brad.

As none of the remaining candidates have reached the quota, Brad, the candidate with the fewest votes, is excluded from the count. All of his votes have Carter as the next-place choice, and are reallocated to Carter. This gives Carter 20 votes and he is elected, filling 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

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.

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.