There are many ways to count votes in Single Transferable Vote elections.
Contents 
If a class of children were choosing representatives, say, they could lineup 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 
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 wholenumber 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.
The most common formula for the quota used now is the Droop quota which is most often given as:
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.
When Thomas Hare originally conceived his version of Single Transferable Vote, he envisioned using the quota:
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 noncontinuing 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.
The singletransferablevote 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 votegetter). Adjust for those freedup 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.
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 DugganSchwartz 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.
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 ballotpapers 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 nth ballot paper is selected, where is the fraction to be selected.
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?
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 childrenvoting example above. Some people consider it fair in that, with 200 required for election, the group of 230 with firstpreference Y get to influence other preferences, whereas the group of only 190 with firstpreference 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.
20 votes are drawn at random from all 220 votes. This is used in Cambridge, Massachusetts (where every 11th vote ( (220200)/220 = 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.
The original HareClark method used random choices. This nonrandom 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:
Example: If the quota Q is 200 and someone has 272 firstchoice 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 secondchoice, 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 HareClark 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 handcounting; if done by computer, it would be better to handle those surpluses simultaneously.
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 20 over 220  40, i.e. 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 of a vote each. In this case, these 150 ballots would now be retransferred with a compounded fractional value of . 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 labourintensive, 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 8seat district council elections), and up to 700,000 votes counted (in 3seat 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
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 alreadyelected 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 previouslyelected candidate X receive 20 transfers from a newlyelected 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 everdecreasing fractions applicable.
In 1969, B.L. Meek devised an algorithm based on Senatorial rules, which uses an iterative approximation to shortcircuit 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 
which is repeated until 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
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 (1a)b, and C will retain (1a)(1b)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 midprocess. The quota is found by
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).^{[1]}
In 1994, C.H.E. Warren proposed another method of passing on subsequent surplus to previouslyelected candidates.^{[2]} Warren's method is essentially identical to Meek's except in the amounts of votes retained by previouslyelected 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 (1a) if (1a)<b), and C will retain c (or (1ab) if (1ab)<c — or 0 if (1ab) 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.^{[3]}
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.^{[4]}^{[5]}^{[6]}
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 ^{[7]}
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 upperhouse elections.^{[8]}
Unlike the Western Australian upperhouse 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 withoutvalue 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.
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:
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.
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.
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 (Fullvalue 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 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.^{[9]}
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.
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.
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).
A Quota Breakpoint is the highest Running Sum value that is less than half of the Quota
A Running Breakpoint is the highest Candidate's total vote (Score) that is less than the associated Running Sum value.
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 Abovetheline group voting method.)
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).
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  J1  LNP  Liberal  345559  Quota  
HOGG, John Joseph  O1  ALP  Australian Labor Party  345559  Quota  
BOYCE, Sue  J2  LNP  Liberal  345559  Quota  
MOORE, Claire  O2  ALP  Australian Labor Party  345559  Quota  
BOSWELL, Ron  J3  LNP  Liberal  284488  539459  Contest 
WATERS, Larissa  O3  ALP  The Greens  254971  431482  Contest 
FURNER, Mark  M1  GRN  Australian Labor Party  176511  278103  Contest 
HANSON, Pauline  R1  HAN  Pauline  101592  154430  Contest 
BUCHANAN, Jeff  H1  FFP  Family First  52838  98233  Contest 
BARTLETT, Andrew  I1  DEM  Democrats  45395  65672  Contest 
SMITH, Bob  G1  AFLP  The Fishing Party  20277  39358  Quota Breakpoint 
COLLINS, Kevin  P1  FP  Australian Fishing and Lifestyle Party  19081  36364  Contest 
BOUSFIELD, Anne  A1  WWW  What Women Want (Australia)  17283  30140  Contest 
FEENEY, Paul Joseph  L1  ASP  The Australian Shooters Party  12857  21559  Contest 
JOHNSON, Phil  C1  CCC  Climate Change Coalition  8702  15957  Contest 
JACKSON, Noel  V1  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  D1  ALP  Australian Labor Party  46693  Quota  
COLBECK, Richard M  F1  LP  Liberal  46693  Quota  
BROWN, Bob  B1  GRN  The Greens  46693  Quota  
BROWN, Carol  D2  ALP  Australian Labor Party  46693  Quota  
BUSHBY, David  F2  LP  Liberal  46693  Quota  
BILYK, Catryna  D3  ALP  Australian Labor Party  37189  Contest  
MORRIS, Don  F3  LP  Liberal  28586  Contest  
WILKIE, Andrew  B2  GRN  The Greens  12193  27607  Running Breakpoint 
PETRUSMA, Jacquie  K1  FFP  Family First  6471  15414  Quota Breakpoint 
CASHION, Debra  A1  WWW  What Women Want (Australia)  2487  8943  Applied Breakpoint 
CREA, Pat  E1  DLP  D.L.P.  Democratic Labor Party  2027  6457  
OTTAVI, Dino  G1  UN3  1347  4430  
MARTIN, Steve  C1  UN1  848  3083  
HOUGHTON, Sophie Louise  B3  GRN  The Greens  353  2236  
LARNER, Caroline  J1  CEC  Citizens Electoral Council  311  1883  
IRELAND, Bede  I1  LDP  LDP  298  1573  
DOYLE, Robyn  H1  UN2  245  1275  
BENNETT, Andrew  K2  FFP  Family First  174  1030  
ROBERTS, Betty  K3  FFP  Family First  158  856  
JORDAN, Scott  B4  GRN  The Greens  139  698  
GLEESON, Belinda  A2  WWW  What Women Want (Australia)  135  558  
SHACKCLOTH, Joan  E2  DLP  D.L.P.  Democratic Labor Party  116  423  
SMALLBANE, Chris  G3  UN3  102  307  
COOK, Mick  G2  UN3  74  205  
HAMMOND, David  H2  UN2  53  132  
NELSON, Karley  C2  UN1  35  79  
PHIBBS, Michael  J2  CEC  Citizens Electoral Council  23  44  
HAMILTON, Luke  I2  LDP  LDP  21  21  
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:
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 nextplace 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 
