Counting single transferable votes
The single transferable vote 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 by reaching quota or eliminated, surplus votes are transferred from winners to remaining candidates 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 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 is the number of votes a candidate must receive to be elected. The Hare quota and the Droop quota are commonly used to determine the quota.Hare quota
When Thomas Hare originally conceived his version of Single Transferable Vote, he envisioned using the quota:In the unlikely event that each successful candidate receives exactly the same number of votes, not enough candidates can meet the quota and fill the available seats in one count. Thus the last candidate cannot not meet the quota, and it may be fairer to eliminate that candidate.
To avoid this situation, it is common instead to use the Droop quota, which is always lower than the Hare quota.
Droop quota
The most common quota formula is the Droop quota, which given as:Droop produces a lower quota than Hare. If each ballot has a full list of preferences, Droop guarantees that every winner meets the quota rather than being elected as the last remaining candidate after lower candidates are eliminated. The fractional part of the resulting number, if any, is dropped
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 majority of the voters can be guaranteed to elect a majority of the seats when there is an odd number of seats; 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 and p is a number of second preferences for the given candidate. Meek's counting method recomputes the quota on each iteration of the count.
Example
Two seats need to be filled among four candidates: Andrea, Brad, Carter, and Delilah. 57 voters cast ballots with the following preference orderings:The quota is calculated as.
In the first round, Andrea receives 40 votes and Delilah 17. Andrea is elected with 20 surplus votes. Ignoring how the votes are valued for this example, 20 votes are reallocated according to their second preferences. 12 of the reallocated votes go to Carter, 8 to Brad.
As none of the hopefuls have reached the quota, Brad, the candidate with the fewest votes, is excluded. All of his votes have Carter as the next-place choice, and are reallocated to Carter. This gives Carter 20 votes and he fills the second seat.
Thus:
Counting rules
Under the single transferable vote system, votes are successively transferred to hopefuls from two sources:- Surplus votes of elected candidates.
- All votes of eliminated candidates.
- Compute the quota.
- Assign votes to candidates by first preferences.
- Declare as winners all candidates who received at least the quota.
- Transfer the excess votes from winners to hopefuls.
- Repeat 3–4 until no new candidates are elected.
- 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.
- Transfer the votes of the losers to remaining hopeful candidates.
- Repeat 3–7 until all seats are full.
Surplus allocation
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 th ballot is selected, where 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 elections in the Republic of Ireland to Dáil Éireann, to local government, to the European Parliament, and to the university constituencies in Seanad Éireann. 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.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 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 nominated.For each successful candidate that exceeds the quota threshold, calculate the ratio of that candidate's surplus votes 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
The UK's Electoral Reform Society recommends essentially this method. Every preference continues to count until the choices on that ballot have been exhausted or the election is complete. Its main disadvantage is that given large numbers of votes, candidates and/or seats, counting is administratively burdensome for a manual count due to the number of interactions. This is not the case with the use of computerised distribution of preference votes.
From May 2011 to June 2011, The Proportional Representation Society of Australia reviewed the Wright System noting:
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.- Separate all ballots according to their first preferences.
- Count the votes.
- Declare as winners those hopefuls whose total is at least Q.
- For each winner, compute surplus as total minus Q.
- For each winner, in order of descending surplus:
- # Assign that candidate's ballots to hopefuls according to each ballot's preference, setting aside exhausted ballots.
- # Calculate the ratio of surplus to the number of reassigned ballots or 1 if the number of such ballots is less than surplus.
- # For each hopeful, multiply ratio * the number of that hopeful's reassigned votes and add the result to the hopeful's tally.
- Repeat 3–5 until winners fill all seats, or all ballots are exhausted.
- 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.
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, or the Gregory method 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. 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. In this case, these 150 ballots would now be retransferred with a compounded fractional value of.
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. In Northern Ireland, the Gregory Method has been used since 1973 for all STV elections, with up to 7 fractional transfers, and up to 700,000 votes counted.
An alternative means of expressing Gregory in calculating the Surplus Transfer Value applied to each vote is
The Unweighted Inclusive Gregory Method is used for the Australian Senate.
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.Or 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, e.g. Stack Exchange Network portals.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 | |
| Excluded | |
| 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 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:
For example, consider a ballot with top preferences A, B, C, where the weightings of the three candidates are,, respectively. From this ballot A will retain, B will retain, and C will retain.
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
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.
Warren
In 1994, C. H. E. Warren proposed another method of passing surplus to previously-elected candidates. 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, where the weightings are a, b, and c. Under Warren's method, A will retain a, B will retain b if, and C will retain c if <c — or 0 if.
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.
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 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 to reduce distortion and limit the subsequent value of a transfer from a candidate elected as result of a segmented transfer.
- * FIFO —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.
- 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
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.
- Applied Breakpoint—The highest running total that is less than the difference between the highest candidate's tally and the quota. All candidates above an applied breakpoint continue in the next iteration.
Example
Quota breakpoint| 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 | 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
| 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 | 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 | 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 |