Counting single transferable votes

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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, provides approximately proportional representation, and enables votes to be explicitly cast for individual candidates rather than for closed party lists. A variety of algorithms (methods) carry out these transfers.

Voting[edit]

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

2 Andrea
1 Carter
4 Brad
3 Delilah

Quota[edit]

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[edit]

Main article: 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[edit]

Main article: 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[edit]

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[edit]

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 successful 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 2-4 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:

  1. 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.
  2. Transfer the votes of the losers to continuing candidates are declared to be losers.
  3. Repeat 2-7 until all seats are full.

Surplus allocation[edit]

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[edit]

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 nth ballot is selected, where \begin{matrix} \frac {1} {n} \end{matrix} is the fraction to be selected.

Hare[edit]

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[edit]

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's[edit]

This is a simplified explanation of the Wright System below.

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[edit]

This is a variation on the original Hare method which 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[edit]

Another method, known as Senatorial rules (after its use for most seats in Irish Senate elections), or 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 and members of Parliament. 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[edit]

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[edit]

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.

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%).[2]

Warren[edit]

In 1994, C.H.E. Warren proposed another method of passing surplus to previously-elected candidates.[3] 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.[4]

Wright[edit]

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 JFH "Jack" Wright, author of the book Mirror of the Nation's Mind and past president of the Proportional Representation Society of Australia) as an alternative counting technique.[5][6][7]

Wright is a refinement of the Australian Senate system that replaces the method of distribution and segmentation of preferences with a reiterative linear counting system where the count is reset and restarted each time a candidate is excluded from the count..

Wright fulfills the two principles identified by Meek[8]

  • Once a candidate is eliminated from the count, all ballots are treated as if that candidate had never stood.
  • Winners retain a fixed proportion of every vote received, and transfer the remainder to the next nominated preference for a continuing candidate.

Wright adopts Droop Quota (the integer value of the total number of votes divided by the number of vacant positions plus one) and the Gregory method of weighted surplus transfer. The transfer value is then multiplied by the value of each vote received by the candidates whose votes are to be redistributed, as is the case in the Western Australian upper-house elections.[9]

Wright proposes an iterative counting process that differs from Meek in method of segmentation and distribution of losers' votes.

On every exclusion of a candidate from the count, the count is reset and all valid votes are redistributed to continuing 'hopeful' candidates.

In each iteration votes are first distributed according to each ballot's next 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 reaches quota is provisionally declared elected and those ballots' surplus values are distributed according to the ballots' subsequent preference. If all seats are filled in the first count, the count is over.

If vacancies remain after all surplus votes are distributed then the candidate with the lowest tally becomes a loser. Distribution restarts following the next available preference allocated to a hopeful. This process repeats itself until all vacancies are filled in a single count without further exclusions.

Wright incorporates optional preferential voting in that any votes that do not express a valid preference for a continuing candidate are set aside and the quota is recalculated on each iteration of the count following the distribution of surplus ballots. Ballots that exhaust as a result of surplus transfer are set aside along with the value associated with the transfer in which they exhausted.

Wright's main advantage is that each vote has proportionally equal weight and is treated in the same manner as every other vote.

Under the Australian Senate system a ballot whose first preference is for a minor candidate and whose second preference is for a major candidate who won in the first count is set aside for subsequent counts. Under Wright the voter's second preference is counted and is redistributed.

Distribution of excluded candidate preferences[edit]

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.[10]
  • 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[edit]

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[edit]

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[edit]

References[edit]

  1. ^ Single Transferable Vote Rules UK Electoral Reform Society
  2. ^ Hill, I. David; B. A. Wichmann and D. R. Woodall (1987). "Algorithm 123 — Single Transferable Vote by Meek's Method". The Computer Journal 30 (2): 277–281. doi:10.1093/comjnl/30.3.277. 
  3. ^ Warren, C.H.E., "Counting in STV Elections", Voting matters 1 (1994), paper 4.
  4. ^ Hill, I.D. and C.H.E. Warren, "Meek versus Warren", Voting matters 20 (2005), pp 1-5.
  5. ^ van der Craats, Anthony. "The Wright System - Submission for reform to the Australian Parliament Joint Standing Committee on Electoral Matters". JSCEM. 
  6. ^ van der Craats, Anthony. "Rules and procedures for a Reiterative Proportional Representation Single Transferable Vote counting system". JSCEM. 
  7. ^ van der Craats, Anthony. "One Vote One Value - Change that Counts". JSCEM. 
  8. ^ Meek, Brian. "A New Approach to the Single Transferable Vote". Voting matters. 
  9. ^ "Electoral Act 1907". Western Australia Legislation. 
  10. ^ "Rules of the Proportional Representation Society of Australia for conducting election by Quota-Preferential". 

External links[edit]