- Counting Single Transferable Votes
The single transferable vote (STV) is a voting system based on proportional representation and preferential voting. Under STV, an elector's vote is initially allocated to his or her most preferred candidate. After candidates have been either elected or eliminated, any surplus or unused votes are transferred to another according to the voter's stated 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) can be used to carry out these transfers. The votes an elected candidate receives in excess of the quota constitute a surplus.
When using an STV ballot, the voter ranks the candidates on the ballot. For example:
2 Andrea 1 Maskill 4 Brad 3 Delilah
When Thomas Hare originally conceived his version of Single Transferable Vote, he envisioned using the quota:
The Hare Quota
Except in the unlikely event that each successful candidate receives exactly the same number of votes) there will not be enough candidates who meet the quota to fill the available seats. Thus the last candidate to be elected 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.
The most common formula for the quota is the Droop quota which is most often given as:
The Droop Quota
This produces a lower quota than the Hare quota. If each voter expresses a full list of preferences, the Droop quota guarantees that every candidate elected 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.
The Droop Quota 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 successful candidate's surplus votes transfer to other candidates according to the later preferences. In Meek's method, the quota must be recalculated throughout the count.
Counting the votes
Under the single transferable vote system, votes are successively transferred to the remaining candidates (i.e. those who have neither been elected yet nor eliminated yet) 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 one system is best, and the choice of exact method may affect the results.
Step 1: Compute the quota.
Step 2: Assign votes to candidates by first preferences.
Step 3: Declare as elected all candidates who received at least the quota.
Step 4: Transfer the excess votes from winners to remaining candidates.
Step 5: Repeat 2-4 until no new candidates are elected. (Under some systems, votes could initially be transferred in this step to candidates who have already been elected or eliminated. This might make a difference to the result.)
If all seats have winners, the process is complete. Otherwise:
Step 6: Eliminate one or more candidates. Typically either the lowest candidate or all candidates whose combined votes are less than the vote of the lowest remaining candidate.
Step 7: Transfer the votes of the eliminated candidates to continuing candidates.
Step 8: Repeat 2-7 until all seats are full.
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, who receives the votes and the order in which excesses 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.
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 is the fraction to be selected.
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.
Reallocation ballots are drawn at random from all of the candidate's votes. This method is used in Cambridge, Massachusetts (where every 11th vote is selected for transfer). This method is more likely than the Hare method to be representative, and less likely to suffer from exhausted ballots. The starting point for counting to eleven is arbitrary. Under a recount the same sample 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).
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.
This is a simplified explanation of the transfer method described under Wright System below.
For each successful candidate, set aside ballots with no remaining preferences. 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 preference who is still in contention. 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 transfer.)
This is essentially the method recommended by the Electoral Reform Society in the UK. Every 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 or votes, candidates and/or seats, counting is administratively burdensome.
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.
This is a variation on the original Hare method which used random choices. It is used in elections in Australia. It allows votes to transferred repeated. In the following explanation, Q is the quota required for election. Winners are candidates whose total reaches Q. Losers are those who haven't won after all the winners have been chosen. Hopefuls are those whose status is not final.
- Separate all ballots according to their first choices.
- 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, ignoring ballots with no remaining hopefuls.
- 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 (rounded down) to the hopeful's tally.
- Repeat 3-5 until winners fill all seats, or all ballots have no remaining hopefuls.
- 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 the quota 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.
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 in the example. 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 of a vote each. In this case, these 150 ballots would now be retransferred with a compounded fractional value of .
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 the application of the Gregory method in calculating the Surplus Transfer Value applied to each vote is
Secondary preferences for prior winners
Suppose a ballot is to be transferred and its next preference is for a winner in a prior round. The Hare method and the Cincinnati method 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. This is messy; in the case of the Senatorial rules, since all votes are transferred at all stages, the recursion is infinite, with ever-decreasing fractions.
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
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
- 1 − 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
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).
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
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 where 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 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 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 system used in the Australian Senate a voter whose first preference is for a minor candidate and whose second preference is for a major candidate who 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 can affect the outcome 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
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.
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.
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.
- 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).
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 Above-the-line 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 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 an 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 .
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.
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
- 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.
- ^ Single Transferable Vote Rules UK Electoral Reform Society
- ^ 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. http://www.dia.govt.nz/diawebsite.NSF/Files/meekm/$file/meekm.pdf.
- ^ Warren, C.H.E., "Counting in STV Elections", Voting matters 1 (1994), paper 4.
- ^ Hill, I.D. and C.H.E. Warren, "Meek versus Warren", Voting matters 20 (2005), pp 1-5.
- ^ van der Craats, Anthony. "The Wright System - Submission for reform to the Australian Parliament Joint Standing Committee on Electoral Matters". JSCEM. http://www.aph.gov.au/house/committee/em/elect07/subs/sub051.pdf.
- ^ van der Craats, Anthony. "Rules and procedures for a Reiterative Proportional Representation Single Transferable Vote counting system". JSCEM. http://www.aph.gov.au/house/committee/em/elect07/subs/sub051.1.pdf.
- ^ van der Craats, Anthony. "One Vote One Value - Change that Counts". JSCEM. http://www.aph.gov.au/house/committee/em/elect07/subs/sub051.2.pdf.
- ^ Meek, Brian. "A New Approach to the Single Transferable Vote". Voting matters. http://www.votingmatters.org.uk/ISSUE1/P1.HTM.
- ^ "Electoral Act 1907". Western Australia Legislation. http://www.slp.wa.gov.au/legislation/statutes.nsf/main_mrtitle_290_homepage.html.
- ^ "Rules of the Proportional Representation Society of Australia for conducting election by Quota-Prefernetial". http://www.prsa.org.au/rule1977.htm.
- 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.
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