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57-505: The system attempts to ensure political parties are represented proportionally without official party lists by having each winner elected with the same or about the same number of votes. There are several variants of the Single Transferable Vote, each having different properties. When using an STV ballot, the voter ranks candidates on the ballot. For example: Some, but not all single transferable vote systems require

114-423: A political spectrum , with factions on the far-left, center-left, center, center-right, and far-right. Then, the three moderate groups will not form a solid coalition, because some members of the center-right may not rank the center-left candidate above the far-right candidate. In the following let n {\displaystyle n} be the number of voters, k {\displaystyle k} be

171-399: A ballot paper in which the voter is required to write numbers 1, 2, 3, etc. opposite the name of the candidate who is their first, second, third, etc. preference. In OPV and semi-optional systems, candidates not explicitly ranked by the voter are implicitly ranked lower than all numbered candidates. Some OPV jurisdictions permit a ballot expressing a single preference to use some other mark than

228-444: A coalition of voters must rank all candidates within the same party first before candidates of other parties. And PSC does not guarantee proportional representation if voters rank candidates of different parties together (as they will no longer form a solid coalition ). In party-list systems , proportional representation guarantees each party a number of representatives proportional to its number of votes. In systems without parties,

285-402: A complete party list prepared by one of the parties, instead of manually entering personal preferences marked for individual candidates "below the line". Ranked vote systems vary in that some cast aside, at the start of the vote count, a ballot not correctly filled out but other systems allow a vote even if not fully and correctly marked to be used until the first mistake annuls the ballot. That

342-469: A formula ( p/t)*s , 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. This is the whole-vote method used in Ireland and Malta national elections. Transfers are done using whole votes, with some of the votes that are directed to another candidate left behind with

399-595: A group consisting of at least half of all voters is guaranteed to win control of at least half of all seats, which is not always the case using the Hare quota. Until all seats have been filled, votes are successively transferred to one or more "hopeful" candidates (those who are not yet elected or eliminated) from two sources: (In either case, some votes may be non-transferable as they bear no marked back-up preferences for any non-elected, non-eliminated candidate.) The possible algorithms for doing this differ in detail, e.g., in

456-518: A lower quota, such as the Droop quota. The most common quota formula is the Droop quota , which is a number larger than this amount: Droop quota is a smaller number of votes than Hare. Because of this difference, under Droop it is more likely that every winner meets the quota rather than being elected as the last remaining candidate after lower candidates are eliminated. But in real-life elections, if it

513-619: A manual count of paper ballots, this is the easiest method to implement. Votes are transferred as whole votes. Fractional votes are not used. This system is close to Thomas Hare 's original 1857 proposal. It is used in elections in the Republic of Ireland to Dáil Éireann (the lower chamber), to local government , to the European Parliament , and to the university constituencies in Seanad Éireann (the upper chamber). This

570-433: A preference to be expressed for every candidate, or for the voter to express at least a minimum number of preferences. Others allow a voter just to mark one preference if that is the voter's desire. The vote will be used to elect just one candidate at the most, in the end. The quota (sometimes called the threshold) is the number of votes that guarantees election of a candidate. Some candidates may be elected without reaching

627-436: A system using the Hare quota and mandatory full marking of the ballot, with optional marking and Droop quota, a certain percentage of ballots are not used to elect anyone. As a result, it is said by some that the Droop quota tends to be strongly disproportional (and is in fact the most-biased quota possible). However, most STV systems used in the world use Droop as many say it is more fair to large parties than Hare. Under Droop,

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684-431: A task which is mathematically equivalent to establishing a share of surplus votes to be transferred to a hopeful candidate based on the overall vote for an eliminated candidate. Example: If the quota is 200 and a winner has 272 first-choice votes, then the surplus is 72 votes. If 92 of the winner's 272 votes have no other hopeful listed, then the remaining 180 votes have a second-choice selection and can be transferred. Of

741-443: A time, imposing a spurious ordering on the votes. To prevent all transferred ballots coming from the same precinct, every n {\displaystyle n} th ballot is selected, where 1 n {\displaystyle {\begin{matrix}{\frac {1}{n}}\end{matrix}}} is the fraction to be selected. The Wright system is a reiterative linear counting process where on each candidate's exclusion

798-409: A value of 1 5 {\displaystyle \textstyle {\frac {1}{5}}} . In this case, these 150 ballots would now be retransferred with a compounded fractional value of 1 5 × 4 10 = 4 50 {\displaystyle \textstyle {\frac {1}{5}}\times {\frac {4}{10}}={\frac {4}{50}}} . In the Republic of Ireland ,

855-455: Is a criterion of proportionality for ranked voting systems . It is an adaptation of the quota rule to voting systems in which there are no official party lists , and voters can directly support candidates. The criterion was first proposed by the British philosopher and logician Michael Dummett . PSC is a relatively minimal definition of proportionality. To be guaranteed representation,

912-582: Is administratively burdensome for a manual count due to the number of interactions. This is not the case with the use of computerized distribution of preference votes. From May 2011 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

969-412: Is allowed for valid ballots to not bear full rankings, it is common even under Droop for one or two candidates to be elected with partial quota at the end, as the field of candidates is thinned to the number of remaining open seats and as the valid votes still in play become scarcer. The use of Droop leaves a full quota's worth of votes held by unsuccessful candidates, which are effectively ignored. Unlike

1026-486: Is attained by the expanding approvals rule, but violated by the single transferable vote . It can be decided in polynomial time whether a given committee satisfies Rank-PJR+. Optional preferential voting One of the ways in which ranked voting systems vary is whether an individual vote must express a minimum number of preferences to avoid being considered invalid ("spoiled" or "informal" or "rejected"). Possibilities are: Ranked-voting systems typically use

1083-520: Is no requirement to complete the entire ballot paper (rank all the candidates). Elections for all other Australian lower houses use full-preferential voting. In the New South Wales Legislative Council , semi-optional preferential voting has been used since 1978, with a minimum 10 preferences required for 15 seats before 1991, and 15 preferences for 21 seats since. Voters also have the option since 1984 of voting "above

1140-592: Is not in C {\displaystyle C} . When a voter is part of a solid coalition that prefers some set of candidates, they are said to be "solidly supporting" or "solidly committed to" that set of candidates. Any voter who ranks a candidate as their first-preference solidly supports that candidate. Note that a solid coalition may be "nested" within another solid coalition, so there may be some faction of voters that can further be split into subfactions. However, solid coalitions cannot cut across different factions. For example, say voters are organized along

1197-414: Is sometimes described as "random" because it does not consider later back-up preferences but only the next usable one. Through random drawing of the votes to make up the transfer, statistically the transfers often reflect the make-up of the votes held by the successful candidate. Sometimes, ballots of the elected candidate are manually mixed. In Cambridge, Massachusetts , votes are counted one precinct at

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1254-412: Is that any solid coalition with a majority will always be able to elect at least half of seats. However, this comes at the cost of a substantial seat bias in favor of larger parties. This means a coalition of smaller parties who together win a majority of the vote can nevertheless fail to reach a majority in the legislature. Examples of quota-proportional methods include the expanding approvals rule ,

1311-523: Is used for the Australian Senate . The effect of the Gregory system can be replicated without using fractional values by a party-list proportional allocation method, such as D'Hondt , Webster/Sainte-Laguë or Hare-Niemeyer . A party-list proportional representation electoral system allocates a share of the seats in a legislature to a political party in proportion to its share of the votes,

1368-776: Is used in elections in the Australian state of New South Wales . It was used in Queensland's system of instant-runoff voting from 1992 to 2015, when it was replaced by full-preferential voting. OPV was adopted in the Northern Territory in 2016. In both the Tasmanian House of Assembly and the Tasmanian Legislative Council , semi-optional voting is used, with a minimum number of preferences required to be expressed. There

1425-502: Is used to transfer the value of surplus votes, denying the other voters who contributed to a candidate's success a say in the surplus distribution. In the following explanation, Q is the quota required for election. 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 57 2 + 1 = 19 {\displaystyle {57 \over 2+1}=19} . In

1482-782: The Northern Ireland constituency from 1979 to 2020). An alternative means of expressing Gregory in calculating the Surplus Transfer Value applied to each vote is surplus transfer value = ( total value of candidate's votes − quota total value of candidate's votes ) × value of each vote {\displaystyle {\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}}} The Unweighted Inclusive Gregory Method

1539-496: The method of equal shares , and the single transferable vote . Aziz and Lee define a property called generalized PSC, and another property, called inclusion PSC, that apply also to weak rankings (rankings with indifferences). Their expanding approvals rule satisfies these generalizations of PSC. Brill and Peters define a fairness property called Rank-PJR+, which also applies to weak rankings, but makes positive guarantees also to coalitions that are only partially solid. Rank-PJR+

1596-411: The 180 votes which can be transferred, 75 have hopeful X as their second-choice, 43 have hopeful Y as their second-choice, and 62 have hopeful Z as their second-choice. The D'Hondt system is applied to determine how the surplus votes would be transferred - successive quotients are calculated for each hopeful candidate, one surplus vote is transferred to the hopeful candidate with the largest quotient, and

1653-540: The Australian Capital Territory and in Tasmania. Under some systems, a fraction of the vote is transferred, with a fraction left behind with the winner. As all votes are transferred (but at fractional value), there is no randomness and exact reduction of the successful candidate's votes are guaranteed. However the fractions may be tedious to work with. If the transfer is of surplus received in

1710-662: 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 (the Irish Parliament). In Northern Ireland , the Gregory Method has been used since 1973 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 for

1767-422: The Hare quota, i.e. j {\displaystyle j} Droop quotas entitle a solid coalition to j {\displaystyle j} candidates. It is a generalization of the majority criterion in the sense that it relates to groups of supported candidates (solid coalitions) instead of just one candidate, and there may be more than one seat to be filled. An advantage of Droop proportionality

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1824-653: 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

1881-592: The Republic of Ireland , and in Northern Ireland use OPV. The ranked-choice voting system used in Maine , United States, Instant-runoff voting , can be considered optional-preferential as voters are allowed to rank just one candidate. The system also allows voters to skip one ranking (e.g. marking a first choice and a third choice, but not a second choice). In that case, the next ranking would be advanced to

1938-578: The STV system used in Cincinnati (1924-1957) and in Cambridge city elections, votes received by a winning candidate were numbered sequentially, then if the surplus votes made up one quarter of the votes held by the successful candidate, each vote that was numbered a multiple of four was extracted and moved to the next usable marked preference on each of those votes. (In British, Irish and Canadian uses of STV

1995-543: The Wright system. This variation is used in Tasmanian and ACT lower house elections in Australia. The Gregory method (transferring fractional votes) is used but the allocation of transfers is based just on the next usable preference marked on the votes of the last bundle transferred to the successful candidate. The last bundle transfer method has been criticized as being inherently flawed in that only one segment of votes

2052-421: The digit '1', such as a cross or tick-mark, opposite the preferred candidate's name, on the basis that the voter's intention is clear; other do not, arguing for example that an 'X' might be an expression of dislike. FPV may not be possible if write-in candidates are allowed. In a transferable-vote system like the single transferable vote (STV) or instant runoff voting (IRV), a ballot is initially allocated to

2109-455: The first count, transfers are done in reference to all the votes held by the successful candidate. If the transfer is of surplus received after the first count through transfer from another candidate, transfers are done in reference to all the votes held by the successful candidate or merely in reference to the most recent transfer received by the successful candidate. Reallocation ballots are drawn at random from those most recently received. In

2166-544: The first round, Andrea is elected, and her 20 surplus votes are transferred. In the second, Carter, the candidate with the fewest votes, is excluded. His 8 votes have Brad as the next choice. This gives Brad 20 votes (exceeding the quota), electing him to the second seat: Other systems, such as the ones used in Ashtabula, Kalamazoo, Sacramento and Cleveland, prescribed that the votes to be transferred would be drawn at random but in equal numbers from each polling place. In

2223-402: The first-preference candidate but if the first preference candidate is elected or found to be un-electable, the vote may be transferred one or more times to successively lower preferences. If there is no lower preference available when such a transfer is applicable, the ballot is said to be exhausted . FPV prevents exhausted ballots. On the other hand, FPV increases the risk of invalid ballots:

2280-533: The hopeful candidate's quotient is recalculated; this is repeated until all surplus votes have been transferred. Proportionality for solid coalitions Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results Proportionality for solid coalitions ( PSC )

2337-530: The line". In the Victorian Legislative Council , semi-optional preferential voting is used if a voter chooses to vote below the line. Voting above the line requires only a '1' being placed in one box, and group voting ticket s voting has applied since 1988. Around 1999 Melbourne's Albert Langer engaged in a campaign to assist voters to mark fewer than all the candidates but still have their vote not deemed informal and rejected. This

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2394-488: The more numbers a voter is required to mark, the greater the opportunity for mistakes, by repeating or skipping numbers or skipping candidates. The Australian election systems used in almost all the state lower houses (all except Tasmania and New South Wales) and in the ACT mandate FPV but they reduce the number of informal (invalid) votes by adding group voting tickets "above the line" on ballot papers. These allow voters to select

2451-512: The natural analogue of a "party" is a solid coalition . A solid coalition is a group of voters who prefer any candidate within a certain set of candidates over any candidate not in the set. A set of voters V {\displaystyle V} is a solid coalition for a set of candidates C {\displaystyle C} , if every voter in V {\displaystyle V} ranks every candidate in C {\displaystyle C} ahead of every candidate that

2508-741: The number of seats to be filled and j {\displaystyle j} be some positive integer. k {\displaystyle k} –PSC or Hare-PSC is defined with respect to the Hare quota n / k {\displaystyle n/k} . It says that if there is a solid coalition for a set of candidates with at least j {\displaystyle j} Hare quotas, then at least j {\displaystyle j} candidates from this set must be elected. (If C {\displaystyle C} has less than j {\displaystyle j} candidates, all of them must be elected). This criterion

2565-424: The order of the steps. There is no general agreement on which is best, and the choice of method used may affect the outcome. To minimize wasted votes , surplus votes are transferred to other candidates if possible. 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

2622-420: 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 been 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

2679-532: The quota, but any candidate who receives quota is elected. The Hare quota and the Droop quota are the common types of quota. The quota is typically set based on the number of valid votes cast, and even if the number of votes in play decreases through the vote count process, the quota remains as set through the process. Meek's counting method recomputes the quota on each iteration of the count, as described below. When Thomas Hare originally conceived his version of single transferable vote, he envisioned using

2736-455: The quota: The Hare quota is mathematically simple. The Hare quota's large size means that elected members have fewer surplus votes and thus other candidates do not get benefit from vote transfers that they would in other systems. Some candidates may be eliminated in the process who may not have been eliminated under systems that transfer more surplus votes. Their elimination may cause a degree of dis-proportionality that would be less likely with

2793-399: The ranking of the two least-popular candidates. In systems where exhausted votes can exist such as optional preferential voting , if the number of votes bearing a next usable marked preference are fewer than the surplus votes, then the transferable votes are simply transferred based on the next usable preference. If the transferable votes surpass the surplus, then the transfer is done using

2850-403: The relevant fraction is 72 272 − 92 = 4 10 {\displaystyle \textstyle {\frac {72}{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

2907-457: 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.) 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

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2964-466: The votes and the order in which surpluses from two or more winners are transferred. Transfers are attempted when a candidate receives more votes than the quota. Excess votes are transferred to remaining candidates, where possible. A winner's surplus votes are transferred according to their next usable marked preference. Transfers are only done if there are still seats to fill. In some systems, surplus votes are transferred only if they could possibly re-order

3021-522: The whole-vote method outlined above was used.) The Gregory method, known as 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 ) eliminates all randomness. Instead of transferring a fraction of votes at full value, transfer each of the votes at a fractional value. In the above example,

3078-404: The winner and others of the same sort of votes moved in whole to the indicated candidate. Lower preferences piggybacked on the ballots may not be perfectly random and this may affect later transfers. This method can be made easier if only the last incoming parcel of votes is used to determine the transfer, not all of the successful candidate's votes. Such a method is used to elect the lower houses in

3135-504: Was a way to use the vote as if under the optional preferential voting system. The Australian Senate voting reform of 2016 switched from full preferential voting to optional preferential voting. A minimum number is specified in the instructions on the ballot paper. Since in the past a single number '1' above the line was valid, that is still a formal vote even though voters are encouraged to rank six candidates. STV Elections in Malta , in

3192-455: Was proposed by Michael Dummett . In the single-winner case, it is equivalent to the unanimity criterion , as a Hare quota in the single-winner case includes all voters. k + 1 {\displaystyle k+1} –PSC, also called Droop-PSC , is defined like k {\displaystyle k} –PSC but using the Droop quota n / ( k + 1 ) {\displaystyle n/(k+1)} instead of

3249-670: Was the case in federal Australian elections prior to 1998 (a Langer vote ) but after 1998 Australia classified those votes as informal (invalid). Australia now allows ballots to express less than complete preferences ( optional preferential voting ) for Senate elections but full preferential voting is still used in the instant-runoff voting system used to elect members of Australia's House of Representatives. The terms OPV and FPV are used mainly in Australia, in relation to elections at state and territory and Commonwealth (federal) level, which use proportional representation ( Single transferable voting /STV) or preferential voting (IRV). OPV

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