Condorcet methods
71-555: 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 In the study of electoral systems , the Droop quota (sometimes called the Hagenbach-Bischoff , Britton , or Newland-Britton quota )
142-515: A . For the Nauru system, the first preference a is worth one and the common difference d between adjacent denominators is also one. Numerous other harmonic sequences can also be used in positional voting. For example, setting a to 1 and d to 2 generates the reciprocals of all the odd numbers (1, 1/3, 1/5, 1/7, …) whereas letting a be 1/2 and d be 1/2 produces those of all the even numbers (1/2, 1/4, 1/6, 1/8, …). The harmonic variant used by
213-406: A = N , the number of candidates. The value of the first preference need not be N . It is sometimes set to N – 1 so that the last preference is worth zero. Although it is convenient for counting, the common difference need not be fixed at one since the overall ranking of the candidates is unaffected by its specific value. Hence, despite generating differing tallies, any value of a or d for
284-475: A Borda count election will result in identical candidate rankings. The consecutive Borda count weightings form an arithmetic progression . Common systems for evaluating preferences, other than Borda, are typically "top-heavy". In other words, the method focuses on how many voters consider a candidate one of their "favourites". Under first-preference plurality (FPP), the most-preferred option receives 1 point while all other options receive 0 points each. This
355-495: A Droop quota's worth of votes is guaranteed to win a seat in a multiwinner election . Besides establishing winners, the Droop quota is used to define the number of excess votes , i.e. votes not needed by a candidate who has been declared elected. In proportional quota-based systems such as STV or expanding approvals , these excess votes can be transferred to other candidates, preventing them from being wasted . The Droop quota
426-447: A candidate who does not need them. If seats remain open after the first count, any surplus votes are transferred. This may generate the necessary winners. As well, least popular candidates may be eliminated as way to generate winners. The specific method of transferring votes varies in different systems (see § Vote transfers and quota ). Transfer of any existing surplus votes is done before eliminations of candidates. This prevents
497-434: A district. The key to STV's approximation of proportionality is that each voter effectively only casts a single vote in a district contest electing multiple winners, while the ranked ballots (and sufficiently large districts) allow the results to achieve a high degree of proportionality with respect to partisan affiliation within the district, as well as representation by gender and other descriptive characteristics. The use of
568-403: A geometric progression with a common ratio of one-half ( r = 1/2). Such weightings are inherently valid for use in positional voting systems provided that a legitimate common ratio is employed. Using a common ratio of zero, this form of positional voting has weightings of 1, 0, 0, 0, … and so produces ranking outcomes identical to that for first-past-the-post or plurality voting . Alternatively,
639-415: A given rank position ( n ) is defined below; where the value of the first preference is a . w n = a 2 a + ( n − 1 ) d = a 1 + ( n − 1 ) d a , {\displaystyle w_{n}={\frac {a^{2}}{a+(n-1)d}}={\frac {a}{1+{\frac {(n-1)d}{a}}}},} where w 1 =
710-448: A high value and all the remaining options with a common lower value. The two validity criteria for a sequence of weightings are hence satisfied. For an N -candidate ranked ballot, let the permitted number of favoured candidates per ballot be F and the two weightings be one point for these favoured candidates and zero points for those not favoured. When analytically represented using positional voting, favoured candidates must be listed in
781-522: A large number of effective votes – 19 votes were used to elect the successful candidates. (Only the votes for Oranges at the end were not used to select a food. The Orange voters have satisfaction of seeing their second choice – Pears – selected, even if their votes were not used to select any food.) As well, there was general satisfaction with the choices selected. Nineteen voters saw either their first or second choice elected, although four of them did not actually have their vote used to achieve
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#1732766242711852-485: A mathematical sequence such as an arithmetic progression ( Borda count ), a geometric one ( positional number system ) or a harmonic one ( Nauru/Dowdall method ). The set of weightings employed in an election heavily influences the rank ordering of the candidates. The steeper the initial decline in preference values with descending rank, the more polarised and less consensual the positional voting system becomes. Positional voting should be distinguished from score voting : in
923-598: A method for converting sets of individual preferences (ranked ballots) into one collective and fully rank-ordered set. It is possible and legitimate for options to be tied in this resultant set; even in first place. Consider a positional voting election for choosing a single winner from three options A, B and C. No truncation or ties are permitted and a first, second and third preference is here worth 4, 2 and 1 point respectively. There are then six different ways in which each voter may rank order these options. The 100 voters cast their ranked ballots as follows: After voting closes,
994-453: A party from losing a candidate in the early stage who might be elected later through transfers. When surplus votes are transferred under some systems, some or all of the votes held by the winner are apportioned fractionally to the next marked preference on the ballot. In others, the transfers to the next available marked preference is done using whole votes. When seats still remain to be filled and there are no surplus votes to transfer (none of
1065-540: A party representing less than half of the voters to take a majority of seats in a constituency. The Droop quota is today the most popular quota for STV elections. Positional voting 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 Positional voting
1136-448: A quota means that, for the most part, each successful candidate is elected with the same number of votes. This equality produces fairness in the particular sense that a party taking twice as many votes as another party will generally take twice the number of seats compared to that other party. Under STV, winners are elected in a multi-member constituency (district) or at-large, also in a multiple-winner contest. Every sizeable group within
1207-546: A replacement for the Hare quota (votes/seats). Their quota was meant to produce more proportional result by having the quota as low as thought to be possible. Their quota was basically votes/seats plus 1, plus 1, the formula on the left on the first row. This formula may yield a fraction, which was a problem as early STV systems did not use fractions. Droop went to votes/seats plus 1, plus 1, rounded down (the variant on top right). Hagenbach-Bischoff went to votes/seats +1, rounded up,
1278-451: A second one is given 10 points. The next eight consecutive preferences are awarded 8, 7, 6, 5, 4, 3, 2 and 1 point. All remaining preferences receive zero points. In positional voting, the weightings ( w ) of consecutive preferences from first to last decline monotonically with rank position ( n ). However, the rate of decline varies according to the type of progression employed. Lower preferences are more influential in election outcomes where
1349-437: A transfer if the first-preference food is chosen with a surplus of votes. The 23 guests at the party mark their ballots: some mark first, second and third preferences; some mark only two preferences. When the ballots are counted, it is found that the ballots are marked in seven distinct combinations, as shown in the table below: The table is read as columns: the left-most column shows that there were three ballots with Orange as
1420-564: A voter's subsequent preferences if necessary. Under STV, no one party or voting bloc can take all the seats in a district unless the number of seats in the district is very small or almost all the votes cast are cast for one party's candidates (which is seldom the case). This makes it different from other commonly used candidate-based systems. In winner-take-all or plurality systems – such as first-past-the-post (FPTP), instant-runoff voting (IRV), and block voting – one party or voting bloc can take all seats in
1491-483: A worse-ranked candidate must receive fewer points than a better-ranked candidate. The classic example of a positional voting electoral system is the Borda count . Typically, for a single-winner election with N candidates, a first preference is worth N points, a second preference N – 1 points, a third preference N – 2 points and so on until the last ( N th) preference that is worth just 1 point. So, for example,
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#17327662427111562-466: Is 100, and there are 3 seats. The Droop quota is therefore 100 3 + 1 = 25 {\textstyle {\frac {100}{3+1}}=25} . These votes are as follows: First preferences for each candidate are tallied: Only Washington has strictly more than 25 votes. As a result, he is immediately elected. Washington has 20 excess votes that can be transferred to their second choice, Hamilton. The tallies therefore become: Hamilton
1633-425: Is Hamburgers, so the three votes are transferred to Hamburgers. Hamburgers is elected with 7 votes in total. Hamburgers now has a surplus vote, but this does not matter since the election is over. There are no more foods needing to be chosen – three have been chosen. Result: The winners are Pears, Cake, and Hamburgers. Orange ends up being neither elected nor eliminated. STV in this case produced
1704-466: Is a ranked voting electoral system in which the options or candidates receive points based on their rank position on each ballot and the one with the most points overall wins. The lower-ranked preference in any adjacent pair is generally of less value than the higher-ranked one. Although it may sometimes be weighted the same, it is never worth more. A valid progression of points or weightings may be chosen at will ( Eurovision Song Contest ) or it may form
1775-452: Is a multi-winner electoral system in which each voter casts a single vote in the form of a ranked ballot . Voters have the option to rank candidates, and their vote may be transferred according to alternative preferences if their preferred candidate is eliminated or elected with surplus votes, so that their vote is used to elect someone they prefer over others in the running. STV aims to approach proportional representation based on votes cast in
1846-442: Is calculated by a specified method (STV generally uses the Hare or Droop quota ), and candidates who accumulate that many votes are declared elected. In many STV systems, the quota is also used to determine surplus votes, the number of votes received by successful candidates over and above the quota. Surplus votes are transferred to candidates ranked lower in the voters' preferences, if possible, so they are not wasted by remaining with
1917-439: Is elected, so his excess votes are redistributed. Thanks to Hamilton's support, Jefferson receives 30 votes to Burr's 20 and is elected. If all of Hamilton's supporters had instead backed Burr, the election for the last seat would have been exactly tied, requiring a tiebreaker; generally, ties are broken by taking the limit of the results as the quota approaches the exact Droop quota. There are at least six different versions of
1988-484: Is eliminated. In accordance with the next preference marked on the vote cast by the voter who voted Strawberry as first preference, that vote is transferred to Oranges. In accordance with the next preference marked on the two votes cast by the Pear–Strawberry–Cake voters (which had been transferred to Strawberry in step 2), the two votes are transferred to Cake. (The Cake preference had been "piggy-backed" along with
2059-469: Is more favourable to candidates with many first preferences than the conventional Borda count. It has been described as a system "somewhere between plurality and the Borda count, but as veering more towards plurality". Simulations show that 30% of Nauru elections would produce different outcomes if counted using standard Borda rules. The Eurovision Song Contest uses a first preference worth 12 points, while
2130-402: Is straightforward. All the preferences cast by voters are awarded the points associated with their rank position. Then, all the points for each option are tallied and the one with the most points is the winner. Where a few winners ( W ) are instead required following the count, the W highest-ranked options are selected. Positional voting is not only a means of identifying a single winner but also
2201-458: Is the minimum number of supporters a party or candidate needs to receive in a district to guarantee they will win at least one seat in a legislature . The Droop quota is used to extend the concept of a majority to multiwinner elections , taking the place of the 50% bar in single-winner elections. Just as any candidate with more than half of all votes is guaranteed to be declared the winner in single-seat election, any candidate who holds more than
Droop quota - Misplaced Pages Continue
2272-400: Is the most top-heavy positional voting system. An alternative mathematical sequence known as a geometric progression may also be used in positional voting. Here, there is instead a common ratio r between adjacent weightings. In order to satisfy the two validity conditions, the value of r must be less than one so that weightings decrease as preferences descend in rank. Where the value of
2343-575: Is therefore guaranteed to win a seat. Modern variants of STV use fractional transfers of ballots to eliminate uncertainty. However, STV elections with whole vote reassignment cannot handle fractional quotas, and so instead will round up or round down . For example: ⌈ total votes k + 1 ⌉ {\displaystyle \left\lceil {\frac {\text{total votes}}{k+1}}\right\rceil } The Droop quota can be derived by considering what would happen if k candidates (who we call "Droop winners") have achieved
2414-455: The largest remainder method . The Droop quota for a k {\displaystyle k} -winner election is given by the expression: total votes k + 1 {\displaystyle {\frac {\text{total votes}}{k+1}}} Sometimes, the Droop quota is written as a share of all votes, in which case it has value 1 ⁄ k +1 . A candidate who, at any point, holds more than one Droop quota's worth of votes
2485-441: The 23 guests. STV is chosen to make the decision, with the whole-vote method used to transfer surplus votes. The hope is that each guest will be served at least one food that they are happy with. To select the three foods, each guest is given one vote – they each mark their first preference and are also allowed to cast two back-up preferences to be used only if their first-preference food cannot be selected or to direct
2556-1658: The Droop quota to appear in various legal codes or definitions of the quota, all varying by one vote . Some claim that, depending on which version is used, a failure of proportionality in small elections may arise. Common variants include: Historical: ⌊ votes seats + 1 + 1 ⌋ ⌈ votes seats + 1 ⌉ ⌊ votes seats + 1 + 1 ⌋ Accidental: ⌊ votes + 1 seats + 1 ⌋ Unusual: ⌊ votes seats + 1 ⌋ ⌊ votes seats + 1 + 1 2 ⌋ {\displaystyle {\begin{array}{rlrl}{\text{Historical:}}&&{\phantom {\Bigl \lfloor }}{\frac {\text{votes}}{{\text{seats}}+1}}+1{\phantom {\Bigr \rfloor }}&&\left\lceil {\frac {\text{votes}}{{\text{seats}}+1}}\right\rceil &&{\Bigl \lfloor }{\frac {\text{votes}}{{\text{seats}}+1}}+1{\Bigr \rfloor }\\{\text{Accidental:}}&&{\phantom {\Bigl \lfloor }}{\frac {{\text{votes}}+1}{{\text{seats}}+1}}{\phantom {\Bigr \rfloor }}\\{\text{Unusual:}}&&\left\lfloor {\frac {\text{votes}}{{\text{seats}}+1}}\right\rfloor &&\left\lfloor {\frac {\text{votes}}{{\text{seats}}+1}}+{\frac {1}{2}}\right\rfloor \end{array}}} Droop and Hagenbach-Bischoff derived new quota as
2627-451: The Droop quota. However, some jurisdictions fail to correctly specify this in their election administration laws. The Droop quota is often confused with the Hare quota . While the Droop quota gives the number of voters needed to mathematically guarantee a candidate's election, the Hare quota gives the number of voters represented by each winner by exactly linear proportionality. As a result,
2698-447: The Droop quota. The goal is to identify whether an outside candidate could defeat any of these candidates. In this situation, if each quota winner's share of the vote equals 1 ⁄ k +1 plus 1, while all unelected candidates' share of the vote, taken together, would be less than 1 ⁄ k +1 votes. Thus, even if there were only one unelected candidate who held all the remaining votes, they would not be able to defeat any of
2769-428: The Droop winners. Newland and Britton noted that while a tie for the last seat is possible, such a situation can occur no matter which quota is used. The following election has 3 seats to be filled by single transferable vote . There are 4 candidates: George Washington , Alexander Hamilton , Thomas Jefferson , and Aaron Burr . There are 102 voters, but two of the votes are spoiled . The total number of valid votes
2840-435: The Hare quota is said to give somewhat more proportional outcomes, by promoting representation of smaller parties, although sometimes under Hare a majority group will be denied the majority of seats, thus denying the principle of majority rule in such settings as a city council elected at-large. By contrast, the Droop quota is more biased towards large parties than any other admissible quota . The Droop quota sometimes allows
2911-461: The binary number system, a common ratio greater than one-half must be employed. The higher the value of r , the slower the decrease in weightings with descending rank. Although not categorised as positional voting electoral systems, some non-ranking methods can nevertheless be analysed mathematically as if they were by allocating points appropriately. Given the absence of strict monotonic ranking here, all favoured options are weighted identically with
Droop quota - Misplaced Pages Continue
2982-441: The binary, ternary, octal and decimal number systems use a radix R of 2, 3, 8 and 10 respectively. The value R is also the common ratio of the geometric progression going up in rank order while r is the complementary common ratio descending in rank. Therefore, r is the reciprocal of R and the r ratios are respectively 1/2, 1/3, 1/8 and 1/10 for these positional number systems when employed in positional voting. As it has
3053-407: The chosen progression employs a sequence of weightings that descend relatively slowly with rank position. The more slowly weightings decline, the more consensual and less polarising positional voting becomes. This figure illustrates such declines over ten preferences for the following four positional voting electoral systems: To aid comparison, the actual weightings have been normalised; namely that
3124-472: The decimal point are employed rather than fractions. (This system should not be confused with the use of sequential divisors in proportional systems such as proportional approval voting , an unrelated method.) A similar system of weighting lower-preference votes was used in the 1925 Oklahoma primary electoral system . For a four-candidate election, the Dowdall point distribution would be this: This method
3195-486: The denominators of the above fractional weightings could form an arithmetic progression instead; namely 1/1, 1/2, 1/3, 1/4 and so on down to 1/ N . This further mathematical sequence is an example of a harmonic progression . These particular descending rank-order weightings are in fact used in N -candidate positional voting elections to the Nauru parliament . For such electoral systems, the weighting ( w n ) allocated to
3266-451: The district where it is used, so that each vote is worth about the same as another. STV is a family of proportional multi-winner electoral systems . They can be thought of as a variation on the largest remainders method that uses solid coalitions rather than party lists . Surplus votes belonging to winning candidates (those in excess of an electoral quota ) may be thought of as remainder votes – they are transferred to
3337-400: The district wins at least one seat: the more seats the district has, the smaller the size of the group needed to elect a member. In this way, STV provides approximately proportional representation overall, ensuring that substantial minority factions have some representation. There are several STV variants. Two common distinguishing characteristics are whether or not ticket voting is allowed and
3408-416: The fewest votes and is eliminated. According to their only voter's next preference, this vote is transferred to Cake. No option has reached the quota, and there are still two to elect with five in the race, so elimination of options will continue next round. Step 4: Of the remaining options, Oranges, Strawberry and Chicken now are tied for the fewest votes. Strawberry had the fewest first preference votes so
3479-598: The first choice and Pear as second, while the right-most column shows there were three ballots with Chicken as first choice, Chocolate as second, and Hamburger as third. The election step-by-step: ELECTED (2 surplus vote) ELECTED (0 surplus votes) ELECTED (1 surplus vote) Setting the quota: The Droop quota formula is used to produce the quota (the number of votes required to be automatically declared elected) = floor(valid votes / (seats to fill + 1)) + 1 = floor(23 / (3 + 1)) + 1 = floor(5.75) + 1 = 5 + 1 = 6 Step 1: First-preference votes are counted. Pears reaches
3550-429: The first preference is a , the weighting ( w n ) awarded to a given rank position ( n ) is defined below. w n = a r n − 1 , 0 ≤ r < 1 {\displaystyle w_{n}=ar^{n-1},\qquad 0\leq r<1} For example, the sequence of consecutively halved weightings of 1, 1/2, 1/4, 1/8, … as used in the binary number system constitutes
3621-418: The first preference is set at one and the other weightings in the particular sequence are scaled by the same factor of 1/ a . The relative decline of weightings in any arithmetic progression is constant as it is not a function of the common difference d . In other words, the relative difference between adjacent weightings is fixed at 1/ N . In contrast, the value of d in a harmonic progression does affect
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#17327662427113692-401: The former, the score that each voter gives to each candidate is uniquely determined by the candidate's rank; in the latter, each voter is free to give any score to any candidate. In positional voting, voters complete a ranked ballot by expressing their preferences in rank order. The rank position of each voter preference is allotted a specific fixed weighting. Typically, the higher the rank of
3763-479: The island nation of Nauru is called the Dowdall system as it was devised by Nauru's Secretary for Justice (Desmond Dowdall) in 1971. Here, each voter awards the first-ranked candidate with 1 point, while the 2nd-ranked candidate receives 1 ⁄ 2 a point, the 3rd-ranked candidate receives 1 ⁄ 3 of a point, etc. When counting candidate tallies in Nauru, decimal numbers rounded to three places after
3834-470: The last parcel of votes received by winners in accordance with the Gregory method. Systems that use the Gregory method for surplus vote transfers are strictly non-random. In a single transferable vote (STV) system, the voter ranks candidates in order of preference on their ballot. A vote is initially allocated to the voter's first preference. A quota (the minimum number of votes that guarantees election)
3905-609: The manner in which surplus votes are transferred. In Australia, lower house elections do not allow ticket voting; some but not all state upper house systems do allow ticket voting. In Ireland and Malta, surplus votes are transferred as whole votes (there may be some random-ness) and neither allows ticket voting. In Hare–Clark , used in Tasmania and the Australian Capital Territory , there is no ticket voting and surplus votes are fractionally transferred based on
3976-420: The points are respectively 4, 3, 2 and 1 for a four-candidate election. Mathematically, the point value or weighting ( w n ) associated with a given rank position ( n ) is defined below; where the weighting of the first preference is a and the common difference is d . w n = a − ( n − 1 ) d {\displaystyle w_{n}=a-(n-1)d} where
4047-428: The points awarded by the voters are then tallied and the options ranked according to the points total. Therefore, having the highest tally, option A is the winner here. Note that the election result also generates a full ranking of all the options. For positional voting, any distribution of points to the rank positions is valid, so long as the points are weakly decreasing in the rank of each candidate. In other words,
4118-439: The preference, the more points it is worth. Occasionally, it may share the same weighting as a lower-ranked preference but it is never worth fewer points. Usually, every voter is required to express a unique ordinal preference for each option on the ballot in strict descending rank order. However, a particular positional voting system may permit voters to truncate their preferences after expressing one or more of them and to leave
4189-408: The quota with 8 votes and is therefore elected on the first count, with 2 surplus votes. Step 2: All of the voters who gave first preference to Pears preferred Strawberry next, so the surplus votes are awarded to Strawberry. No other option has reached the quota, and there are still two to elect with six options in the race, so elimination of lower-scoring options starts. Step 3: Chocolate has
4260-411: The quota) or until there are only as many remaining candidates as there are unfilled seats, at which point the remaining candidates are declared elected. Suppose an election is conducted to determine what three foods to serve at a party. There are seven choices: Oranges, Pears, Strawberries, Cake (of the strawberry/chocolate variety), Chocolate, Hamburgers and Chicken. Only three of these may be served to
4331-501: The rate of its decline. The higher its value, the faster the weightings descend. Whereas the lower the value of the common ratio r for a geometric progression, the faster its weightings decline. The weightings of the digit positions in the binary number system were chosen here to highlight an example of a geometric progression in positional voting. In fact, the consecutive weightings of any digital number system can be employed since they all constitute geometric progressions. For example,
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#17327662427114402-498: The remaining candidates' votes have surplus votes needing to be transferred), the least popular candidate is eliminated. The eliminated candidate's votes are transferred to the next-preferred candidate rather than being discarded; if the next-preferred choice has already been eliminated or elected, the procedure is iterated to lower-ranked candidates. Counting, eliminations, and vote transfers continue until enough candidates are declared elected (all seats are filled by candidates reaching
4473-541: The remaining options unranked and consequently worthless. Similarly, some other systems may limit the number of preferences that can be expressed. For example, in the Eurovision Song Contest only their top ten preferences are ranked by each country although many more than ten songs compete in the contest. Again, unranked preferences have no value. In positional voting, ranked ballots with tied options are normally considered as invalid. The counting process
4544-486: The result. Four saw their third choice elected. Fifteen voters saw their first preference chosen; eight of these 15 saw their first and third choices selected. Four others saw their second preference chosen, with one of them having their second and third choice selected. Note that if Hamburger had received only one vote when Chicken was eliminated, it still would have won because the only other remaining candidate, Oranges, had fewer votes so would have been declared defeated in
4615-428: The smallest radix, the rate of decline in preference weightings is slowest when using the binary number system. Although the radix R (the number of unique digits used in the number system) has to be an integer, the common ratio r for positional voting does not have to be the reciprocal of such an integer. Any value between zero and just less than one is valid. For a slower descent of weightings than that generated using
4686-401: The third row), it is possible for one more candidate to reach the quota than there are seats to fill. However, as Newland and Britton noted in 1974, this is not a problem: if the last two winners both receive a Droop quota of votes, it would mean a tie. Rules are in place to break a tie, and ties can occur regardless of which quota is used. Spoiled ballots should not be included when calculating
4757-885: The top F rank positions in any order on each ranked ballot and the other candidates in the bottom N - F rank positions. This is essential as the weighting of each rank position is fixed and common to each and every ballot in positional voting. Unranked single-winner methods that can be analysed as positional voting electoral systems include: And unranked methods for multiple-winner elections (with W winners) include: Single transferable vote 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 The single transferable vote ( STV ) or proportional-ranked choice voting ( P-RCV ),
4828-459: The transfer to Strawberry.) Cake reaches the quota and is elected. Cake has no surplus votes, no other option has reached the quota, and there is still one choice to select with three in the race, so the vote count proceeds, with the elimination of the least popular candidate. Step 5: Chicken has the fewest votes and is eliminated. The Chicken voters' next preference is Chocolate but Chocolate has already been eliminated. The next usable preference
4899-452: The use of fractions in fractional STV systems, now common today. As well, it is un-necessary to ensure the quota is larger than vote/seats plus 1, as in the historical examples, the variant on the second row, and the formula on the right on the bottom row. When using the exact Droop quota (votes/seats plus 1) or any variant where the quota is slightly less than votes/seats plus 1, such as in votes/seats plus 1, rounded down (the left variant on
4970-458: The variant in the middle of the top row. Hagenbach-Bischoff proposed a quota that is "the whole number next greater than the quotient obtained by dividing m V {\displaystyle mV} , the number of votes, by n + 1 {\displaystyle n+1} " (where n is the number of seats). Some hold the misconception that these rounded-off variants of the Droop and Hagenbach-Bischoff quota are still needed, despite
5041-669: Was first suggested by the English lawyer and mathematician Henry Richmond Droop (1831–1884) as an alternative to the Hare quota , which is a basic component of single transferable voting , a form of proportional representation . Today, the Droop quota is used in almost all STV elections, including those in Australia , the Republic of Ireland , Northern Ireland , and Malta . It is also used in South Africa to allocate seats by
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