Condorcet methods
62-398: 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 Single non-transferable vote or SNTV is an electoral system used to elect multiple winners. It
124-511: 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
186-406: 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
248-475: 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
310-497: A city's seats, a thing common in First past the post elections. The number of wasted votes in an SNTV election is generally lower than in First past the post elections as well. Under SNTV, parties often do not receive representation exactly proportional to their strength, because it is difficult to accurately judge their strength when deciding how many candidates to field ( strategic nomination ) and difficult to direct party supporters as
372-406: A different candidate. The design of many voting rules creates incentives for dishonesty among voters. First-preference methods like first-past-the-post and ranked-choice runoff voting (RCV) have a strong tendency to force voters into supporting the lesser of two evils , i.e. lying about who their favorite candidate is. If a voter's most preferred candidate is unlikely to win the election,
434-403: 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,
496-415: 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 =
558-448: 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
620-505: A hundred years later than Saint-Just's proposal. In 1880s Japan adopted SNTV for provincial politicians and in 1900 for national politicians. SNTV is used for elections in Puerto Rico , Kuwait , Indonesia , Japan , Taiwan , Thailand , Libya , Iraq , Hong Kong and Vanuatu . In Puerto Rico , SNTV is known as at-large representation ("representación por acumulación" in Spanish), SNTV
682-406: A kind of semi-proportional representation at the district level, meaning small parties, as well as large parties, have a chance to be represented. Under certain assumptions, such as perfect tactical voting SNTV is equivalent to proportional representation by the D'Hondt method . SNTV retains many of the problems of first-past-the-post voting (first-preference plurality voting), and as a result
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#1732852177240744-485: 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
806-598: 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,
868-485: A multi-candidate race for multiple offices. Posts are filled by the candidates with the most votes ( plurality voting ). Thus, in a three-seat constituency, the three candidates receiving the largest numbers of votes would win office. SNTV, like single transferable vote , can be used with non-partisan ballots, in election contests where there are no parties. Three seats are to be filled among five candidates: A, B, C, D and E fielded by 3 parties X, Y and Z. E, D and C are
930-451: 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
992-422: A single candidate. Party A has about 35% support among the electorate (with one particularly well-liked candidate), Party B around 25% (with two well-liked candidates) and the remaining voters primarily support independent candidates, but mostly lean towards party B if they have to choose between the two parties. All voters vote sincerely ; there is no tactical voting. Percent of votes under MNTV and Limited Voting
1054-565: A system of vote allocation which had been adopted by all parties for the 2004 ROC legislative elections . Single Non-transferable Voting was first proposed in solid form by Saint-Just in 1793, in a proposal to the French National Convention . He proposed having the whole country as one multi-seat district; but the idea was not adopted in France at that time. Japan was the first country to adopt SNTV for election of government members
1116-409: A whole to spread their votes efficiently. If they field too many, supporters' votes might be split across too many candidates. The party votes might spread their vote numbers to the point where all of a party's candidates lose to a less thinly spread opposing party. If a party fields too few candidates, they might elect all their candidates but not win seats proportional to their level of support, and
1178-479: 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,
1240-466: 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
1302-577: Is a semi-proportional variant of first-past-the-post voting , applied to multi-member districts where each voter casts just one vote. It can also be seen as a variant of STV but with no vote transfers. SNTV generally makes it unlikely that a single party will take all seats in a city, as generally happens with winner-take-all systems . SNTV can be considered a variant of dot voting where each voter has only one point to assign. Unlike block voting or limited voting , where voters can cast multiple votes, under SNTV each voter casts just one. This produces
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#17328521772401364-751: Is common even after STV vote transfers to elect the same as would be elected under SNTV. But not having transfers, SNTV sees more votes wasted than under STV due to votes being placed on un-electable candidate or due to surpluses received by successful candidate over and above the quota used in STV elections that are not able to be transferred under SNTV. SNTV produces representation that is most proportional ( proportional representation ) when political parties have accurate information about their relative levels of electoral support, and nominate candidates in accordance with their respective levels of electoral support or when all parties suffer from poor information of that sort. Knowing
1426-510: Is elected does not mean that another specific candidate will not be. They both can be elected. Because running on issues may lead to a situation in which a candidate becomes too popular and therefore draws votes away from other allied candidates, SNTV may encourage legislators to join factions that consist of patron-client relationships in which a powerful legislator can apportion votes to his or her supporters. In addition, parties will do best if their supporters evenly distribute their votes among
1488-493: Is more chance in SNTV than a more orderly system of PR, such as list PR or STV.) If either party had risked trying to win all three seats, causing more vote splitting among supporters of Parties Y and Z, then A of Party X might have won a seat and either party Y or Z would then have taken one fewer seat. For example, 10,000 voters vote to elect three members. Cumulative voting is not used so each voter may not cast more than one vote for
1550-465: 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
1612-453: Is sometimes viewed skeptically by social choice theorists . However, its extreme simplicity and easy vote-counting makes the system particularly popular for small elections to offices such as city councils , particularly when compared to the more-intricate single transferable vote (STV) system, and has resulted in the method becoming commonly used for ordering open party lists . In any election, each voter casts one vote for one candidate in
1674-402: 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
1736-400: 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
1798-423: Is the percent of voters who voted for the candidate, not the percent of votes cast. SNTV facilitates minority representation, that is, it produces mixed representation of large and small parties where no party takes all the seats. In fact, SNTV would elect the same people as are elected in STV contests where the vote transfers do not move an initially-lower-placing candidate over an initially-higher-placing one. It
1860-417: Is the practice of casting a vote that provides more support to a worse outcome than a better one, i.e. one that involves voters lying about whether they prefer candidate A or B. It is sometimes called misaligned , deceptive , or dishonest voting . For example, in a first-past-the-post election , a sincere voter would support the candidate they think is best, whereas an insincere voter may instead support
1922-509: Is used to elect the 11 at-large members in each of the House of Representatives and the Senate. Under at-large representation , political parties vary the ballot order of their candidates across electoral divisions, in order to ensure each candidate has a roughly equal chance of success. Since most voters choose the candidates placed at the top of their party lists on their ballots, at-large candidates from
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1984-461: 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
2046-441: 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
2108-407: 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
2170-467: 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
2232-482: 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
2294-429: 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
2356-418: 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
2418-401: 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
2480-445: The information offered in campaigns (polls, reporting, fundraising totals, endorsements, etc.), to rationally decide who the most viable candidates are and then vote for them. SNTV can result in complicated intra-party dynamics because in a SNTV system, a candidate runs against candidates from their own party as well as against candidates from the other party. SNTV elections are not zero-sum contests. Just because one particular candidate
2542-474: 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
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2604-787: The legislatures of Japan , South Korea and the Republic of China ( Taiwan ), but its use has been discontinued for the most part. It is still used in Japan for some seats in the House of Councillors (Sangi-in), prefectural assemblies and municipal assemblies. 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
2666-403: The much smaller Puerto Rican Independence Party runs a single-candidate slate for the at-large members in the Senate and the House of Representatives. The SNTV-elected members are a small part of the chambers compared to the members elected in the sixteen Senate districts, elected by block voting, and the forty House districts, elected by first-past-the-post voting . SNTV was once used to elect
2728-409: The party was not as popular as they thought. If every party does that, all suffer the same inefficiency and the final result is proportional. If one party is more prudent, it may do better than the others. Because votes cannot be transferred, there is more chance of vote wastage than under STV. But in elections that use SNTV, representation is usually mixed. It is rare for one party to make a sweep of
2790-635: The party's candidates. Historically, in Taiwan , the Kuomintang did this by sending members a letter telling them which candidate to vote for. With the Democratic Progressive Party , vote sharing is done informally, as members of a family or small group will coordinate their votes. The New Party had a surprisingly effective system by asking party supporters to vote for the candidate whose identification number corresponded to their birthdate. This led to
2852-420: 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
2914-428: 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,
2976-714: The portion of the votes a party can take allows it to avoid vote waste due to lessening the chance of vote splitting and inefficient placement of party support. Under 'perfect' tactical voting and strategic nomination, SNTV would be equivalent to the D'Hondt (Jefferson) method of proportional representation. But under SNTV even inefficient distribution of votes allows more balanced representation than would be elected under either single-member plurality or block voting . Given n {\displaystyle n} candidates to be elected, Candidate A can guarantee success by receiving one more than 1 n + 1 {\displaystyle {\frac {1}{n+1}}} of
3038-439: 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
3100-501: 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,
3162-531: 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
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#17328521772403224-542: The same party usually obtain approximately equal vote totals. When the party's candidates are equally supported, the most-popular party is often able to take six seats of the 11. The two major Puerto Rican political parties, the Popular Democratic Party and the New Progressive Party , usually each run six candidates for the 11 at-large members in each of the House of Representatives and the Senate, while
3286-505: The side of too many candidates, hoping to gain as many seats as possible, perhaps even winning more than its proportion of the electorate if they can edge out candidates from larger parties with just a few votes. As well, a small party running just one candidate would not suffer from vote spitting, while a larger party running four or more may suffer from that. SNTV electoral systems, like STV and proportional electoral systems generally, typically produce more proportional electoral outcomes as
3348-588: The size of the electoral districts (number of seats in each constituency) increases. The potential for tactical voting in a single non-transferable vote system is large. Casting only one vote, a rational voter wanting to maximize the number of seats captured by his party should vote for a candidate of the party that has a chance of winning, but one that will not win by too great a margin and thus take votes away from party colleagues. This creates opportunities for tactical nominations, with parties nominating candidates similar to their opponents' candidates in order to split
3410-428: 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
3472-537: 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: Sincere voting In political science , social choice , and game theory , insincere voting
3534-634: The vote tallies of others, not with a theoretical threshold or quota. In the 2020 Vanuatuan general election , using SNTV, as few as 5 percent of the vote was enough to be elected in a seven-seat district, where about 13 percent is Droop quota. Vote splitting due to poor information on voters' behaviour may deny a popular party its due share of representation. (Single Transferable Voting does not suffer from this handicap as votes are transferable and many are transferred and used that are wasted under SNTV.) Parties organizing slates of multiple candidates may nominate many candidates and then learn on election night that
3596-486: The vote. Like all multiple-winner selections, parties find it advantageous to run a range of candidates in SNTV elections. SNTV has been measured through the lens of such concepts as decision-theoretic analysis . Professor Gary W. Cox , an expert on SNTV, has studied this system's use in Japan. Cox has an explanation of real-world data finding the, "two systems [plurality and semi-proportional] are alike in their strategic voting equilibria." His research found that voters use
3658-455: The voter is instead incentivized to support the "least bad" of the candidates they consider viable. By contrast, systems that satisfy independence of irrelevant alternatives (such as score , approval , and highest medians ) tend to exhibit very low rates of insincere voting, and can even satisfy the sincere favorite criterion (which means voters are never forced to choose between the lesser of two evils). This political science article
3720-507: The votes (the Droop quota ), because n {\displaystyle n} +1 other candidates cannot each receive more than Candidate A (too many would not pass Droop quota) But as SNTV is a plurality system and votes are wasted (not used to elect anyone), it is possible to win with less than Droop quota (but never possible to lose if you have at least Droop). To determine the successful candidates, candidates' vote tallies are compared with
3782-513: The winning candidates would have more support than necessary and thus wasting votes. The risks of poor strategic nomination are not equal for parties of various strengths. A large party would have much more to lose from the split vote effect than to gain from avoiding the wasted vote effect, and so would likely decide to err on the side of fielding fewer candidates (but probably not less than their existing number of seats). A small party with little representation would be more risk-tolerant and err on
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#17328521772403844-652: The winning candidates. Thus, Party Z gets two seats and Party Y gets one seat. No one party took all the seats as might have been the result under first past the post or plurality block voting . But counting the votes by party gives these vote tallies: Party Y has more votes than Party Z, but receives fewer seats because of an inefficient spread of votes across the candidates. If Party Y's two candidates had had more equal vote tallies, it would have won two seats and Party Z only one. Or if Party Z's candidates had received less equal vote tallies, Party Y would have won two seats even if its candidates were not equally popular. (There
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