Condorcet methods
37-508: 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 An electoral list is a grouping of candidates for election, usually found in proportional or mixed electoral systems, but also in some plurality electoral systems . An electoral list can be registered by
74-515: 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
111-406: 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
148-416: A political party (a party list ) or can constitute a group of independent candidates. Lists can be open , in which case electors have some influence over the ranking of the winning candidates, or closed , in which case the order of candidates is fixed at the registration of the list. Electoral lists are required for party-list proportional representation systems. An electoral list is made according to
185-475: 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
222-403: 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,
259-415: 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 =
296-448: 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
333-485: 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
370-598: 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,
407-451: 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
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#1732780728876444-420: A selection meeting, where new candidates may be added, or existing candidates may be moved or removed from the list. When the internal process is over, the final list is made public. The list may be printed on the ballot paper cast by voters at the election, or on a separate voter information paper. When an elected representative vacates their seat, the casual vacancy in a list-PR system is typically filled by
481-483: 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,
518-466: 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
555-648: Is a ranked list of candidates nominated by parties at the state level, used in Germany's mixed-member proportional (MMP) system to allocate seats based on the party's share of the second vote ( Zweitstimme ). 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
592-469: 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
629-402: 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
666-400: 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
703-578: The "party list" refers to the closed list of candidates nominated by a registered party for election to the New Zealand Parliament . Voters cast two votes: one for an electorate candidate and one for the party at large. The "party vote" generally determines the overall distribution of seats in Parliament, with candidates from the party list being elected based on their ranking on the list. The state list ( German : Landesliste )
740-464: The applying nomination rules and election rules . Depending on the type of election, a political party , a general assembly , or a board meeting, may elect or appoint a nominating committee that will add, and if required, prioritize list-candidates according to their preferences. Qualification, popularity, gender, age, geography, and occupation are preferences that may influence the committee's work. The committee's proposed list may then be changed in
777-461: 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
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#1732780728876814-441: 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
851-407: 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
888-472: 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
925-486: 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
962-429: 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
999-418: 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
1036-401: 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
1073-506: The highest-ranked candidate on the departed representative's list who was not already elected. For personal or party-strategic reasons, this person may choose to cede the place to a lower-ranked colleague. Replacement lists are sometimes used to fill casual vacancies in single transferable vote electoral systems. An example is European Parliament elections in Ireland since 1984 . In New Zealand's mixed-member proportional (MMP) system ,
1110-479: 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
1147-420: 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
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1184-428: 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,
1221-439: 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
1258-501: 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,
1295-541: 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
1332-428: 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
1369-516: 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: Fractional social choice Too Many Requests If you report this error to
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