Condorcet methods
45-429: 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 Huntington–Hill method , sometimes called method of equal proportions , is a highest averages method for assigning seats in
90-515: 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
135-406: 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
180-422: A division by zero error. In the U.S. House of Representatives, this is ensured by guaranteeing each state at least one seat; in party-list representation , small parties would likely be eliminated using some electoral threshold , or the first divisor can be modified. Consider an example to distribute 8 sits between three parties A, B, C having respectively 100,000, 80,000 and 30,000 voices. Each eligible party
225-473: A legislature to political parties or states . Since 1941, this method has been used to apportion the 435 seats in the United States House of Representatives following the completion of each decennial census . The method minimizes the relative difference in the number of constituents represented by each legislator. In other words, the method selects the algorithm such that no transfer of
270-475: 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
315-441: A class of voting methods that allow voters to state how strongly they support a candidate, by giving each one a grade on a separate scale. The distribution of ratings for each candidate—i.e. the percentage of voters who assign them a particular score—is called their merit profile . For example, if candidates are graded on a 4-point scale, one candidate's merit profile may be 25% on every possible rating (1, 2, 3, and 4), while
360-403: 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,
405-415: 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 =
450-448: 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
495-451: A large number of voters, the strategic Myerson-Weber equilibria for such methods are the same as for methods where only extreme ballots are allowed. In this setting, the optimal strategy for Range voting is the same as for approval voting, and the optimal strategy for cumulative voting is the same as for first-past-the-post . For approval voting (and thus Range voting), the optimal strategy involves approving (or rating maximum) everybody above
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#1732781051824540-485: 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
585-598: 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,
630-400: A perfect candidate would have a merit profile where 100% of voters assign them a score of 4. Since rated methods allow the voters to express how strongly they support a candidate, these methods are not covered by Arrow's impossibility theorem , and their resistance to the spoiler effect becomes a more complex matter. Some rated methods are immune to the spoiler effect when every voter rates
675-465: A seat from one state to another can reduce the percent error in representation for both states. In this method, as a first step, each of the 50 states is given its one guaranteed seat in the House of Representatives, leaving 385 seats to assign. The remaining seats are allocated one at a time, to the state with the highest average district population , to bring its district population down . However, it
720-451: 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
765-483: 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,
810-466: 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
855-418: Is assigned one seat. With all the initial seats assigned, the remaining five seats are distributed by a priority number calculated as follows. Each eligible party's (Parties A, B, and C) total votes is divided by √ 2 • 1 ≈ 1.41 , then by approximately 2.45, 3.46, 4.47, 5.48, 6.48, 7.48, and 8.49. The 5 highest entries, marked with asterisks, range from 70,711 down to 28,868 . For each,
900-469: 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
945-424: Is not clear if we should calculate the average before or after allocating an additional seat, and the two procedures give different results. Huntington-Hill uses a continuity correction as a compromise, given by taking the geometric mean of both divisors, i.e.: where P is the population of the state, and n is the number of seats it currently holds before the possible allocation of the next seat. Consider
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#1732781051824990-402: 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
1035-400: 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
1080-570: The Condorcet winner criterion , usually by combining cardinal voting with a first stage (as in Smith//Score ). Like all (deterministic, non-dictatorial, multicandidate) voting methods, rated methods are vulnerable to strategic voting, due to Gibbard's theorem . Cardinal methods where voters give each candidate a number of points and the points are summed are called additive . Both range voting and cumulative voting are of this type. With
1125-487: The actual apportionment, Kulanu would have lost one seat, while The Jewish Home would have gained one seat. 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
1170-461: 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
1215-441: 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
1260-406: The candidates on an absolute scale, but they are not when the voters' rating scales change based on the candidates who are running. There are several voting systems that allow independent ratings of each candidate, which allow them to be immune to the spoiler effect given certain types of voter behavior. For example: However, other rated voting methods have a spoiler effect no matter what scales
1305-407: 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
1350-480: The corresponding party gets another seat. The Knesset ( Israel 's unicameral legislature), are elected by party-list representation with apportionment by the D'Hondt method. Had the Huntington–Hill method, rather than the D'Hondt method, been used to apportion seats following the elections to the 20th Knesset , held in 2015, the 120 seats in the 20th Knesset would have been apportioned as follows: Compared with
1395-472: 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
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1440-486: 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
1485-429: 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
1530-418: 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
1575-401: 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
1620-479: 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
1665-420: 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
1710-428: 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,
1755-439: 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
1800-501: 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,
1845-482: The reapportionment following the 2010 U.S. census: after every state is given one seat: This process continues until all remaining seats are assigned. Each time a state is assigned a seat, n is incremented by 1, causing its priority value to be reduced. Unlike the D'Hondt and Sainte-Laguë systems, which allow the allocation of seats by calculating successive quotients right away, the Huntington–Hill system requires each party or state have at least one seat to avoid
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1890-541: 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
1935-428: 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
1980-854: 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: Rated 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 Rated , evaluative , graded , or cardinal voting rules are
2025-530: The voters use: In addition, there are many different proportional cardinal rules, often called approval-based committee rules. Ratings ballots can be converted to ranked/preferential ballots, assuming equal ranks are allowed. For example: Arrow's impossibility theorem does not apply to cardinal rules. Psychological research has shown that cardinal ratings (on a numerical or Likert scale, for instance) convey more information than ordinal rankings in measuring human opinion. Cardinal methods can satisfy
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