ranked voting
The term ranked voting (also known as preferential voting or ranked choice voting) refers to any voting system in which voters ranking, rank their candidates (or options) in a sequence of first or second (or third, etc.) on their respective ball ...
electoral system
An electoral system or voting system is a set of rules that determine how elections and Referendum, referendums are conducted and how their results are determined. Electoral systems are used in politics to elect governments, while non-political ...
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
The Eurovision Song Contest (), sometimes abbreviated to ESC and often known simply as Eurovision, is an international songwriting competition organised annually by the European Broadcasting Union (EBU), featuring participants representing pr ...
) or it may form a mathematical sequence such as an arithmetic progression (
Borda count
The Borda count is a family of positional voting rules which gives each candidate, for each ballot, a number of points corresponding to the number of candidates ranked lower. In the original variant, the lowest-ranked candidate gets 0 points, the ...
), a geometric one (
positional number system
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
) 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
Score voting or range voting is an electoral system for single-seat elections, in which voters give each candidate a score, the scores are added (or averaged), and the candidate with the highest total is elected. It has been described by various ...
: in the former, the score that each voter gives to each candidate is uniquely determined by the candidate's rank; in the latter, the each voter is free to give any score to any candidate.
Voting and counting
In positional voting, voters complete a
ranked ballot
A ranking is a relationship between a set of items such that, for any two items, the first is either "ranked higher than", "ranked lower than" or "ranked equal to" the second.
In mathematics, this is known as a weak order or total preorder of ...
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 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 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
The Eurovision Song Contest (), sometimes abbreviated to ESC and often known simply as Eurovision, is an international songwriting competition organised annually by the European Broadcasting Union (EBU), featuring participants representing pr ...
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 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 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.
Example
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, 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.
Point distributions
For positional voting, any distribution of points to the rank positions is valid provided that they are common to each ranked ballot and that two essential conditions are met. Firstly, the value of the first preference (highest rank position) must be worth more than the value of the last preference (lowest rank position). Secondly, for any two adjacent rank positions, the lower one must not be worth more than the higher one. Indeed, for most positional voting electoral systems, the higher of any two adjacent preferences has a value that is greater than the lower one so satisfying both criteria.
However, some non-ranking systems can be mathematically analysed as positional ones provided that implicit ties are awarded the same preference value and rank position; see
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
.
The classic example of a positional voting electoral system is the
Borda count
The Borda count is a family of positional voting rules which gives each candidate, for each ballot, a number of points corresponding to the number of candidates ranked lower. In the original variant, the lowest-ranked candidate gets 0 points, the ...
. 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 (Nth) preference that is worth just 1 point. So, for example, the points are respectively 4, 3, 2 and 1 for a four-candidate election.
Mathematically, the point value or weighting (wn) 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'.
::wn = a-(n-1)d where 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 a Borda count election will result in identical candidate rankings.
The consecutive Borda count weightings form an
arithmetic progression
An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
. An alternative mathematical
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
known as a
geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For e ...
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 the first preference is ‘a’, the weighting (wn) awarded to a given rank position (n) is defined below.
::wn = arn-1 where 0 ≤ r < 1
For example, the sequence of consecutively halved weightings of 1, 1/2, 1/4, 1/8, … as used in the
binary number
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one).
The base-2 numeral system is a positional notatio ...
system constitutes 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
Plurality voting refers to electoral systems in which a candidate, or candidates, who poll more than any other counterpart (that is, receive a plurality), are elected. In systems based on single-member districts, it elects just one member per ...
.
Alternatively, 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 (wn) allocated to a given rank position (n) is defined below; where the value of the first preference is ‘a’.
::wn = a2/(a+(n-1)d) = a/(1+(n-1)d/a) where w1 = a2/(a+(1-1)d) = a
For the Nauru (
Dowdall
Dowdall is an Irish surname. The earliest forms of spelling were: Dowdell, Dowdale and Dowdle. Dowdall was first used as a surname in Yorkshire, certainly by the time of the Norman conquest of England.
The Irish Dowdalls came from the valley of Do ...
) 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, …).
Apart from these three standard types of mathematical progression (arithmetic, geometric and harmonic), there are countless other sequences that may be employed in positional voting. The two validity criteria only require that a sequence monotonically decreases with descending rank position. Such a sequence is a ‘strict’ one when no two adjacent weightings are equal in value. There are many integer sequences that increase monotonically so by taking the reciprocal of each integer a monotonically decreasing sequence is thereby generated. For example, taking the reciprocal of every number in the
Fibonacci sequence
In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...
(excepting the 0 and 1 start numbers) produces a valid positional voting sequence of 1, 1/2, 1/3, 1/5, 1/8 and so on.
Mathematical progression formulas are needed to define the preference weightings of a positional voting electoral system where the number of options or candidates is undefined or unlimited. In actual elections however the number of preferences is finalised in advance of voting so an arbitrary weighting may be assigned to each rank position provided that the resulting sequence is valid. A classic example of this approach is the unique positional voting system used in the
Eurovision Song Contest
The Eurovision Song Contest (), sometimes abbreviated to ESC and often known simply as Eurovision, is an international songwriting competition organised annually by the European Broadcasting Union (EBU), featuring participants representing pr ...
. Here, the value ‘a’ of a first preference is worth 12 points while 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. Although this sequence of preferences is monotonic as all valid ones must be, it is not a ‘strict’ one as all the lowest weightings are equal in value (zero). Like the Nauru system, this method is sometimes referred to as a 'variant' of the Borda count.
Comparison of progression types
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 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:
*Borda count (where a = N = 10 and d = 1)
*Binary number system (where a = 1 and r = 1/2)
*Nauru method (where a = 1 and d = 1)
*Eurovision Song Contest (non-zero preferences only)
To aid comparison, the actual weightings have been normalised; namely that 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 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, the binary, ternary, octal and decimal number systems use a
radix
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...
‘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 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 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.
Analysis of non-ranking systems
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 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 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:
*
Plurality voting
Plurality voting refers to electoral systems in which a candidate, or candidates, who poll more than any other counterpart (that is, receive a plurality), are elected. In systems based on single-member districts, it elects just one member per ...
(FPTP): The most preferred option receives 1 point; all other options receive 0 points each. =1*
Anti-plurality voting
Anti-plurality voting describes an electoral system in which each voter votes ''against'' a single candidate, and the candidate with the fewest votes against wins. Anti-plurality voting is an example of a positional voting method.
Example
In ...
: The least preferred option receives 0 points; all other options receive 1 point each. =N-1
And unranked methods for multiple-winner elections (with W winners) include:
*
Single non-transferable vote
Single non-transferable vote or SNTV is an electoral system used to elect multiple winners. It is a generalization of first-past-the-post, applied to multi-member districts with each voter casting just one vote. Unlike FPTP, which is a single-winn ...
: The most preferred option receives 1 point; all other options receive 0 points each. =1*
Limited voting
Limited voting (also known as partial block voting) is a voting system in which electors have fewer votes than there are positions available. The positions are awarded to the candidates who receive the most votes absolutely. In the special case ...
: The X most preferred options (where 1 < X < W) receive 1 point each; all other options receive 0 points each. =X* Bloc voting: The W most preferred options receive 1 point each; all other options receive 0 points each. =W
In
approval voting
Approval voting is an electoral system in which voters can select many candidates instead of selecting only one candidate.
Description
Approval voting ballots show a list of the options of candidates running. Approval voting lets each voter i ...
, voters are free to favour as many or as few candidates as they wish so F is not fixed but varies according to the individual ranked ballots being cast. As rank positions would then have different weightings on different ballots, approval voting is not a positional voting system; nor can it be analysed as such.
Comparative examples
Where wn is the weighting of the nth preference, the following table defines the resultant tally calculation for each city:
For a first preference worth w1 = 1, the table below states the value of each of the four weightings for a range of different positional voting systems that could be employed for this election:
These five positional voting systems are listed in progression type order. The slower the decline in weighting values with descending rank order, the greater is the sum of the four weightings; see end column. Plurality declines the fastest while anti-plurality is the slowest.
For each positional voting system, the tallies for each of the four city options are determined from the above two tables and stated below:
For each potential positional voting system that could be used in this election, the consequent overall rank order of the options is shown below:
This table highlights the importance of progression type in determining the winning outcome. With all voters either strongly for or against Memphis, it is a very ‘polarized’ option so Memphis finishes first under plurality and last with anti-plurality. Given its central location, Nashville is the ‘consensus’ option here. It wins under the Borda count and the two other non-polarized systems
Consistency
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
*
Participation
Participation or Participant may refer to:
Politics
*Participation (decision making), mechanisms for people to participate in social decisions
*Civic participation, engagement by the citizens in government
*e-participation, citizen participation ...
Monotonicity
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
*
Pareto efficiency
Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engine ...
But it fails to satisfy the following criteria:
*
Independence of Irrelevant Alternatives
The independence of irrelevant alternatives (IIA), also known as binary independence or the independence axiom, is an axiom of decision theory and various social sciences. The term is used in different connotation in several contexts. Although it a ...
(IIA)
*
Independence of Clones
In voting systems theory, the independence of clones criterion measures an election method's robustness to strategic nomination. Nicolaus Tideman was the first to formulate this criterion, which states that the winner must not change due to the ...
(IoC)
*
Condorcet winner
An electoral system satisfies the Condorcet winner criterion () if it always chooses the Condorcet winner when one exists. The candidate who wins a majority of the vote in every head-to-head election against each of the other candidatesthat is, a ...
*
Condorcet loser
In single-winner voting system theory, the Condorcet loser criterion (CLC) is a measure for differentiating voting systems. It implies the majority loser criterion but does not imply the Condorcet winner criterion.
A voting system complying wit ...
(except the Borda count)
*
Reversal symmetry
Reversal symmetry is a voting system criterion which requires that if candidate A is the unique winner, and each voter's individual preferences are inverted, then A must not be elected. Methods that satisfy reversal symmetry include Borda count, r ...
(except the Borda count)
*
Majority
A majority, also called a simple majority or absolute majority to distinguish it from #Related terms, related terms, is more than half of the total.Dictionary definitions of ''majority'' aMerriam-Webster Arrow’s impossibility theorem, no ranked voting system can satisfy all of the following four criteria when collectively ranking three or more alternatives:
* Non-dictatorship
* Unrestricted domain
*
Pareto efficiency
Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engine ...
*
Independence of Irrelevant Alternatives
The independence of irrelevant alternatives (IIA), also known as binary independence or the independence axiom, is an axiom of decision theory and various social sciences. The term is used in different connotation in several contexts. Although it a ...
(IIA)
Prior to voter preferences being cast, voting systems that treat all voters as equals and all candidates as equals pass the first two criteria above. So, like any other ranking system, positional voting cannot pass both of the other two. It is
Pareto efficient
Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engin ...
but is not independent of irrelevant alternatives. This failure means that the addition or deletion of a non-winning (irrelevant) candidate may alter who wins the election despite the ranked preferences of all voters remaining the same.
IIA example
Consider a positional voting election with three candidates A, B and C where a first, second and third preference is worth 4, 2 and 1 point respectively. The 12 voters cast their ranked ballots as follows:
The election outcome is hence:
Therefore, candidate A is the single winner and candidates B and C are the two losers. As an irrelevant alternative (loser), whether B enters the contest or not should make no difference to A winning provided the voting system is IIA compliant.
Rerunning the election without candidate B while maintaining the correct ranked preferences for A and C, the 12 ballots are now cast as follows:
The rerun election outcome is now:
Given the withdrawal of candidate B, the winner is now C and no longer A. Regardless of the specific points awarded to the rank positions of the preferences, there are always some cases where the addition or deletion of an irrelevant alternative alters the outcome of an election. Hence, positional voting is not IIA compliant.
IoC example
Positional voting also fails the
independence of clones
In voting systems theory, the independence of clones criterion measures an election method's robustness to strategic nomination. Nicolaus Tideman was the first to formulate this criterion, which states that the winner must not change due to the ...
(IoC) criterion. The
strategic nomination Strategic nomination refers to the entry of a candidate into an election with the intention of changing the ranking of other candidates. The name is an echo of ‘tactical voting’ and is intended to imply that it is the candidates rather than the ...
of clones is quite likely to significantly affect the outcome of an election and it is often the intention behind doing so. A clone is a nominally identical candidate to one already standing where voters are unable to distinguish between them unless informed as to which of the two is the clone. As tied rankings are not permitted, these two candidates must be ranked by voters in adjacent positions instead. Cloning may well promote or demote the collective ranking of any non-cloned candidate.
Consider a positional voting election in which three candidates may compete. There are just 12 voters and a first, second and third preference is worth 4, 2 and 1 point respectively.
In this first scenario, two candidates A and B are nominated but no clone enters the contest. The voters cast their ranked ballots as follows:
The election outcome is hence:
Given equal support, there is an evitable tie for first place between A and B.
Suppose B, anticipating this tie, decided to enter a clone of itself. The nominated candidates are now A, B1 and B2. As the voters are unable to distinguish between B1 and B2, they are just a likely to rank B1 over B2 as to prefer B2 over B1. In this second scenario, the 12 ballots are now cast as follows:
The new election outcome is now:
By adding a clone of itself, B has handed victory to candidate A. This counter-productive ‘spoiler’ effect or act of self-harm is called
vote-splitting
Vote splitting is an electoral effect in which the distribution of votes among multiple similar candidates reduces the chance of winning for any of the similar candidates, and increases the chance of winning for a dissimilar candidate.
Vote spl ...
.
To promote itself into first place, B should instead instruct all its supporters to always prefer one of its candidates (say B1) over the other (B2). In this third scenario, the 12 ballots are now cast as follows:
The revised election outcome is now:
By ‘team’ B signalling to its own supporters - but not to A supporters - which of its two candidates it wants to win, B has achieved its objective of gaining victory for B1. With no clone, A and B tie with equal numbers of first and second preferences. The introduction of clone B2 (an irrelevant alternative) has pushed the second preferences for A into third place while preferences for ‘team’ B (B or B1) are unchanged in the first and third scenarios. This wilful act to ‘bury’ A and promote itself is called teaming. Note that if A signals to its own supporters to always prefer B2 over B1 in a tit-for-tat retaliation then the original tie between A and ‘team’ B is re-established.
To a greater or lesser extent, all positional voting systems are vulnerable to teaming; with the sole exception of a plurality-equivalent one. As only first preferences have any value, employing clones to ‘bury’ opponents down in rank never affects election outcomes. However, precisely because only first preferences have any value, plurality is instead particularly susceptible to vote-splitting. To a lesser extent, many other positional voting systems are also affected by ‘spoiler’ candidates. While inherently vulnerable to teaming, the Borda count is however invulnerable to vote-splitting.
Notes
Donald G. Saari
Donald Gene Saari (born March 1940) is an American mathematician, a Distinguished Professor of Mathematics and Economics and former director of the Institute for Mathematical Behavioral Sciences at the University of California, Irvine.
His resear ...
has published various works that mathematically analyse positional voting electoral systems. The fundamental method explored in his analysis is the Borda count.