Positional voting

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 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 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.

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.[1] 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.

The classic example of a positional voting electoral system is the Borda count.[1] 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 (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 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.[1]

The consecutive Borda count weightings form an arithmetic progression. 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 the first preference is ‘a’, the weighting (w_n) awarded to a given rank position (n) is defined below.

w_n = ar^n-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 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.

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 (w_n) allocated to a given rank position (n) is defined below; where the value of the first preference is ‘a’.

w_n = a²/(a+(n-1)d) = a/(1+(n-1)d/a) where w₁ = a²/(a+(1-1)d) = a

For the Nauru (Dowdall) 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 (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. 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.

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.

Relative decline in preference weightings with descending rank order for four positional voting electoral systems

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 ‘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.

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.[1] 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 (FPTP): The most preferred option receives 1 point; all other options receive 0 points each. [F=1]
• Anti-plurality voting: The least preferred option receives 0 points; all other options receive 1 point each. [F=N-1]

And unranked methods for multiple-winner elections (with W winners) include:

• Single non-transferable vote: The most preferred option receives 1 point; all other options receive 0 points each. [F=1]
• Limited voting: The X most preferred options (where 1 < X < W) receive 1 point each; all other options receive 0 points each. [F=X]
• Bloc voting: The W most preferred options receive 1 point each; all other options receive 0 points each. [F=W]

In approval voting, 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.

Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities and that everyone wants to live as near to the capital as possible.

The candidates for the capital are:

• Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
• Nashville, with 26% of the voters, near the center of the state
• Knoxville, with 17% of the voters
• Chattanooga, with 15% of the voters

The preferences of the voters would be divided like this:

Where w_n 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

As a class of voting systems, positional voting can be evaluated against objective mathematical criteria to evaluate its strengths and weaknesses in comparison with other single-winner electoral methods.

Positional voting satisfies the following criteria:

• Non-dictatorship
• Unrestricted domain
• Summability (with order N)
• Consistency
• Participation
• Resolvability
• Monotonicity
• Pareto efficiency

But it fails to satisfy the following criteria:

• Independence of Irrelevant Alternatives (IIA)
• Independence of Clones (IoC)
• Condorcet winner
• Condorcet loser (except the Borda count)
• Reversal symmetry (except the Borda count)
• Majority (except when equivalent to plurality)

According to 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
• Independence of Irrelevant Alternatives (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 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.

Donald G. Saari has published various works that mathematically analyse positional voting electoral systems. The fundamental method explored in his analysis is the Borda count.

• Economic Theory, Vol. 15, Issue 1, 2000: Mathematical Structure of Voting Paradoxes: II. Positional Voting, Donald G. SAARI