Positional voting
Positional voting 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 or it may form a mathematical sequence such as an arithmetic progression, a geometric one or a harmonic one. 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 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.
Voting and counting
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 are instead required following the count, the 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 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:| Number of ballots | First preference | Second preference | Third preference |
| 24 | A | B | C |
| 18 | A | C | B |
| 12 | B | A | C |
| 16 | B | C | A |
| 20 | C | A | B |
| 10 | C | B | A |
After voting closes, the points awarded by the voters are then tallied and the options ranked according to the points total.
| Option | Points to be tallied | Total | Overall rank |
| A | x 4 + x 2 + x 1 | 258 | First |
| B | x 4 + x 2 + x 1 | 218 | Third |
| C | x 4 + x 2 + x 1 | 224 | Second |
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, so long as the points are weakly decreasing in the rank of each candidate. In other words, a worse-ranked candidate must not receive more points than a better-ranked candidate.Borda (Unbiased)
The classic example of a positional voting electoral system is the Borda count. Typically, for a single-winner election with candidates, a first preference is worth points, a second preference points, a third preference points and so on until the last 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 associated with a given rank position is defined below; where the weighting of the first preference is and the common difference is.
where, the number of candidates.
The value of the first preference need not be. It is sometimes set to 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 or for a Borda count election will result in identical candidate rankings.
The consecutive Borda count weightings form an arithmetic progression.
Top-heavy
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".Plurality voting
Under first-preference plurality, the most-preferred option receives 1 point while all other options receive 0 points each. This is the most top-heavy positional voting system.Geometric
An alternative mathematical sequence known as a geometric progression may also be used in positional voting. Here, there is instead a common ratio between adjacent weightings. In order to satisfy the two validity conditions, the value of must be less than one so that weightings decrease as preferences descend in rank. Where the value of the first preference is, the weighting awarded to a given rank position is defined below.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. 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.
Dowdall system (Nauru)
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. This further mathematical sequence is an example of a harmonic progression. These particular descending rank-order weightings are in fact used in -candidate positional voting elections to the Nauru parliament. For such electoral systems, the weighting allocated to a given rank position is defined below; where the value of the first preference is.where.
For the Nauru system, the first preference is worth one and the common difference between adjacent denominators is also one. Numerous other harmonic sequences can also be used in positional voting. For example, setting to 1 and to 2 generates the reciprocals of all the odd numbers whereas letting be 1/2 and be 1/2 produces those of all the even numbers.
The harmonic variant used by the island nation of Nauru is called the Dowdall system as it was devised by Nauru's Secretary for Justice in 1971. Here, each voter awards the first-ranked candidate with 1 point, while the 2nd-ranked candidate receives a point, the 3rd-ranked candidate receives of a point, etc. When counting candidate tallies in Nauru, decimal numbers rounded to three places after the decimal point are employed rather than fractions. 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:
| Ranking | Candidate | Formula | Points |
| 1st | Andrew | 1/1 | 1.000 |
| 2nd | Brian | 1/2 | 0.500 |
| 3rd | Catherine | 1/3 | 0.333 |
| 4th | David | 1/4 | 0.250 |
This method 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.
Eurovision
The Eurovision Song Contest uses a first preference 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.Sports and awards
Positional voting methods are used in some sports, either for combining rankings in different events or for judging contestants. For instance, points systems are used to keep score in Formula One and for the Major League Baseball Most Valuable Player Award. These applications tend to also be top-heavy: both the F1 and baseball MVP points systems favor the top end.Comparison of progression types
In positional voting, the weightings of consecutive preferences from first to last decline monotonically with rank position. 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
- Binary number system
- Nauru method
- Eurovision Song Contest
The relative decline of weightings in any arithmetic progression is constant as it is not a function of the common difference. In other words, the relative difference between adjacent weightings is fixed at. In contrast, the value of 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 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 of 2, 3, 8 and 10 respectively. The value is also the common ratio of the geometric progression going up in rank order while is the complementary common ratio descending in rank. Therefore, is the reciprocal of and the 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 has to be an integer, the common ratio 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, the slower the decrease in weightings with descending rank.