Positional Voting System - Point Distributions

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 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 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 (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) 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 = a-(n-1)d

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.

For the Nauru parliament, N-candidate elections use a positional voting system where descending rank order preferences are allocated fractional values of 1/1, 1/2, 1/3, 1/4 and so on down to 1/N. The Eurovision Song Contest also uses a unique positional voting system. 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 subsequent preferences receive zero points. Like the Nauru system, this voting method is sometimes referred to as a 'variant' of the Borda count.