Voting and countingEdit
In positional voting, voters cast their preferences using a conventional ranked ballot. For each option, the points corresponding to the voters' preferences are tallied. The option with the most points is the winner. Where a few winners (W) are instead required, then the W highest ranked options are selected.
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.
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 (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 electoral 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 electoral 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 electoral system is sometimes referred to as a 'variant' of the Borda count.
Analysis of non-ranking systemsEdit
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. Despite the absence of ranking here, favoured options are all treated as belonging to the higher of just two rank positions and all remaining options to the lower one. As the higher rank position is awarded a greater value than the lower one, then the two necessary criteria for positional voting are satisfied. Preferences that are given the same rank are not ordered within that rank.
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.
- Anti-plurality voting: The least preferred option receives 0 points; all other options receive 1 point each.
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.
- Limited voting: The X most preferred options (where 1 < X < W) receive 1 point each; all other options receive 0 points each.
- Bloc voting: The W most preferred options receive 1 point each; all other options receive 0 points each.
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.