User:Mencor/Positional Voting

Positional Voting

A positional voting system is a ranked voting method in which the options receive points based on their rank position on each ballot and the option with the most points overall wins.

Voting and counting

In positional voting systems, 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.

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.

Arithmetic

The classic example of a positional voting system is the Borda count in which an arithmetic progression of decreasing points is distributed to declining rank positions. Typically, for a single-winner election with N competing 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. There is therefore a common difference of one point between adjacent rank position values.

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

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.

Geometric

An alternative approach is to distribute points according to a geometric progression instead of an arithmetic one. Such positional voting systems employ a common ratio rather than a common difference and are known as geometric voting methods. For example, using a common ratio of one half, the points for a four-candidate election would typically be 8, 4, 2 and 1 for the first to last preferences respectively. This particular system is known as consecutively halved positional voting (CHPV) as each rank position is worth half that of the preceding one.

To qualify as a geometric voting system, a common ratio (r) in the range 0 ≥ r > 1 is required. Mathematically, the weighting (w_n) associated with a given rank position (n) is defined below; where the weighting of the first preference is 'a'.

w_n = ar^n-1

The weightings of the first, second, third, fourth and last preferences are then worth a, ar, ar², ar³ and ar^N-1 for an N-candidate election. For counting purposes, any suitable value for the first preference may be employed as the overall ranking of the candidates is unaffected by this initial weighting. However, the choice of the common ratio greatly affects an election outcome.

Geometric voting with a common ratio of zero (r = 0) is directly equivalent to the first-past-the-post (FPTP) system. The Borda count and geometric voting with a common ratio approaching one (r → 1) also produce equivalent candidate rankings; but not tallies. CHPV adopts the central common ratio (r = ½) between its two extreme values and it has features that are also intermediate between the 'consensus' Borda count and the 'polarized' FPTP systems. Consecutively halved positional voting has critical threshold properties in relation to teaming and vote splitting.

Other

For the Nauru parliament, N-candidate elections use a positional voting system where descending rank order preferences are allocated fractional values of w_n = 1/n or 1/1, 1/2, 1/3, 1/4 and so on down to 1/N. Although the denominators form an arithmetic progression, this sequence of fractional weightings is neither arithmetic nor geometric. 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. Although such variants are valid positional voting systems, they are not strictly Borda count ones as there is no common difference between adjacent rank position weightings.

Analysis of non-ranking systems

Although not categorised as positional voting 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 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.

Note

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

External links

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