Positional voting

The steeper the initial decline in preference values with descending rank, the more polarised and less consensual the positional voting system becomes.

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.

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.

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.

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.

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.

These particular descending rank-order weightings are in fact used in N-candidate positional voting elections to the Nauru parliament.

[2][3] For such electoral systems, the weighting (wn) allocated to a given rank position (n) is defined below; where the value of the first preference is a.

For a four-candidate election, the Dowdall point distribution would be this: This method is more favourable to candidates with many first preferences than the conventional Borda count.

[5] Simulations show that 30% of Nauru elections would produce different outcomes if counted using standard Borda rules.

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.

Lower preferences are more influential in election outcomes where the chosen progression employs a sequence of weightings that descend relatively slowly with rank position.

In fact, the consecutive weightings of any digital number system can be employed since they all constitute geometric progressions.

As it has the smallest radix, the rate of decline in preference weightings is slowest when using the binary number system.

For a slower descent of weightings than that generated using the binary number system, a common ratio greater than one-half must be employed.

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.

Unranked single-winner methods that can be analysed as positional voting electoral systems include: And unranked methods for multiple-winner elections (with W winners) include: 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.

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.

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.

As an irrelevant alternative (loser), whether B enters the contest or not should make no difference to A winning provided the voting system is IIA compliant.

Regardless of the specific points awarded to the rank positions of the preferences, there are always some cases where the addition or deletion of an irrelevant alternative alters the outcome of an election.

The voters cast their ranked ballots as follows: The election outcome is hence: Given equal support, there is an evitable tie for first place between A and B.

Note that if A signals to its own supporters to always prefer B2 over B1 in a tit-for-tat retaliation then the original tie between A and ‘team’ B is re-established.

To a greater or lesser extent, all positional voting systems are vulnerable to teaming; with the sole exception of a plurality-equivalent one.

[1] Donald G. Saari has published various works that mathematically analyse positional voting electoral systems.