Single distributed vote: Difference between revisions

'''Single distributed vote''' (SDV) is an [[electoral system]] that extends the concept of [[Sequentialsequential proportional approval voting]] to [[Scorescore voting]] ballots. It is general [[Cardinal voting systems|Cardinalcardinal voting system]] which reducedreduces to [[Sequentialsequential proportional approval voting]] with [[approval voting]] ballots. Proposed by [[Keith Edmonds]] in 2020,<ref> {{Cite web|url=https://forum.electionscience.org/t/unifying-thiele-and-unitary-philosophy/604|title=Unifying Thiele and Unitary philosophy|last=|first=|date=2020-02-16|website=The Center for Election Science forum|language=en-US|url-status=live|archive-url=|archive-date=|access-date=2020-04-16}}</ref> as a way to improve [[Reweighted Range Voting]] to be more inlinein line with the desire to preserve vote weight. As such, it uses a similar but different vote conserving mechanism to [[Votevote unitarity]]. It a natural extension of the [[Webster]] or [[Jefferson Method]] to a [[Multi-Member System]].

# AcquireDetermine candidate scorewith formmax voterssummed and(over normalizevoters) them in [0,1] into a matrixscore. Voter, v, ratesThat candidate, c, with score,is <math>S_{v,c}</math>elected.

#: </math>\begin{equation}

The first winner is the [[Scorescore voting]] winner. Then we allow ballot weight to be distributed between the first winner and all potential next winners according to the score given. The second winner is the candidate who has the highest sum score when it is down -weighted by the ballot weight they are supported with by each voter. Then we allow ballot weight to be distributed between the first two winnerwinners and all potential next winners according to the score given. The the third winner is the candidate who has the highest sum score when it is down -weighted by the ballot weight they are supported with by each voter. And so on…

In a 5 winner race, the following example will show the progression of redistribution at each step for the a single voter. This voter scored the winners 0.4 | 0.7 | 110 | 0 | 0.2 in the sequence in which they won. The ballot weight formula is

<math>\beginfrac{equationS_{v,c}}{ S_{v,c}} = 1</math>

SoNote that the ballot weight is always a value in [0,1] and will sum to 1 across all candidates for each voter at all times. The score matrix is the original one and the winner is the same as the [[Scorescore voting]] winner. In the next round, the ballot weight is redistributed to the next potential winners in proportion to the score given to them. For the second winner, this amount would be

So atAt the end of this round the second winner has 7/15 of the ballot weight leaving 8 /15 with the first. Following the same formula the third winner has 10/32 of the ballot weight at the end of the third round. This is because 8/32 and 14/32 needed to be left with the first and second winners respectively. Note that later winners can receive more ballot weight if they are scored higher than prior winners. The fourth round has no effect since the voter gave a score of zero to that winner. In the final round, the winner receives 2/44 since the first three winners had to hold 8/44, 14/44 and 20/44 of the ballot weight. Another, example can be seen pictorially in the accompanying image.

As with all [[Highesthighest averages method]] based systems there is a [[D'Hondt method | D'Hondt]] / [[Jefferson method]] as well as a [[Sainte-Laguë method | Sainte-Laguë/Webster method]] variant. The method described above is the [[Sainte-Laguë method | Sainte-Laguë/Webster method]]. The [[D'Hondt method | D'Hondt]] / [[Jefferson method]] variant can be achieved by removal of the 2 giving:

The most general form of the reweigtingreweighting of the score matix <math>S_{v,c}</math> is

A, B, C, D, and E are all constants to be determined. When no winners are elected for a voter we want no reweighting to be applied. This implies

In this form [[Reweighted Range Voting]] is when A=1 and B=0 and [[Singlesingle distributed vote]] is when A=0 and B=1.

A desirable property of a sequential system is to conserved vote power across rounds. The most literal interpretation of this concept is [[Vote unitarity]]. An alternative idea is that instead of spending the amount of ballot it is distributed between previous winners and potential winners. The distribution would follow the rule that the total is preserved. Whichever potential winner has the most distributed vote power available to them wins. This concept is maintained in SPAV and highest average party list systems. Each voter has their vote split between all their approved winners and the next potential one. The formula for their ballot weight is 1/(1+W). It is not clear if this conservation of ballot weight was what lead Thiele to proposingpropose SPAV. However, such systems have later been characterized this way using the concept of Maximin Support. <ref>https://arxiv.org/abs/1609.05370</ref>

RRV gets around this because ballot weight is multiplied by the score so when the score is 0 there is no issue mathematically. RRV under the above conservation theory would assign ballot weight to those where the voter scored them 0. This is thought to be undesirable. It is a theory argument not really a practical one. RRV works it is just hard to motivate from theory. Even when ignoring the zero score candidates the ballot weight distributed to each candidate does not add to 1.

For simplicity, one can express the ratio of the constants as a single constant E/B = K giving the final form

This last undetermined constant, K, was discussed above in the variant section. The value of 2 would give Webster reweighting. This is a common debate when designing a system intended to produce something like [[Proportionalproportional representation]].

The key is to consider what the natural threshold should be in specific scenarios. Consider a 2 seat race with two factions Red and Blue. Assume Red is the larger party so it will win one of the seats. What fraction of votes does Red need to win the second seat.? This all depends on the system but they all come down to two options for threshold.

! Party list !! [[Monroe's method | Quota systems]] !! Psi/Harmonic voting !! [[Single distributed vote]] !! Threshold

| Sainte-Laguë/Webster || Hare || Δ=½ || K=2 || 3/4

| d’Hondt/Jefferson || Droop || Δ=1 || K=1 || 2/3

This system is very similar to Reweighted Range Voting. The logic for the theoretical motivation is laid out above. While the concept of an underlying conserved ballot might be compelling theoretically it is worth considering the practical differences. While results are similar in most situation it is worth noting that [[Single distributed vote]] downweightesdown-weights vote power the same or more than [[Reweighted Range Voting]].