Single distributed vote

Single distributed vote (SDV) is an electoral system that extends the concept of Sequential proportional approval voting to Score voting ballots. It is general Cardinal voting system which reduced to Sequential proportional approval voting with approval voting ballots. Proposed by Keith Edmonds in 2020, as a way to improve Reweighted Range Voting to be more inline with the desire to preserve vote weight. As such, it uses a similar but different vote conserving mechanism to Vote unitarity.

Procedure

1. Acquire candidate score form voters and normalize them in [0,1] into a matrix. Voter v scores candidate c with score S_{v,c}.
2. Determine candidate with max summed (over voters) score. He is elected.
3. Update the entire score matrix. The v,c entry of the new matrix is

   Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} \frac{S_{v,c}^2 }{ S_{v,c} + 2 \times \Sigma_{winners j} S_{v,j}} \end{equation}}

   where $S_{v,c}$ is the original score matrix not the one from the prior round.

4. Go back to step 2 until desired number of winners have been elected.

The first winner is the score winner. Then we allow ballot weight to be distributed between the first winner and all potential next winners according to the score given. The 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 winner 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…

Variants

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

                        Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}                          \frac{S_{v,c}^2 }{ S_{v,c} + \Sigma_{winners j} S_{v,j}}                          \end{equation}}

Generalization

The most general form of the rewiegting of the score matix $S_{v,c}$

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} \frac{S_{v,c} \times (A + B \times S_{v,c})}{ C + D \times S_{v,c} + E \times \Sigma_{winners j} S_{v,j}} \end{equation}}

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

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} 1 = \frac{A + B \times S_{v,c}}{C + D \times S_{v,c} } \end{equation}}

So we know A = C and B =D. This gives

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} \frac{S_{v,c} \times (A + B \times S_{v,c})}{ A + B \times S_{v,c} + E \times \Sigma_{winners j} S_{v,j}} \end{equation}}

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

Motivation for specific parameter choice

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 proposing SPAV.

This rule is that the sum over all the reweighted ballots past and potentials = 1 for each voter. That is, the winners for each voter get some fraction of the ballot weight and the ballot weight is conserved.

To continue in the above derivation it would seem logical that when the score given is 0 we do not want the ballot to have any weight assigned to that candidate.

ie if $S_{v,c}=0$ then

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} \frac{S_{v,c} \times (A + B \times S_{v,c})}{ A + B \times S_{v,c} + E \times \Sigma_{winners j} S_{v,j}} = 0 \end{equation}}

This implies A = 0.

RRV gets around this because ballot weight is multiplied by the score so when 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.

With A = 0 which is equal to S_{v,c} * (S_{v,c}) / (S_{v,c} + (E/B)*SUM_{winners j} S_{v,j})

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} \frac{S_{v,c} \times (B \times S_{v,c})}{ B \times S_{v,c} + E \times \Sigma_{winners j} S_{v,j}} \end{equation}}

which is equal to

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} \frac{S_{v,c} \times ( S_{v,c})}{ \times S_{v,c} + (E/B) \times \Sigma_{winners j} S_{v,j}} \end{equation}}

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

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} \frac{S_{v,c}^2 }{ S_{v,c} + K \times \Sigma_{winners j} S_{v,j}} \end{equation}}

Proportionality Threshold

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 Proportional 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 it need to win the second seat. This all depends on the system but they all come down to two options for threshold.

Party list	Quota systems	Psi/Harmonic voting	Single distributed score	Threshold
Sainte-Laguë/Webster	Hare	Δ=½	K=2	3/4
d’Hondt/Jefferson	Droop	Δ=1	K=1	2/3

The 3/4 threshold systems are of related to the common Monroe's method interpretation for multi-member systems. This would mean K is just a tuning parameter and 2 is the correct value to get it to line up with Hare Quotas.

Comparison to Reweighted Range Voting

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 downweighted the same or more than Reweighted Range Voting.

One clear benefit is that Single distributed vote is scale invariant and is not Reweighted Range Voting. The Kotze-Pereira transformation can be used to add scale invariance to Reweighted Range Voting but it is an added complication to the system. The details can be seen on the page here.