Single distributed vote: Difference between revisions
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'''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 systems|Cardinal voting system]] which reduced to [[Sequential proportional approval voting]] with [[approval voting]] ballots. Proposed by [[Keith Edmonds]] in 2020,<ref> https://forum.electionscience.org/t/unifying-thiele-and-unitary-philosophy/604</ref> as a way to improve [[Reweighted Range Voting]] to be more
==Procedure==
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===Verbal Description===
The first winner is the [[Score 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
===Distribution Example===
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\end{equation}</math>
In a 5 winner race, the following example will show the progression of redistribution at each step for
<math>\begin{equation}
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\end{equation}</math>
So the score matrix is the original one and the winner is the same as the [[Score 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
<math>\begin{equation}
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\end{equation}</math>
So at 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.
When selecting the winner at each round the ballot weight for the next potential winners all depend on the score given to them. The winner is selected as the sum of scores multiplied by the ballot weight from each voter.
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==Generalization==
The most general form of the
<math>\begin{equation}
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\end{equation}</math>
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
<math>\begin{equation}
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==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
This rule is that for each voter the sum of all ballot fractions given to the prior winners is equal to 1 when combined with each potential winner. That is, the winners for each voter get some fraction of the ballot weight and the ballot weight is conserved.
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This implies A = 0.
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.
With A = 0
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\end{equation}</math>
For simplicity, one can express the ratio of the constants as a single constant E/B = K giving the final form
<math>\begin{equation}
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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 Red need to win the second seat
{| class="wikitable"
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==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]]
One clear benefit is that [[Single distributed vote]] is scale
==References==
[[Category:Cardinal voting methods]]
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