Single distributed vote: Difference between revisions
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'''Single distributed vote''' (SDV) is an [[electoral system]] that extends the concept of [[
==Procedure==
[[File:SDV Procedure.svg|thumb]]
#Acquire candidate scores form voters and represent them in a matrix. Voter, v, rates candidate, c, with score, <math>S_{v,c}</math>.
#
▲# Update the entire score matrix. The v,c entry of the new matrix is
▲ \frac{S_{v,c}^2 }{ S_{v,c} + 2 \times \Sigma_{winners j} S_{v,j}}
▲#: where <math>S_{v,c}</math> is the '''original''' score matrix not the one from the prior round.
▲# Go back to step 2 until desired number of winners have been elected.
===Verbal Description===
The first winner is the [[
===Distribution Example===
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To illustrate how ballot weight is distributed it will now be shown how the scores are distributed across candidates. The formula for score matrix reweighting can be properly viewed as the original score matrix times a score dependant ballot weight.
<math>\frac{S_{v,c}^2 }{ S_{v,c} + 2 \times \Sigma_{winners j} S_{v,j}} = S_{v,c} \times \frac{S_{v,c}}{ S_{v,c} + 2 \times \Sigma_{winners j} S_{v,j}} </math>▼
▲\frac{S_{v,c}^2 }{ S_{v,c} + 2 \times \Sigma_{winners j} S_{v,j}} = S_{v,c} \times \frac{S_{v,c}}{ S_{v,c} + 2 \times \Sigma_{winners j} S_{v,j}}
In a 5 winner race, the following example will show the progression of redistribution at each step for a single voter. This voter scored the winners
<math>\frac{S_{v,c}}{ S_{v,c} + 2 \times \Sigma_{winners j} S_{v,j}}</math>▼
▲\frac{S_{v,c}}{ S_{v,c} + 2 \times \Sigma_{winners j} S_{v,j}}
When selecting the first winner there are no prior winners so the ballot weight formula reduces to
<math>\
Note 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 [[
<math>\frac{S_{v,c}}{ S_{v,c} + 2 \times \Sigma_{winners j} S_{v,j}} = \frac{
▲\frac{S_{v,c}}{ S_{v,c} + 2 \times \Sigma_{winners j} S_{v,j}} = \frac{0.7}{ 0.7 + 2 \times 0.4} = \frac{7}{ 15}
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. Another, example can be seen pictorially in the accompanying image.
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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==Variants==
As with all [[
▲ \frac{S_{v,c}^2 }{ S_{v,c} + \Sigma_{winners j} S_{v,j}}
==Generalization==
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The most general form of the reweighting of the score matix <math>S_{v,c}</math> is
<math>\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}}</math>▼
▲\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}}
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
▲ 1 = \frac{A + B \times S_{v,c}}{C + D \times S_{v,c} }
So we know A = C and B =D. This gives
<math>\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}}</math>▼
▲\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}}
In this form [[Reweighted Range Voting]] is when A=1 and B=0 and [[
==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 propose SPAV. However, such systems have later been characterized this way using the concept of Maximin Support. <ref>https://arxiv.org/abs/1609.05370</ref>
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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ie if <math> S_{v,c} = 0 </math> then
<math>\frac{(A + B \times S_{v,c})}{ A + B \times S_{v,c} + E \times \Sigma_{winners j} S_{v,j}} = 0</math>▼
▲\frac{(A + B \times S_{v,c})}{ A + B \times S_{v,c} + E \times \Sigma_{winners j} S_{v,j}} = 0
This implies A = 0.
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With A = 0
<math>\frac{S_{v,c} \times (B \times S_{v,c})}{ B \times S_{v,c} + E \times \Sigma_{winners j} S_{v,j}}
▲\frac{S_{v,c} \times (B \times S_{v,c})}{ B \times S_{v,c} + E \times \Sigma_{winners j} S_{v,j}}
which is equal to
▲\frac{S_{v,c} \times ( S_{v,c})}{ S_{v,c} + (E/B) \times \Sigma_{winners j} S_{v,j}}
For simplicity, one can express the ratio of the constants as a single constant E/B = K giving the final form
<math>\frac{S_{v,c}^2 }{ S_{v,c} + K \times \Sigma_{winners j} S_{v,j}}</math>▼
▲\frac{S_{v,c}^2 }{ S_{v,c} + K \times \Sigma_{winners j} S_{v,j}}
===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 [[
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.
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The 3/4 threshold systems are 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 [[Quota]]s.
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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]] down-weights vote power the same or more than [[Reweighted Range Voting]].
One clear benefit is that [[Single distributed vote]]
==References==
<references />
[[Category:Cardinal voting methods]]
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[[Category:Multi-winner voting methods]]
[[Category:Cardinal PR methods]]
[[Category:Highest averages-reducing voting
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