Single distributed vote: Difference between revisions
m
→Variants
Dr. Edmonds (talk | contribs) (→Procedure: formatting) |
m (→Variants) |
||
(22 intermediate revisions by 9 users not shown) | |||
Line 1:
'''Single distributed vote''' (SDV) is an [[electoral system]] that extends the concept of [[
==Procedure==
[[File:SDV Procedure.svg|thumb]]
#
</math>
===Verbal Description===
▲# Acquire candidate score form voters and normalize them in [0,1] into a matrix. Voter v scores candidate c with score <math>S_{v,c}</math>.
▲# Determine candidate with max summed (over voters) score. He is elected.
▲# 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.
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
===Distribution Example===
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
<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
<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>\
<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}
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.
Line 47 ⟶ 40:
==Variants==
As with all [[
▲ \frac{S_{v,c}^2 }{ S_{v,c} + \Sigma_{winners j} S_{v,j}}
==Generalization==
The most general form of the
<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
This rule is that for each voter the sum
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.
Line 85 ⟶ 70:
ie if <math> S_{v,c} = 0 </math> then
<math>\frac{
▲\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
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
With A = 0 which is equal to S_{v,c} * (S_{v,c}) / (S_{v,c} + (E/B)*SUM_{winners j} S_{v,j})▼
<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})}{ \times 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
===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
{| class="wikitable"
|-
!
|-
|
|-
|
|}
The 3/4 threshold systems are
▲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]]
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 [[Kotze-Pereira_transformation#Scale_Invariance_example_for_RRV | here]].▼
▲One clear benefit is that [[Single distributed vote]]
==References==
<references />
[[Category:Cardinal voting methods]]
[[Category:Proportional voting methods]]
[[Category:Multi-winner voting methods]]
[[Category:Cardinal PR methods]]
[[Category:Highest averages-reducing voting methods]]
|