Single distributed vote: Difference between revisions
add illustrative example
Psephomancy (talk | contribs) (→Variants: Do you just want indentation here?) |
Dr. Edmonds (talk | contribs) (add illustrative example) |
||
Line 14:
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…
==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 candidate and voter dependant ballot weight.
<math>\begin{equation}
\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}}
\end{equation}</math>
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 | 1 | 0 | 0.2 in the sequence in which they won. The ballot weight formula is
<math>\begin{equation}
\frac{S_{v,c}}{ S_{v,c} + 2 \times \Sigma_{winners j} S_{v,j}}
\end{equation}</math>
When selecting the first winner there are no prior winners so the ballot weight formula reduces to
<math>\begin{equation}
\frac{S_{v,c}}{ S_{v,c}} = 1
\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}
\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}
\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.
==Variants==
| |||