Kotze-Pereira transformation: Difference between revisions
Scale invariance
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==Scale Invariance for RRV==
Aside from making approval systems be aple to be run on score ballots there is an interesting effect on [[Reweighted Range Voting]]. This example will use Jefferson RRV because it's marginally easier to work with, but it's basically the same with any divisors.
Let's say we have 5 candidates to elect and there are multiple A and B candidates, and we have the following approval ballots.
200 voters: A
100 voters: B
In this case A would win 3, B would win 1 out of the first 4, and there would be a tie for the final seat. After the initial 4 have been elected, the weighted approvals for A would be 200/4 = 50, and for B it would be 100/2 = 50. So a tie. But now let's move onto score voting (out of 10)
200 voters: A1=10, A2=10, A3=9, A4=9
100 voters: B1=10, B2=9
A1, A2, and B1 would be elected as the first three. This gives us the proportional 2:1 ratio and this is the same as approval currently (because they had max scores). The A votes are now worth 1/3, and the B votes worth 1/2. A3 would be the next elected. But because of the score of 9, there isn't a full deweighting. Each A vote would now be worth 1/(1 + 29/10) = 10/39 or about 0.256. Both A4 and B2 have been given a score of 9 by their voters. The total score for A4 is 200 x 0.9 x 10/39 = 46.15. For B2 it is 100 x 0.9 x 0.5 = 45. So A4 would be elected instead of there being a tie. OK, so it's only a tie being broken, but if there were 101 B voters, then B2 should win the final seat but 101 x 0.9 x 0.5 = 45.45, so it wouldn't be enough to take the final seat. The wrong candidate would win the final seat.
With KP, the initial ballots would be transformed to:
200 ballots: A1, A2
1800 ballots: A1, A2, A3, A4
100 ballots: B1
900 ballots: B1, B2
Imagine A1, A2, B1, A3 are already elected. The new weighted total approvals for A4 would be 1800 x 1/4 = 450. For B2 it would be 900 x 1/2 = 450. This is the exact tie we want. So scale invariance is preserved.
==Further Reading==
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