To show that the complex Sainte-Lague method automatically does the
strategy that we'd prefer voters not to have to do manually, here's a
rough evaluation of c({A: 6, M: 1, B: 3}, {A: 5, M: 1, B: 4}) for the
party ballots
549: pA
102: pM
349: pB
10: pM > pB > pA
10 seats.
Call the first council cA, the second council cB. We want cB to win so
the 10 preferential voters don't have to strategize.
Imagine pA means "A1>A2>...A10" and the same for pM and pB. Then:
rep(cA, cB, "549: pA"):
The 10 ranking numbers are all for pA members. cA has 6 of them, cB has
5 of them.
so rep(cA, cB, "549: pA") = 549 - 549/f(6)
rep (cB, cA, "549: pA") = 549 - 549/f(5)
Similarly, for the other plumpers:
cA gets 102 - 102/f(1) + 349 - 349 / f(3).
cB gets 102 - 102/f(1) + 349 - 349 / f(4).
For the preferential voters, rep(cA, cB, "10: pM > pB > pA"):
M1 M2 M3 M4 M5 M6 M7 M8 M9 MA B1 B2 B3 B4 B5 B6 B7 B8 B9 BA A1 A2 A3
A4 A5 A6 A7 A8 A9 AA
0 1 2 3 4 5 5 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
23 24 25 26 27 28 29
cA A1 A2 A3 A4 A5 A6 M1 B1 B2 B3
20 21 22 23 24 25 0 10 11 12
cB A1 A2 A3 A4 A5 M1 B1 B2 B3 B4
20 21 22 23 24 0 10 11 12 13
So the ranking numbers that are used are: 0 10 11 12 13 20 21 22 23 24
25
Truncating to 10 (the number of seats), we get: 0 10 11 12 13 20 21 22
23 24
So sA is 9 and sB is 10.
cA gets 10 - 10/f(9), cB gets 10 - 10/f(10).
The grand sum is:
cA: 549 - 549/f(6) + 102 - 102/f(1) + 349 - 349 / f(3) + 10 - 10/f(9)
= 549 - 549/13 + 102 - 102/3 + 349 - 349/7 + 10 - 10/19
= 1527374/1729
= 883.38578
cB: 549 - 549/f(5) + 102 - 102/f(1) + 349 - 349 / f(4) + 10 - 10/f(10)
= 549 - 549/11 + 102 - 102/3 + 349 - 349/9 + 10 - 10/21
= 614578/693
= 886.83694
so cB wins as desired.
Let's also check that it doesn't give everything to the preferential
voters. So let's try giving the B-party yet another seat more:
c({A: 5, M: 1, B: 4}, {A: 4, M: 1, B: 5})
cA:
for the A-voters: 549 - 549 / f(5)
for the M-voters: 102 - 102 / f(1)
for the B-voters: 349 - 349 / f(4)
for the preferential voters:
cA A1 A2 A3 A4 A5 M1 B1 B2 B3 B4
20 21 22 23 24 0 10 11 12 13
cB A1 A2 A3 A4 M1 B1 B2 B3 B4 B5
20 21 22 23 0 10 11 12 13 14
so sA is 9 and sB is 10, so 10 - 10 / f(9)
cB:
for the A-voters: 549 - 549 / f(4)
for the M-voters: 102 - 102 / f(1)
for the B-voters: 349 - 349 / f(5)
for the preferential voters: 10 - 10 / f(10)
cA grand sum: 549 - 549 / f(5) + 102 - 102 / f(1) + 349 - 349 / f(4) +
10 - 10 / f(9)
= 549 - 549 / 11 + 102 - 102 / 3 + 349 - 349 / 9 + 10 - 10 / 19
= 1668046/1881
= 886.78681
cB grand sum: 549 - 549 / f(4) + 102 - 102 / f(1) + 349 - 349 / f(5) +
10 - 10 / f(10)
= 549 - 549 / 9 + 102 - 102 / 3 + 349 - 349 / 11 + 10 - 10 / 21
= 203926/231
= 882.796537
so the preferential voters only get a single seat more.
----
Election-Methods mailing list - see http://electorama.com/em for list info
