[EM] Even more complex Sainte-Lague: showing that compromising is fixed

[Kristofer Munsterhjelm]

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.