[EM] Sainte-Lage, continued

[MIKE OSSIPOFF]

I said that SL is the unbiased monotonic seat allocation method. It's also 
the most proportional allocation. How are those statements justified?:

When we divide each party's votes by the same quota, and round off to the 
nearest seat, we're putting each party as close as possible, in its 
seat-count, to what is called for by the ideal uniform votes/seat ratio 
embodied by the quota.

I'll call that ideal votes/seat ratio "the middle". When I speak of a 
party's distance from the middle, I'm referring to the factor by which its 
seat-count differs from the ideal seat number that the quota would give it, 
before rounding off.

Sainte-Lague is the only method that has this property: If, starting with an 
SL seat allocation, we take a seat from one party and give it to another, 
that will inevitably cause those two parties' votes/seat to differ by a 
greater factor than they differed before the transfer. Why?:

We take a seat from A and give it to B.

Say A & B are on opposite sides of the middle.

Say A has fewer seats than the fractional amount that the middle would give 
t, and that B has more than the middle would give it. Obviously, then, 
taking a seat from A and giving it to B increases the factors by which each 
differs from the middle, and therefore from eachother.

Say A has more seats than the middle would give it, and that B has fewer 
than the middle would give it. A & B are already as close to the middle as 
possible, so when we transfer that seat, and they trade sides across the 
middle, each will be farther from the middle than before, differing from it 
by a greater factor, and therefore differing from eachother by a greater 
factor.

Say A & B are on the same side of the middle, with more seats than the 
middle calls for. If A is closer to the middle, giving the seat will put 
itit across the middle which will make it farther from the middle than it 
was, since it was already as close as it could be, in seat-count.  Giving 
its seat to B will put B farther from the middle. Obviously that increases 
the factor by which A and B differ, in their votess/seat. If A is farther 
from the middkle than B is.

Say A & B are on the same side of the middle with fewer seats than the 
middle calls for. By giving a seat to B, A puts itself even farther from the 
middle. By receiving the seat, B crosses the middle, because, again, the 
parties were already as close as they could be to the middle, in seat-count. 
Obviously this puts the parties farther apart.

So in every case, a seat transfer increases the factor by which the 2 
parties' votes/seat differ.

Someone might say, "But when A gives a seat to B, might that not make A 
closer to some C?" Sure. Could there be an allocation for which that 
couldn't be said? But, since we're talking about A giving a seat to B, 
what's relevant is the relative deservingness of A and B. How they compare 
to eachother.

Now, those two parties, A and B could be a large party and a small party. 
Therefore, even with large vs small parties, SL has every pair 
proportionally right with respect to seat transfers between the pair. SL is 
unbiased.

d'Hondt, which rounds down instead of rounding off, thereby lowers small 
parties by a greater factor, and therefore favors large parties over small, 
when compared to SL. Therefore d'Hondt is biased in favor of large parties.

Largest Remainder is unbiased, as long as there's no thereshold. But it 
sporadically unnecessarily deviates from proportioniality. Some say that LR 
is more obvious and intuitive than SL. No. LR, when the Hare quota doesn't 
give the right number of seats, goes to another procedrue, abandoning the 
use of a quota. SL merely picks a quote that does exactly what the Hare 
quota was suspposed to do.

to be continued...

Mike Ossipoff

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