[EM] Joe--thanks fsor comments

[Michael Ossipoff]

It's true that not everyone puts so much importance on transfer properties. 
Joe mentioned that they aren't used in Europe. And, in apportionment, 
there's just no excuse for allowing bias, when it can be avoided. We can't 
justify intentionally systematically giving more seats per quota to small 
states than to big states.

If only the 5 traditional methods have the transfer properties, then that 
means that the transfer properties are incompatible with unbias. So, drop 
the transfer properties.

And bias isn't something that different people will disagree on. If, in PR, 
very large parties have an incentive to split, or very small parties have an 
incentive to coalesce, in order to maximze their s/q, then we'd all agree 
that bias is there. Hill is biased. No one would deny that.

Of my 4 methods, only one  isn't a "divisor method" in the strictest sense. 
Adjusted-Rounding doesn't have an unchanging roundoff point for each cycle, 
and I suppose that disqalifies it from being  a strict divisor method. But 
it's a divisor method in the sense that it differs from the other one only 
by not having a fixed rounding point in each cycle, because each cycle's 
rounding point is instead chosen so as to give that cycle an over s/q of 1, 
as nearly as possible.

Joe wrote:

Another thing mathematics can do is for each "reasonable" method Z find a 
list of axioms that are satisfied by method Z but no other method. When one 
has such axioms for each method that one might think is appealing one can 
see more clearly what one gains and loses by adopting one method rather than 
another.

I reply:

Of course that's what we've been doing here, with regard to single-winner 
methods, except that we use the term "criteria" instead of "axioms".

I claim that the only critreion that is absolutely crucial in apportionment 
is unbias. Cycle-Webster and Adjusted-Rounding achieve it independent of the 
frequency distribution. Weighted Bias-Free tries to take the distribution 
into account to avoid measured correlation between q and s/q. Ordinary 
Bias-Free doesn't try to stop distribution-caused q  & s/q correlation, but 
I've said that there's a good case for saying that when it's 
disribution-caused, that isn't unfair in the sense that it's unfair when 
it's method-caused.

Mike Ossipoff