# [EM] apportionment, some replies to Ossip. and Malk.

Warren Smith wds at math.temple.edu
Fri Jan 19 12:28:38 PST 2007

```> O: I made no attack. I merely stated that B/(q+1) is a function that can
approximate the density function of states over the range of populations. I
don't claim it to be more than a rough approximation.

--well, no.   Non-normalizable "probability distributions" are not probability
distributions, and the do not approximate probability distributions in any reasonable sense.
They are simply something that should never be used.  When one does use them,
that is a symptom of underlying rot in one's thinking.

> O: But, for example, nonmonotonicity was never what bothered me about IRV. It
doesn't bother the IRVists, or the Australians or Irish. Yes, it resulted in
Hamilton's rejection. But Hamilton didn't have the unconditional unbias

--"nonmonotonicity" means two different things for IRV and in apportionment and in the latter
it is far more serious.   Second, it would help if Ossipoff were to define whatever it
is he means by "unconditional unbias", and he also stated to me once he'd proved
"cycle webster" was "unbiased" but again I saw neither proof nor a definition of
what unbias might mean.

> O: I don't know whether or not AR is in the class of methods that B & Y proved
nonmonotonic.

--it is nonmonotonic.  "Divisor methods" are methods where a pop-p state gets round(p/q)
seats, where round is a rounding function that is upward stairstepping with integer heights.
AR is not a divisor method.  (Here q is chosen to make the total #seats come out right and
is called the "divisor".)   Ossipoff's "bias free Webster" (or whatever he prefers to call it)
involving the A^A-using formula for the stairstep locations, IS a divisor method.

The "Ossipoff-Smith method"  (which Ossipoff informs me he still has not bothered to read,
along with whatever else I had to say in that page?!)
is the same except for  different formula for the step locations, which reduces to O's
formula in the limit K-->0, where K=#seats/#states.  I have not called it the "OS method"
in the paper/web page, so O can relax, but I did say on that web page that the limiting
K-->0 case was discovered by O.

> O: A probability distribution can take any shape.

--no it cannot.  Probability distributions are non-negative valued and normalizable.
If you propose somethng disobeying these laws, it is not a probability distribution.  Period.

> O: it better not be adjusted-rounding...

--Mike, the idea that I might be "stealing" your work is just so far off the mark, it
is beyond ridiculous.  Anyhow, I have heard quite enough flaming from you even before
examining any of my work.

> Joe Malkevitch: Ties (using a particular
method) can occur not only when two states have the same population
but also when they have different populations but the house size is a
particular value.

--true.  Every divisor method can fail, i.e. a q-value can fail to exist that yields the
right number of seats.  However, this can only happen in cases of "ties" where
you get "hops by 2" that "skip over" te desired #seats.
These failures are rare and could be avoided by random
tiebreaking.  It is a good point that failures can occur even in the absence of direct
population ties, though (kind of indirect ties).

> JM quoting B&Y: Not every measure of inequality gives stable apportionments: for
some measures there exist problems for which every apportionment can
be improved upon by some transfer. Huntington showed that, except for
four such "unworkable" measures, all others resulted in the methods
of either Adams, Dean, Hill, Webster or Jefferson.

--actually, I really object to this B&Y quote, which was perhaps the most misleading statement
they made in their entire book.   (They also have a bunch of pro-Webster arguments which I find
to be rather "spun" and overmuch, although I agree with them Webster is the best of the
classic 5.)  Huntington did not at all show that
"every measure" yielded the methods of either Adams, Dean, Hill, Webster or Jefferson.
He in fact just did a case-analysis of a few measures resulting from certain
rational functions.  There are an infinity of other possible measures which
Huntington never examined, and there are almost certainly an infinity of other divisor methods
that nobody ever examined, all of which are "workable" in Huntington's sense.
Read the Huntington paper (which by the way is very well written) if you
want to see that:   http://rangevoting.org/Hunti28.pdf

> JM: Evaluation and Optimization of Electoral Systems, by Pietro Grilli di
Cortona, Cecilia Manzi, Aline Pennisi, Federica Ricca, and Bruno
Simeone, SIAM Monographs on Discrete Mathematics and Its

--that sounds interesting.  There is a quick summary of some global optimality statements
in   http://rangevoting.org/Apportion.html
and it is open whether the "unbiased" methods I've been deriving and compiling in
http://rangevoting.org/NewAppo.html
also obey global optimality theorems that are analogous.  I suspect they do and
I have some clue about how to figure them out, but I haven't; maybe this book'd help.
Incidentally, I find the global optimality thm obeyed by Webster to be very pleasing and
I think much more pleasing than the similar-looking statement obeyed by Huntington-Hill
(when you think about it using "quantified morality"; the issue is that
huntington's objective fn regards an unreporesented human to be "infintiely bad"
whereas in reality that is only finitely bad, so his objective function is clearly
pretty morally-distorted, whereas Webster's objective function seems to make a lot
of moral sense).

warren D Smith
http://rangevoting.org

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