[EM] replies to recent EM posts re apportionment & voting-design puzzle

Warren Smith wds at math.temple.edu
Sun Jan 21 12:52:35 PST 2007

> Benham: Right. And how does a voter express an "infinitesimal" preference in
the Range 0-99 that you advocate?

--sorry, when I speak of "range voting" in mathematical analysis, I almost always mean
"continuum range voting" where all real numbers in [0,1] are castable votes.
As far as I am concerned, restriction to discrete sets such as {0,1,...,99} is
not really a good idea and is only done for reasons of practicality (interface with
old voting machines, etc).  I therefore prefer it if more and more 9s are allowed.  There is some
reason to believe (in fact, precisely the sort of reason Benham speaks of) that about six 9s
may be desirable.

One of the points of continuum range voting is that you can ADJUST your votes
CONTINUOUSLY, allowing ARBITRARILY SLIGHT prefrences to be expressed in an
arbitrarily slight manner.  With Condorcet, etc [rank based] systems, you are
"discontinuous" and are FORCED to express preferences at strength 1 or 0, with no
intermediate strength permitted.  Therefore, you get problems.  One problem is,
you have to make clones with slight preferences, act as though
they are full preferences, while with range voting, the appropriate defn of cloning is
to actually use equal (or very nearly) clones.
This is not a "problem" for range voting, it is an "advantage."  
Appreciate it.

Further, in case you do not appreciate it enough, see my paper #59 here
titled "Voting schemes based on candidate-orderings or discrete choices regarded as harmful"
in which it is argued that ALL voting schemes with discontinuous "votes" are
inherently handicapped and indeed we can construct counterexample elections in which every such
voting scheme must elect "wrong" winners (including pessimal ones).
It's an interesting, amusing, and short paper.

> Bishop: computer studies of US census

Interesting.  I commend to Bishop, the new apportionment schemes described in
http://rangevoting.org/NewAppo.html .
Although Bishop failed to find better behavior (by his Spearman measure) for "Ossipoff"
(it was not said exactly what "Ossipoff" was) versus Webster, Bishop will definitely
find superior behavior for the scheme advocated at the beginning of
versus Webster - it is merely a matter of wat the best value of the magic constant "d" is.
Bishop can search for the best value of d with 0<d<1 to optimize his Spearman measure.

> Malkevitch: what role does "1/e" play in Ossipoff formula? What is 0^0?

I can answer these (since http://rangevoting.org/NewAppo.html rederives the Ossipoff formula,
among other things)
    1/e plays a very nice role.  The point is that a & b are consecutive integers a=n, b=n+1
and limit behavior as n->infinity of
       (n+1)^(n+1) / n^n
       e * (n + 1/2 - 24/n + O(n^(-2))).
So the e is exactly what we need to make the Ossipoff breakpoint tend to n+1/2,
the same as Webster.
And use 0^0 = 1,  it comes from   lim(as n->0+) n^n = 1.

Warren D Smith

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