[EM] metrics on elections
Stephen Turner
smturner0 at yahoo.es
Sun Mar 21 06:42:20 PDT 2010
Hello. It's been quite a while since
I posted here. I have a question: does
anyone have any good pointers
to material on metrics on elections?
A "metric" is as usual, and an
"election" would be simply an
election profile, that is you have
some set S of permitted ballot types,
and so many ballots of type 1, so many of
type 2, ...
Though these sound like subsets-of-S-with-
repetition-allowed, you can view them
as functions from S to N (N=0,1,2,...),
where for x in S, f(x) is the number of
ballots of type x.
So metrics on the set N^S are what is
of interest.
One obvious way to do this is simply to
take metrics on R^#S (R=reals), restricted
to N^S. However S itself has a metric,
so I was really after metrics on N^S
which reflect the metric on S,
rather like the Hausdorff metric gives
a metric on the finite subsets of an
arbitrary metric space.
[I've appended a summary of the
Hausdorff construction.]
In practice, of course, S is a finite set
but I've already found a couple of
constructions on N^S, incorporating
a metric on S, which
turn out to be themselves metrics
on elections under certain more general
conditions (than when S is finite).
The fact that they actually are metrics
matters because it simplifies the
calculations quite a bit.
I don't currently have access to a
university library, so I'd prefer
some specialised on-line resources.
TIA
Stephen Turner
--------------
Hausdorff metric: let M be a metric space
with metric d:MxM->R. If x is in M and
A,B are two non-empty finite subsets of M, we define
the distance from x to B as usual, namely
d(x,B) := min d(x,y)
where the minimum is taken over all y in B.
Then we could define f(A,B) by
f(A,B) := max d(x,B)
where the maximum is taken over all x in A.
Finally we define
d(A,B) = max (f(A,B),f(B,A)), and this
is called the Hausdorff metric on
the set of (non-empty) finite subsets of M.
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