# [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

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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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