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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