[EM] Question about a criterion for ballot counting
Alex Small
alex_small2002 at yahoo.com
Mon Jun 5 21:08:57 PDT 2006
I'm still working on a paper that I alluded to in a post to the list a few weeks ago. In my work, I imposed a condition on election methods that is innocuous for public elections but mathematically somewhat arbitrary: I assumed that the ballot counting rules can be specified with a finite number of linear inequalities:
In practical terms, this is fairly innocuous. For every election method that's been discussed on this forum you have a short list of rules that can be stated as simple inequalities. For instance, with IRV:
1) If a candidate's tally of first-place votes is greater than half the number of ballots cast then elect him.
2) Otherwise, eliminate a candidate, transfer votes, and then see if there is a candidate whose tally of votes is greater than half the number of ballots cast.
And so forth. (I think we could get into some semantic issues about "ballots cast" and truncation and whatnot, but let's leave that aside.)
The rules outlined above are a simple list of statements saying "If [some linear inequality] is true then elect [whichever candidate]." And there's only a finite number of steps in the process.
Mathematically, of course, this is arbitrary. Mathematically you could define all sorts of arbitrary election rules. For instance, in a 2-way race the candidate with an odd number of votes could win. Or you could do something like Borda count, but you could square the number of first-place ballots received by each candidate and then tally up the points from the rest of the ballots in the normal manner.
These would be ridiculous, of course, but mathematically admissable. And the problem is that I'm doing math. Is there a simple criterion that is widely invoked in the literature, one that I could impose to get rid of weird non-linear methods, or methods with an infinite number of "if-then" statements? I want something that would be understandable by social scientists, not some criterion that uses lots of fancy language from topology or real analysis or whatever.
Thanks for any recommendations that anybody can offer.
Alex Small
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