[EM] A Metric for Issue/Candidate Space
Kevin Venzke
stepjak at yahoo.fr
Thu Dec 10 17:28:47 PST 2020
Hi Forest, max distance between any pair of points (I call it the "broad" measure) at least wouldn't make much difference with 3 candidates. I tried it with 4 candidates and it seemed to behave very similarly.
I can somewhat rationalize minimizing a candidate's distance from the candidate who beat him. Assuming two candidates A and B might both be able to win, ignoring a win of one over the other seems less risky when A/B are similar than when they are different. It should be less aggravating to those who are overruled.
I find it harder to rationalize penalizing a candidate A for being defeated by two different candidates B/C that are different from each other but less different from A. In that case one intuitively wonders if A is spatially between B/C.
I also tried defining distances using the entire ballot (i.e. interpret ranks as grades and infer distances from these, no fractional counting) and that's certainly somewhat different. I'm not sure what kind of theoretical merit or deficiency there is in setting distances in various ways. (Naturally the best would be to determine them outside of the method, but that doesn't leave much of a method to study.)
Kevin
Le mercredi 9 décembre 2020 à 00:30:47 UTC−6, Forest Simmons <fsimmons at pcc.edu> a écrit :
Kevin,
Great ..thanks for incorporating it into your analyses!
My intended definition of diameter was simply the max distance between any pair of points in the set ... as in metric spaces:-)
It would be interesting to see if that makes any difference.
For your similations no harm in inferring the distances from the rankings, but I think it would be more accurate to start with points distributed in a plane with some of them designated as candidates. Then use the distances to calculate the rankings as in a Yee diagram. Clone sets should be relatively small in diameter compared to other distances.
In actual practice the reason for estimating distances independently from preferences is for political neutrality ... saying that x and y are far apart or close together does not tell directly which you prefer. Two voters of opposite persuasions could theoretically come up with identical distance estimates for all pairs of candidates. It seems like that should reduce manipulation somewhat if not altogether.
Also when preference changes do not directly affect distance estimates it is harder to create a monotonicity violation.
Here's my "proof" of monotonicity: raise winner X in the rankings ... that doesn't change the diameter of S(X) . And it changes the diameter of S(Y) only by augmenting S(Y) with X which increases the diameter of S(Y), which reinforces X's win.
Here is my argument for clone winner: replace the winner X with clone set C say . Then S(C) is contained in S(X) union C. So diameter of S(C) is greater than diameter of S(X) only if for some Y, the distance d(X,Y) is equal to the diameter of S(X).
And in that case the diameter of S(C) is no greater than S(X) + diameter(C) which is very close to S(X) since a true clone set is small in diameter compared the absolute difference between the diameters of S(X) and S(Z), say.
I could clean that up, but you get the idea.
Clone loser: if loser Y is replaced with a clone set C, then S(C) is at least as large as S(Y). But can this enlarge S(X)? The only chance of this is if X is pairwise beaten by Y, and at least one member of C increases the diameter of S(Y), which cannot happen by more than an infinitesimal if C is a true clone set. And the difference between diamS(Y) and diamS(X) is a non-infinitesimal positive number.
So if you define true clone sets as having (relatively) infinitesimal diameter, the method is clone independent. Otherwise we say, as for Range Voting, the method is marginally clone free.
I hope that makes sense!
Forest
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