[EM] A Metric for Issue/Candidate Space

Forest Simmons fsimmons at pcc.edu
Tue Dec 8 22:30:38 PST 2020

```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

On Tuesday, December 8, 2020, Kevin Venzke <stepjak at yahoo.fr> wrote:

> Hi Forest, this seems rather good. I implemented it such that the pairs
> are inferred from the top and bottom rankings (fractionalized, but with no
> equal ranking, just truncation), 3 candidates, 4 blocs. I also made it so
> that each pair of candidates has a tiny minimum distance, so that some
> pairwise defeat will always have a greater diameter than a single
> undefeated candidate. I understood the diameter of S(X) to mean the largest
> distance between X and some other candidate in S(X).
>
> Interesting aspects: Seems to have little burial incentive. Truncation
> incentive isn't bad, worse than C//IRV but better than C//A. Compromise is
> not the best but not bad. I found Mono-raise relatively bad, but that's
> probably because I built the distance setting into the method itself.
> Mono-add-top likewise. No Plurality issues yet. If there is only one
> majority contest, it almost always respects it. I'm a little puzzled why
> that should work out nicely like that... It tries to avoid electing
> candidates defeated by "dissimilar" candidates, but that's got no direct
> tie to the defeat strength.
>
> Let's see if this will post right. Might have to copy to a fixed width
> text editor. If it looks bad I'll try viewing it in the archive:
> http://lists.electorama.com/pipermail/election-methods-elect
> orama.com//2020-December/
>
> But here's a couple of maps placing decloned Copeland and this new
> "diameter" method:
>
> . . . . . . . . . . . . . . . . . . IFPP. . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . DSC
> . . . . . . . . . . TACC. IRV . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . C/IRV KML BTRIRV. . . . . . . . . . .
> . . . . . . . . .KOTH dcCop . . . .ChainRO. . . . . . . . .
> . . . . . . . . . . SV. . . . . . . . . . . . . . . . . . .
> . . . . . . . . . .C/KOTH . . . . . . . . . . . . . . . . .
> . . . . . . . BPW . . . . . . Marg. . . . . . . . . . . . .
> . . . . . . . **. . . . . . . . . . . . . . . . . . . . . .
> . . . . . . Diam. . . . . C//A. . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . .DMC.WV . . . . VBV . . . . . . . . . .
> . . . . . . . . . . AWP . **. . **. . . . . . . . . . . . .
> . . . . . . . . . . AER CdlA. IBIFA VBV . . . . . . . . . .
> IRVnoelim . . . . . MMPO. . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . MDDA.MAMPO. . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . DAC . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . Bucklin . . . . . . . . . . . . . .
>
> notes: KML = Kristofer's Linear method fpA-fpC. IBIFA is Chris's method.
> SV and BPW are Eivind Stensholt methods. AER = aka Approval AV. dcCop =
> decloned Copeland. IFPP = Craig Carey's Improved FPP. DAC and DSC are
> Woodall methods. TACC and DMC are other Forest or Jobst methods. AWP is
> James Green-Armytage's Approval-Weighted Pairwise (using MinMax; all
> approval methods are using implicit approval). My methods: KOTH, ChainRO,
> CdlA, VBV (x2), MDDA, MAMPO, ** (unnamed obscure methods).
>
> I will zoom in a bit now:
>
> . . . . . ChainRO . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . IRV . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . BTRIRV. . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . KMLinear. . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> TACC. . C/IRV . . . . . . . . . Marg. . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . VBV .
> . .dcCopeland . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . VBV
> KOTH. SV. . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . C/KOTH. . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . .IBIFA. .
> . . . . . . . . . . . . . . . . .C//A . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . **. . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
> . . BPW . . . . . . . . . AER WV. . . . . . . . . . . . . .
> . . **. . . . . . . . . .AWP.DMC. . CdlA. . . . . . . . . .
> . . . . Diam. . . . . . . . . . . . . . . . . . . . . . . .
> . . . . . . . . . . . . . . **. MMPO. . . . . . . . . . . .
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
>
> Hope this posts OK.
>
> Kevin
>
>
>
> Le lundi 7 décembre 2020 à 23:56:20 UTC−6, Forest Simmons <
> fsimmons at pcc.edu> a écrit :
> >
> >How to define such a metric?
> >
> >How to use it?
> >
> >Ask the voters which issue they felt to be most important, and which pair
> of candidates were the
> >farthest apart on that issue. The pair with the most votes is taken to be
> at unit distance ... the others
> >at distances proportional to their respective mentions.
> >
> >Use it to in conjunction with the pairwise margins matrix as follows: for
> each candidste X, let S(X) be
> >the set of candidates that beat or tie X pairwise. In particular X is
> itself a member of S(X) by virtue of
> >a self tie.
> >
> >Elect the candidate X that minimizes the diameter of S(X).
> >
> >Note that if S(X) has only one member then that member is the Condorcet
> Winner, and the diameter
> >of S(X) is zero, the absolute minimum.
> >
> >So the method is Condorcet Compliant.
> >
> >It seems pretty obvious that it satisfies clone independence.... and mono
> raise.
> >
> >Anybody else like this?
> >
> >How about using it for tournaments? What questions would you ask the fans
> to estimate the
> >distances between teams?
> >----
> >Election-Methods mailing list - see https://electorama.com/em for list
> info
>
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