[EM] A Metric for Issue/Candidate Space

classic Classic list List threaded Threaded
5 messages Options
Reply | Threaded
Open this post in threaded view
|

[EM] A Metric for Issue/Candidate Space

Forest Simmons
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
Reply | Threaded
Open this post in threaded view
|

Re: [EM] A Metric for Issue/Candidate Space

Kevin Venzke
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-electorama.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 <[hidden email]> 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
----
Election-Methods mailing list - see https://electorama.com/em for list info
Reply | Threaded
Open this post in threaded view
|

[EM] A Metric for Issue/Candidate Space

Forest Simmons
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 <[hidden email]> 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-electorama.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 <[hidden email]> 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

----
Election-Methods mailing list - see https://electorama.com/em for list info
Reply | Threaded
Open this post in threaded view
|

Re: [EM] A Metric for Issue/Candidate Space

Kevin Venzke
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 <[hidden email]> 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
----
Election-Methods mailing list - see https://electorama.com/em for list info
Reply | Threaded
Open this post in threaded view
|

Re: [EM] A Metric for Issue/Candidate Space

Forest Simmons
See comments inline ..

On Thursday, December 10, 2020, Kevin Venzke <[hidden email]> wrote:
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.

A would be somewhere roughly between B and C, which would make it more likely for A to be the CW, and the diameter smaller, which should correlate ... small diameter <--> A wins.

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.)

Making distance strictly proportional to distance in rank would introduce the same distortion that Borda exhibits visa vis Range. But even using range scores in this way is a distortion: if A and B are both rated top, then fine ... the rsting difference of zer reflects the likely proximity of the candidates.  But if both are rated zero, they could be anywhere on the perimeter of the set of candidates  For example, the ranking

A>> B =C 

could mean A is near the center of a circle, while  B and C are on opposite ends of a diameterr of that circle. They are both enemies of A but not friends of eachother.

Thanks for helping to clarify these confusing details!

Forest

Kevin




Le mercredi 9 décembre 2020 à 00:30:47 UTC−6, Forest Simmons <[hidden email]> 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

----
Election-Methods mailing list - see https://electorama.com/em for list info