[EM] Improved Copeland (was "A New Spinoff of Our Recent Discussions")
Forest Simmons
fsimmons at pcc.edu
Fri Jun 7 15:04:12 PDT 2019
I knew it was too good to be true!
Kristofer,
I forgot that when we worked on decloning Borda that we tried this fixing
Copeland. That wa a long time ago. We must be getting old!
I think you have found the cure for failure of mono-raise (if not for
mononucleosis itself)..
To make it clone proof let's try using explicit approval cutoffs and
calculate the penalities by fractional approval.
We can appeal to the convention that a true clone is not split by an
approval cutoff. Otherwise, Approval itself based on ranked ballots with
approval cutoffs would be clone dependent.
Another advantage of using fractional approval for the penalty counts is
that the method then distinguishes between
49 C
26 A>>B
25 B
and the same profile with A>>B replaced by A>B>>..
yielding the respective winners C and B in the two cases.
Does that fix everything?
On Fri, Jun 7, 2019 at 1:17 PM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:
> On 06/06/2019 23.11, Forest Simmons wrote:
> > Here's another slightly simpler approach aimed at the lay voter:
> >
> > Tell the audience that the Condorcet ideal is a candidate that is not
> > pairwise beaten by any other candidate.
> >
> > When that is not possible, it is natural to consider a candidate that is
> > beaten pairwise by the fewest other candidates. This idea is the basis
> > of the Copeland Method.
> >
> > There are two problems with the Copeland method: (1) It has a strong
> > tendency to produce ties, and (2) More subtle problems created by
> > cloning certain candidates to increase the number of defeats suffered by
> > certain other candidates without increasing the number of defeats of the
> > cloned candidates.
> >
> > Because of these two problems, Copeland is not considered a serious
> > contender for use in public elections.
> >
> > But what if there were a simple modification of Copeland that would
> > totally resolve these two problems in one fell swoop?
> >
> > There is; instead of counting the number of candidates that defeat
> > candidate X, (and electing the candidate with the smallest count), we
> > add up all of the first place votes of all of the candidates that defeat
> > X, and elect the candidate with the smallest sum.
>
> This sounds like what I called first preference Copeland and you called
> Clone-proofed Copeland back in 2006:
>
> http://lists.electorama.com/pipermail/election-methods-electorama.com/2006-December/117180.html
>
> In the examples below, I'll call the candidates' sums "penalties".
>
> It's not cloneproof. Example outline by Warren Smith
> (https://groups.yahoo.com/neo/groups/RangeVoting/conversations/topics/2934
> ):
>
> 6: A>B>C
> 3: C>A>B
> 4: B>C>A
>
> ABCA cycle, so A's penalty is 3, B's penalty is 6, and C's penalty is 4.
> A wins.
>
> Now clone A in a Condorcet cycle (into D,E,F below):
>
> 2: D>E>F>B>C
> 2: E>F>D>B>C
> 2: F>D>E>B>C
> 3: C>E>F>D>B
> 4: B>C>F>D>E
>
> B is beaten by D, E, and F, so his penalty is 6.
> C is beaten by B, so his penalty is 4.
> D is beaten by C and F, so his penalty is 5.
> E is beaten by C and D, so his penalty is 5.
> F is beaten by C and E, so his penalty is 5.
>
> Cloning A made A lose and C win.
>
> Also, compared to ordinary Copeland, it loses monotonicity:
>
> 3: A>B>C
> 3: B>C>A
> 3: C>A>B
>
> Everybody has a penalty of 3, so we have a tie.
>
> Now raise A to the top on one of the B>C>A ballots:
>
> 4: A>B>C
> 2: B>C>A
> 3: C>A>B
>
> A is beaten by C: penalty 3
> B is beaten by A: penalty 4
> C is beaten by B: penalty 2
>
> so C wins.
>
> The problem here is that putting A top may harm A by concealing a
> candidate (here, B) who would otherwise penalize someone else (here, C)
> enough to keep A from losing.
>
> The fpA - fpC method avoids nonmonotonicity by crediting A with A-top
> ballots. If raising A to the top conceals some penalizer's first
> preferences, that doesn't matter because whatever A loses to the third
> candidate, he regains by the fpA term.
>
> So the fpA-fpC method seems better than first preference Copeland for
> three candidates. If only we could cloneproof it! (And hopefully retain
> its anti-strategy properties, like CD compliance and unmanipulable
> majority)
>
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