<div dir="ltr"><div>I knew it was too good to be true!</div><div><br></div><div>Kristofer,</div><div><br></div><div>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!</div><div><br></div><div>I think you have found the cure for failure of mono-raise (if not for mononucleosis itself)..</div><div><br></div><div>To make it clone proof let's try using explicit approval cutoffs and calculate the penalities by fractional approval.</div><div><br></div><div>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.</div><div><br></div><div>Another advantage of using fractional approval for the penalty counts is that the method then distinguishes between</div><div><br></div><div>49 C</div><div>26 A>>B</div><div>25 B</div><div><br></div><div>and the same profile with A>>B replaced by A>B>>..</div><div><br></div><div>yielding the respective winners C and B in the two cases.</div><div><br></div><div>Does that fix everything?<br></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Fri, Jun 7, 2019 at 1:17 PM Kristofer Munsterhjelm <<a href="mailto:km_elmet@t-online.de">km_elmet@t-online.de</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">On 06/06/2019 23.11, Forest Simmons wrote:<br>
> Here's another slightly simpler approach aimed at the lay voter:<br>
> <br>
> Tell the audience that the Condorcet ideal is a candidate that is not<br>
> pairwise beaten by any other candidate.<br>
> <br>
> When that is not possible, it is natural to consider a candidate that is<br>
> beaten pairwise by the fewest other candidates. This idea is the basis<br>
> of the Copeland Method.<br>
> <br>
> There are two problems with the Copeland method: (1) It has a strong<br>
> tendency to produce ties, and (2) More subtle problems created by<br>
> cloning certain candidates to increase the number of defeats suffered by<br>
> certain other candidates without increasing the number of defeats of the<br>
> cloned candidates.<br>
> <br>
> Because of these two problems, Copeland is not considered a serious<br>
> contender for use in public elections.<br>
> <br>
> But what if there were a simple modification of Copeland that would<br>
> totally resolve these two problems in one fell swoop?<br>
> <br>
> There is; instead of counting the number of candidates that defeat<br>
> candidate X, (and electing the candidate with the smallest count), we<br>
> add up all of the first place votes of all of the candidates that defeat<br>
> X, and elect the candidate with the smallest sum.<br>
<br>
This sounds like what I called first preference Copeland and you called<br>
Clone-proofed Copeland back in 2006:<br>
<a href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2006-December/117180.html" rel="noreferrer" target="_blank">http://lists.electorama.com/pipermail/election-methods-electorama.com/2006-December/117180.html</a><br>
<br>
In the examples below, I'll call the candidates' sums "penalties".<br>
<br>
It's not cloneproof. Example outline by Warren Smith<br>
(<a href="https://groups.yahoo.com/neo/groups/RangeVoting/conversations/topics/2934" rel="noreferrer" target="_blank">https://groups.yahoo.com/neo/groups/RangeVoting/conversations/topics/2934</a><br>
):<br>
<br>
6: A>B>C<br>
3: C>A>B<br>
4: B>C>A<br>
<br>
ABCA cycle, so A's penalty is 3, B's penalty is 6, and C's penalty is 4.<br>
A wins.<br>
<br>
Now clone A in a Condorcet cycle (into D,E,F below):<br>
<br>
2: D>E>F>B>C<br>
2: E>F>D>B>C<br>
2: F>D>E>B>C<br>
3: C>E>F>D>B<br>
4: B>C>F>D>E<br>
<br>
B is beaten by D, E, and F, so his penalty is 6.<br>
C is beaten by B, so his penalty is 4.<br>
D is beaten by C and F, so his penalty is 5.<br>
E is beaten by C and D, so his penalty is 5.<br>
F is beaten by C and E, so his penalty is 5.<br>
<br>
Cloning A made A lose and C win.<br>
<br>
Also, compared to ordinary Copeland, it loses monotonicity:<br>
<br>
3: A>B>C<br>
3: B>C>A<br>
3: C>A>B<br>
<br>
Everybody has a penalty of 3, so we have a tie.<br>
<br>
Now raise A to the top on one of the B>C>A ballots:<br>
<br>
4: A>B>C<br>
2: B>C>A<br>
3: C>A>B<br>
<br>
A is beaten by C: penalty 3<br>
B is beaten by A: penalty 4<br>
C is beaten by B: penalty 2<br>
<br>
so C wins.<br>
<br>
The problem here is that putting A top may harm A by concealing a<br>
candidate (here, B) who would otherwise penalize someone else (here, C)<br>
enough to keep A from losing.<br>
<br>
The fpA - fpC method avoids nonmonotonicity by crediting A with A-top<br>
ballots. If raising A to the top conceals some penalizer's first<br>
preferences, that doesn't matter because whatever A loses to the third<br>
candidate, he regains by the fpA term.<br>
<br>
So the fpA-fpC method seems better than first preference Copeland for<br>
three candidates. If only we could cloneproof it! (And hopefully retain<br>
its anti-strategy properties, like CD compliance and unmanipulable majority)<br>
</blockquote></div>