[EM]Ranked-Pairs (wv) can lose a Cond. Winner

[Elisabeth Varin/Stephane Rouillon]

MIKE OSSIPOFF a écrit :

> Steph--
>
> [a shift key isn't working, and so some letters may be incorrectly
> uncapitalized. i'll use brackets for parentheses.]
>
> here's your example:
>
> A family of examples with (11+2x) (x>=0) voters and 3 candidates is:
> 2+x: A
> 2: A > B > C
> 2: B > A > C
> 1+x: B > C > A
> 4: C
> Ranked pairs with winning votes produces:
> A (6+x) > C (5+x) , B (5+x) > C (4) and A (4+x) > B (3+x).
> A is the Condorcet winner and wins.
> Margins and relative margins produce of course the same result.
> If I am one of the two B > A > C voter, my 2nd (A)
> choice harms my favorite 1st choice (B).
> The proof is, if I and my co-thinker vote B only:
> 2+x: A
> 2: A > B > C
> 2: B   (truncated !)
> 1+x: B > C > A
> 4: C
> Ranked pairs with winning votes produces:
> B (5+x) > C (4), C (5+x) > A (4+x) and A (4+x) > B (3+x) can't lock.
> B wins now.
>
> i reply:
>
> i assume that you're saying that in your example x can have any
> nonzero value without changing the result. For simplicity i let X = 1.
>
> maybe i made an error, but by my count, B is the BeatsAll winner,
> regardless of whether the BAC voters truncate.

You are wrong there. A always beats B by one vote (4+x)-(3+x) =1.
Yes, regardless of the BAC voters truncating, it does not change.
With X=1, A (5) > B (4).

> Would you re-post the example that you intended, preferably with
> exactly-determined numbers of voters?

With X=1:

3: A
2: A > B > C
2: B > A > C
2: B > C > A
4: C
Ranked pairs with winning votes produces:
A (7) > C (6) , B (6) > C (4) and A (5) > B (4).
A is the Condorcet winner and wins.
Margins and relative margins produce of course the same result.
If I am one of the two B > A > C voter, my 2nd (A)
choice harms my favorite 1st choice (B).
The proof is, if I and my co-thinker vote B only:
3: A
2: A > B > C
2: B   (truncated !)
2: B > C > A
4: C
Ranked pairs with winning votes produces:
B (6) > C (4), C (6) > A (5) and A (5) > B (4) can't lock.
B wins now.

The other example I am referring to at:
http://groups.yahoo.com/group/election-methods-list/message/10107
uses X=0.

> if you say that truncation can take victory from a CW in wv, you're
> probably intending the old example in which a poorly-supported CW loses
> to truncation. i've many times said that truncation can work against
> a cw when there's lots of indifference toward that cw. When there's
> no majority voting the cw over the candidate who steals the election.
> A candidate can be CW even if he beats the other candidates with
> sub-majority defeats. And if he beats y with a submajority defeat,
> i make no claim that y can't steal the election by truncation.

Please define a poorly-supported CW. If the definition is criteria (wv or
rm)
independent it should be really helpful. If it is not, we need to compare
the sizes of the poorly-supported CW (wv) class to the size of the same
class
using (rm). This is why the probability of a CW being a poorly-supported CW
depending on the criteria matters.

> if that's your point, then you're right. it's something that i've
> often said.
>
> please note that SFC is about a CW who is preferred to
> candidate y by a majority who vote sincerely. Re-read SFC & GSFC.
> Those criteria, though they don't mention truncation, tell what i
> claim for wv when no one order-reverses.
>
> mike ossipoff

All votes I gave previously were sincere preferences. Only the truncation
that leads
to a better result for the truncaters is unsincere as you were doing.
The CW I used (namely A) was a "CW who is preferred to
candidate y (namely B or C) by a majority who vote sincerely". Thus if this
is the definition of a non "poorly-supported CW", my counter-example fits...

If SFC & GSFC apply to non-poorly supported CW, please explain why the
sincere CW (namely A) can get stolen with an unsincere truncation in my
examples.
I will consider "non poorly-supported CW" as a sincere CW. I hope it is what
you mean.

The only conclusion I was able to obtain by myself, was that your SFC & GSFC
analysis was good when there was no truncation (nor sincere, neither
unsincere) present. In this particular case, margins, relative margins and
winning votes all
produce the same results. Still I think you are right in assuming the
probability of gain from an unsincere truncation is lower with winning votes
(maybe null as you say, I am not sure). I was able to prove it is null for 3
candidates, not further yet.

But your analysis seems to be built on the assumption only one truncation
(unsincere) occurs. I think many can happen, sincere ones as much as
multiple unsincere (many voters could expect an improvement) or both types.
I have just proved to you that
this probability is not null using (wv) with multiple truncations. Even,
relative margins would protect the CW in these examples (except for X=0
where a tie occurs).
Thus the probability to steal a CW when all sincere votes (including sincere
truncations) distributions are considered may be lower or higher with
relative margin than with winning votes. I do not know. But I think sincere
truncations increase with an increasing number of candidates.

Finally, once we know that because of sincere truncations, unsincere
truncations can
lead to (rare I hope) special cases where a CW can get stolen, it increases
the probability of unsincere truncations occuring.

This is why I try to evaluate the probability of truncation occuring and
then
the probability of being able to steal a CW in overall cases.

Steph.

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