[EM] Steph: Truncation example

Sun Nov 24 23:04:23 PST 2002

Steph--

You said:

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.

I reply:

Another example of the old "poorly-supported CW" objection.

Out of 13 voters, only 5 rank A over B. But you still want wv
to guarantee A's win.

You have here an example in which relative
margins doesn't let the truncation take victory from the CW. But
one non-failure example means nothing. Even a majority ranking the
CW over B doesn't keep truncation by B voters from stealing the election
sometimes with relative margins, without any order-reversal. Relative
margins fails to a degree that wv doesn't. Relative margins has a number of 
kinds of majority rule failure that wv doesn't have.

It's an example in which the CW doesn't have a majority over the
candidate who is able to steal the election from him by offensive
truncation. I say he's poorly supported because, though he's CW,
he has a pairwise victory that isn't a majority, either due to sincere
indiferrence, or perhaps someone's truncation strategy. That's what
I meant by "poorly supported CW".

The trouble with your "poorly-supported CW" objection is that you're
asking a lot when you ask me to protect the CW from someone over whom
no majority bother to rank him. However, you could write a
"Strong SFC" that doesn't mention majority. I encourage you to post
a definition of a method that meets such a criterion. No one would
be more pleased than I. In fact, after doing that would be the best
time for criticising wv for not meeting such a criterion.

>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.

You continued:

Please define a poorly-supported CW. If the definition is criteria (wv or
rm)
independent it should be really helpful.

I reply:

My definition of a poorly-supported CW is: A CW who has a pairwise
victory that doesn't have a majority voting him over the other
candidate. The only place that I've used "poorly supported CW" is
in my term "the old 'poorly-supported CW' objection". I don't use
that definition or that term in criteria.

That definition is independent of how defeats are measured.

You continued:

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".

I reply:

What I say in that paragraph that you quote should make it obvious
that the "Y" in that quote is the Y in SFC. In SFC, I say that
if no one falsifies a preference, and if a majority prefer the CW to
Y, and vote sincerely, then Y shouldn't win.

Your CW has victory taken from him by a candidate over whom he
doesn't have a majority. If a majority preferred him to B, and voted
sincerely, then he'd have a majority over B.

The fact that you may be able to find some other candidate over whom
your CW has a majority is irrelevant with regard to your method's
SFC failure.

You continued:

Thus if this
is the definition of a non "poorly-supported CW", my counter-example fits...

I reply:

But, by "poorly-supported CW", I didn't mean a CW who doesn't have
a majority over anyone. In the spirit of SFC. Not having a majority
over one candidiate, B, makes A poorly supported enough that you shouldn't 
expect wv to protect his win.

You continued:

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 reply:

SFC & GSFC make no use of the term "non-poorl-supported CW", or
"poorly-supported CW".

You ask why the sincere CW can have victory stolen from him by
offensive truncation in your example, with wv: It's because he doesn't
have a majority over B. That's why B can steal the election from him
by truncation.

You continue:

I will consider "non poorly-supported CW" as a sincere CW. I hope it is what
you mean.

I reply:

It isn't. In referring to a "poorly-supported CW" objection (my
only use of the term "poorly supported CW"), the term "poorly supported
CW" means: A CW who has a pairwise victory that isn't supported by
a majority. But I re-emphasize that that definition isn't important,
since I only used it in the name that I gave to your objection-example,
an example that I've replied to so many times that it deserves a name.

"Sincere CW" means a candidate who would beat everyone if all the
voters voted sincere rankings. But we just use "CW" to mean the same
thing, for brevity. If someone beats everyone, but not necessarily with
sincere rankings, I call him the BeatsAll winner.

You continued:

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.

I reply:

SFC & GSFC make no mention of truncation. But if you mean that
SFC & GSFC distinguishe between wv, margins, and relative margins only when 
there's truncation, sure, that's true of SFC, and probably GSFC.

But it isn't at all clear why you think that SFC & GSFC aren't "good"
unless there's truncation, because margins only fails when there's
truncation. When there's no truncation, and margins doesn't show a
failure of SFC, SFC is still good. Criteria usually say that a method
fails if there's any instance of it failing the criterion's requirement.
No one says that criteria are "not good" in other instances.

You continued:

In this particular case, margins, relative margins and
winning votes all
produce the same results.

I reply:

The fact that you can find an instance in which your method doesn't
fail a criterion's requirement doesn't mean anything when, as is
always the case, the criterion requires that, when the criterion's
premise is true, the method must never fail the criterion's requirement.

Conceivably someone could write a criterion that isn't written that
way, but I can't name one right now. In any case, SFC & GSFC explicity
require that, under their premise conditions, a method never fail
their requirements. Most or all criteria also do.

You continued:

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 reply:

I don't make claims about probability. But margins and relative
margins let truncation steal elections from CWs under a wider set
of conditions, failing more flagrantly, compared to wv.

You continue:

But your analysis seems to be built on the assumption only one truncation
(unsincere) occurs.

I reply:

SFC & GSFC make no reference to truncation at all, much less any
specification about how many truncations there are, or whether they're
sincere or insincere. Nor have I said anything about that.

You continue:

I think many can happen, sincere ones as much as
multiple unsincere (many voters could expect an improvement) or both types.

I reply:

Truncation will be common in public elections, as it is in committee
rank-ballotings.

You continued:

I have just proved to you that
this probability is not null using (wv) with multiple truncations.

I reply:

That was already proven, and often said by me, that even wv can't
protect an indifferently-supported CW against truncation.

You continued:

Even,
relative margins would protect the CW in these examples (except for X=0
where a tie occurs).

I reply:

One nonfailure example says nothing. There are examples in which
your method lets victory be stolen from a CW, by truncation, when
a majority vote the CW over that candidate, when no one falsifies
a preference. Your method fails the CW more flagrantly than wv does.

You continued:

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.

I reply:

Especially with margins and relative-margins, since with those methods
truncation can even steal victory from a CW in violation of majority
rule, something that won't happen with wv.

Mike Ossipoff

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