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• ## Re: [EM] Problem solved (for pure rank ballots): ICC & AFB incompatible (essentially)

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• ... Abd, I think you are overly hung up on this predictability detail. B wins, then the C A B voter can change his/her vote to make A win only by downranking
Message 1 of 9 , Feb 2, 2007
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--- In RangeVoting@yahoogroups.com, Abd ul-Rahman Lomax <abd@...>
wrote:
>
> Benham wrote me about his earlier response with "did you even read
> it?" Yes, I had read it. I don't always have time to write. In \
> fact, I *never* have time to write, but I steal it.
>
> At 11:00 AM 1/27/2007, cbenhamau wrote:
>
> > > The assumption here is that the C voter gains something by
> > > betraying C. Where did that assumption come from?
> >
> >That "assumption" comes from the C voter's unaltered ballot in
> >Election 1: "C>A>B".
>
> The problem postulates that B wins the election. This is after, of
> course, the three voters have voted. It is assumed that the C voter
> can anticipate B winning based on knowing the sincere preferences
> of all voters and by assuming that they will all vote as predicted.
> Except C. Now, it is legitimate to make the assumption of vote
> stability except for the voter contemplating FB, but the assumption
> that the victory of B *can be predicted* -- which is necessary for
> favorite betrayal -- conflicts with the assumption that the choice
> between the three candidates for victory is random. If it can be
> predicted from the symmetrical situation shown, it isn't random or
> symmetrical. There is a contradiction in the assumptions that was

> If there is a way around this assumption (of predictability), I
> didn't see it.

> C>A>B does not, in itself, predict strategic benefit for the voter
> from FB, unless the probabilities are asymmetric, or the preference
> strengths vary.
>
> Since the probabilities must be equal, the only remaining variable,
> under the stated conditions, is variation in preference strength,
> but Benham denied that preference strength was relevant.

> If preferences expressed are taken as indicating equal preference
> strength, then there is no strategic benefit for the C voter in
> betraying C to elect A. It has, in fact, the same expected utility
> as making the strategic vote. AFB *requires* that there be a
> predictable strategic benefit. (That is, the probability of each
> outcome times the expected utility of that outcome, for FB, must be
> greater than the situation without FB. In this case, FB controls
> the election of A, it is a certainty (with stable votes from the
> others).
>
Abd, I think you are overly hung up on this "predictability" detail.

B wins, then the C>A>B voter can change his/her vote to make A win
only by downranking C to below equal-top; and that is sufficient to
fail FBC in my book. We are not meant to get our knickers in a twist
worrying about why B wins or how the C>A>B voter knew that B wins.

Warren's wording of FBC (that he often refers to as AFB) might seem to
imply "predictability", but Mike Ossipoff's earlier (perhaps
original) version doesn't:

"Favorite-Betrayal Criterion (FBC):
By voting a less-liked candidate over his/her favorite, a voter
should never gain an outcome that he/she likes better than every
outcome that he/she could get without voting a less-liked candidate
over his/her favorite. *** "
>
But to humour you a bit, maybe we can say that the proof is overly
lean in using just three voters instead of three equal-sized large
factions.

33: A>B>C
33: C>A>B
33: B>C>A

Then we can say that by Symmetry each candidate has a 1/3 chance of
winning but by Discrimination we can make an "arbitrarily small"
change to this profile to make one of the candidates win.

So when we say "B wins" we could mean that one more voter arrives and
provisionally, but then the C>A>B voters are privileged to receive
the information that the last eligible voter is about to arrive and
definitely vote for B.

Then this faction can "predict" that B wins, and so they do "improve
their expectation" by changing their votes from C>A>B to A>C>B.

Chris Benham
• ... Detail!!! ... Notice that a precedent action is dependent upon an unpredictable consequence. It is a clear logical fallacy. ... In the scenario given, it
Message 1 of 9 , Feb 2, 2007
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At 05:21 AM 2/2/2007, cbenhamau wrote:
>Abd, I think you are overly hung up on this "predictability" detail.

Detail!!!

>B wins, then the C>A>B voter can change his/her vote to make A win
>only by downranking C to below equal-top; and that is sufficient to
>fail FBC in my book.

Notice that a precedent action is dependent upon an unpredictable
consequence. It is a clear logical fallacy.

> We are not meant to get our knickers in a twist
>worrying about why B wins or how the C>A>B voter knew that B wins.
>
>Warren's wording of FBC (that he often refers to as AFB) might seem to
>imply "predictability", but Mike Ossipoff's earlier (perhaps
>original) version doesn't:
>
>"Favorite-Betrayal Criterion (FBC):
>By voting a less-liked candidate over his/her favorite, a voter
>should never gain an outcome that he/she likes better than every
>outcome that he/she could get without voting a less-liked candidate
>over his/her favorite. *** "
> >

In the scenario given, it is true that the voter may, if the method
considers it, change the expected utility favorably. However, it is
also necessary to consider the probability of and utility of the
other outcomes which are possible.

What Warren has done, and which Benham is agreeing with, looks to me
like the following:

The winning candidate in a two candidate election is decided by coin
toss. A voter is picked and the procedure is that the picked voter
calls out "Heads" or "Tails." If the coin matches the call, then the
voter's favorite wins, and it it does not match, then the other candidate wins.

So if it does not match, the voter simply changes his call.

What has been said is that if there is a tie, there is an equal
probability of victory for each candidate. The winner is declared
*after* whatever random process is used to pick the winner. That
process takes place after voting.
It cannot take place *before* voting.

So, if it comes out B, then the voter can change his vote *in the
next election.* Not in the original one. And in the next election,
the voter *might* increase his expected utility by FB, but only in
the situation that the C>A preference is weaker than the C>B
preference. And this would be the case in the first election.

The voter may indeed alter the outcome by FB, but only if preference
strengths are not equal. And the tie is irrelevant.

What has really been shown is that pure ranked systems that do not
consider preference strength, but which satisfy all the conditions in
the set of assumptions Warren gave, are intrinsically vulnerable to
FB, or will fail ICC. (I think.) We already knew that. This is
because the preference strength may be as close as needed to cause A
and C to be clones within the definition of the ranked method, yet a
rank preference may still be expressed.

If ranks are presumed to represent a single preference strength,
which is the hidden assumption behind pure ranked methods, then this
does not happen. In the scenario given, C does not improve his
expected utility by FB; rather, it is the same.

I gave the probabilities in another post, with a chart showing the
expected outcomes. Essentially, if, or the C voter, the B utility is
-N, the A utility is 0, and the C utility is +N, then the expected
utility is 0 without FB. With FB, the election of A is definite, but
the utility is zero, so the expected utility is again zero. There is
no "strategically forced betrayal," no motive to betray the favorite.

However, this is not true if strength(B>C) is more than
strength(C>B). At the extremes, if (C>B) is zero, then it's obvious.
Given the other votes, C obtains maximum gain by acting such that the
election of a maximized utility candidate takes place.

And it the real world the election would be resolved by an agreement
between two of the three factions. Or at least between some portion
of these factions. It's an interesting situation, given that the
factions are, essentially diametrically opposed, yet each of them is
motivated to *give away* the election to the enemies of their
favorite. All they need to do is to let the supporters of their next
favorite know that they should vote sincerely. Otherwise those
supporters might go through the same process, and thus defeat their
favorite *and* leave the election as a tie between their middle rated
candidate and their least favorite candidate.

("Look, you evil voters who hate our favorite, please agree to not
will defeat B, whom we both agree is not as good as what happens if
you vote sincerely and we compromise with you.")

>But to humour you a bit, maybe we can say that the proof is overly
>lean in using just three voters instead of three equal-sized large
>factions.

I don't agree that is is an improvement. It's the same. For a faction
to change its vote to improve its outcome in the event of a tie is no
different than for a single voter to change his or her vote, since,
under the assumed conditions, a single vote would shift the result.

>33: A>B>C
>33: C>A>B
>33: B>C>A
>
>Then we can say that by Symmetry each candidate has a 1/3 chance of
>winning but by Discrimination we can make an "arbitrarily small"
>change to this profile to make one of the candidates win.

Yes. I got it. You don't get it. There is no difference from the
point of view of the time travel violation between this and the
original scenario. The effect of a single voter remains the same.

Can that voter base his action on the outcome of the election. Gad,
isn't it obvious that a vote can't be dependent upon a random
outcome, resolved after the election?

>So when we say "B wins" we could mean that one more voter arrives and

But then, in the original election, which we presume to be sincere,
that voter has B as a favorite.

> We could pretend that these votes are all made openly and
>provisionally, but then the C>A>B voters are privileged to receive
>the information that the last eligible voter is about to arrive and
>definitely vote for B.

That's interesting. But that violates an assumption which is that the
election is a tie, it seems. I haven't gone all the way down this road.

I would say that a assumption that voters are equal in voting power,
which wasn't stated or implied, I think, would be violated.
Essentially, for this strategy to work, a follower of one candidate
must be privileged to know that the election is a tie if the voter

>Then this faction can "predict" that B wins, and so they do "improve
>their expectation" by changing their votes from C>A>B to A>C>B.

Yes. The expected outcome for the C voter prior to voting is -1/2,
and the outcome after voting is zero.

In other words, the faction which favors a candidate is favored if
that faction (or at least one voter within the faction) has
privileged knowledge, knowledge that was not available to the other
voters when they voted.

If no assumption is made that the voters are equally privileged, then
the proof appears valid. I had assumed that the polls would be closed
and *then* the winner was determined. But wouldn't this privilege
violate symmetry? One candidate has an advantage: the one favored by
the voter who votes last and knows the outcome without his or her vote.

Given that this appears to be a *necessary* condition for AFB to
fail, in addition to the condition of equal ranking strength, it
should be noted in the proof, as should the assumption of unequal
ranking strength as a possibility, (in which case the advance
knowledge isn't necessary, but it does, again, improve the outcome further).
• ... Not neccessarily. Say that for this election (or Warren s 3-voter election) there is a special tie-breaking procedure that is fair and symmetrical: one of
Message 1 of 9 , Feb 2, 2007
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--- In RangeVoting@yahoogroups.com, Abd ul-Rahman Lomax <abd@...>
wrote:
>
> At 05:21 AM 2/2/2007, cbenhamau wrote:
> >Abd, I think you are overly hung up on this "predictability"
> >detail.
>
> Detail!!!
>
> >B wins, then the C>A>B voter can change his/her vote to make A win
> >only by downranking C to below equal-top; and that is sufficient to
> >fail FBC in my book.
>
> Notice that a precedent action is dependent upon an unpredictable
> consequence. It is a clear logical fallacy.
>
> >We are not meant to get our knickers in a twist worrying about why
> >B wins or how the C>A>B voter knew that B wins.
> >
> >Warren's wording of FBC (that he often refers to as AFB) might
> >seem to imply "predictability", but Mike Ossipoff's earlier
> >(perhaps original) version doesn't:

> >
> >"Favorite-Betrayal Criterion (FBC):
> >By voting a less-liked candidate over his/her favorite, a voter
> >should never gain an outcome that he/she likes better than every
> >outcome that he/she could get without voting a less-liked candidate
> >over his/her favorite. *** "
> > >
>
> In the scenario given, it is true that the voter may, if the method
> considers it, change the expected utility favorably. However, it is
> also necessary to consider the probability of and utility of the
> other outcomes which are possible.
>
>>
> >But to humour you a bit, maybe we can say that the proof is overly
> >lean in using just three voters instead of three equal-sized large
> >factions.
>
>
>
> >33: A>B>C
> >33: C>A>B
> >33: B>C>A
> >
> >Then we can say that by Symmetry each candidate has a 1/3 chance of
> >winning but by Discrimination we can make an "arbitrarily small"
> >change to this profile to make one of the candidates win.
>
> Yes. I got it. You don't get it. There is no difference from the
> point of view of the time travel violation between this and the
> original scenario. The effect of a single voter remains the same.
>
> Can that voter base his action on the outcome of the election. Gad,
> isn't it obvious that a vote can't be dependent upon a random
> outcome, resolved after the election?

Not neccessarily. Say that for this election (or Warren's 3-voter
election) there is a special tie-breaking procedure that is fair and
symmetrical: one of the tied winners is randomly selected (X) and
then it is announced that all voters have the option of changing their
votes. If they all decline to do this or if after doing this there is
still a tie, then X wins.
>
> >So when we say "B wins" we could mean that one more voter arrives
>
> But then, in the original election, which we presume to be sincere,
> that voter has B as a favorite.
>
> >We could pretend that these votes are all made openly and
> >provisionally, but then the C>A>B voters are privileged to receive
> >the information that the last eligible voter is about to arrive and
> >definitely vote for B.
>
> That's interesting. But that violates an assumption which is that
> the election is a tie, it seems. I haven't gone all the way down
>
> I would say that a assumption that voters are equal in voting
> power,which wasn't stated or implied, I think, would be violated.
> Essentially, for this strategy to work, a follower of one candidate
> must be privileged to know that the election is a tie if the voter

Having a voter or faction of voters better informed is not a problem
for the formal "equality of voting power" in an election.
>
> >Then this faction can "predict" that B wins, and so they
> >do "improve their expectation" by changing their votes from C>A>B
> >to A>C>B.
>
> Yes. The expected outcome for the C voter prior to voting is -1/2,
> and the outcome after voting is zero.
>
> In other words, the faction which favors a candidate is favored if
> that faction (or at least one voter within the faction) has
> privileged knowledge, knowledge that was not available to the other
> voters when they voted.
>
> If no assumption is made that the voters are equally privileged,
> then the proof appears valid. I had assumed that the polls would be
> closed and *then* the winner was determined. But wouldn't this
> privilege violate symmetry?

> One candidate has an advantage: the one favored by
> the voter who votes last and knows the outcome without his or her
> vote.

No it doesn't. But it isn't necessary for the C>A>B faction to be
privileged with the extra information. If all the voters know that the
extra voter is turning up to vote for B, then the C>A>B faction will
still switch to A>C>B and that will create a stable "A wins" situation
that none of the voters can from their perspective improve on.
>
Happy now?

Chris Benham
• ... In other words, by setting up a special rule, you can get a method to fail AFB. Specifically by inviting it. The attempt in Warren s proof was to show that
Message 1 of 9 , Feb 2, 2007
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At 09:15 PM 2/2/2007, cbenhamau wrote:
>Not neccessarily. Say that for this election (or Warren's 3-voter
>election) there is a special tie-breaking procedure that is fair and
>symmetrical: one of the tied winners is randomly selected (X) and
>then it is announced that all voters have the option of changing their
>votes. If they all decline to do this or if after doing this there is
>still a tie, then X wins.

In other words, by setting up a special rule, you can get a method to
fail AFB. Specifically by inviting it.

The attempt in Warren's proof was to show that *any* method which
satisfied the set of assumptions was subject to AFB violation *or*
ICC violation under at least one election scenario.

What we have seen so far is that this is unconditionally true if
ranks do not represent equal preference strengths.

Then it is conditionally true, even if ranks represent equal
preference strengths, if the voter (at least one) has prior knowledge
-- prior to fixing his vote permanently -- of who will be chosen by
the tie-breaking procedure.

The above is really only a variation of that, one which appears to
satisfy symmetry. However, I'd suggest that we would need to consider
the new election as one which has a special bias: X is selected if
there is a tie. All voters now have the option of changing their votes.

So we have

A>B>C
B>C>A
C>A>B
with the provision that B wins if there is a tie. In other words, the
above election has B as the winner.

The C voter may change his vote, to be sure, to A>C>B. The B voter
has no motive to change his vote, it would seem, and the A voter is
reasonably content. However, the B voter will notice, perhaps, that
the C voter is motivated to shift his vote, thus converting the
election from his favorite to his least favorite. The B voter is
going to be quite unhappy that the result of the random selection was
B. The B voter cannot avoid the disaster of the A and C voters colluding.

It's kind of like winning a Yankee swap when you are the first
player. You get the prize you want. Except that if anyone else wants
it, they will take it from you. And, in this case, you get the worst
outcome possible. Every candidate will be wishing to lose the
selection, because it invites the other two factions to unite against them.

This raises the whole question of how AFB is judged. It is clear that
by changing his vote, the C voter in the original concept can raise
utility. However, in the vast majority of real election situations,
this does not raise utility in a fixed way, but in a way that is
modified by the likelihood of similar shifts by other voters. By
allowing the isolation of a single voter shifting his vote, while all
other voters are presumed to keep their votes the same, we can set up
a situation where AFB is violated, that is, the voter can, under some
circumstances, anticipate a gain. But that is a highly constricted situation.

criteria are defective, in general, where they are not directly
connected with utility outcome. The purpose of elections is,
properly, to select candidates who will be best for society, and the
democratic assumption is that this is best done through aggregating
the choices of free people. Various election methods may satisfy or
fail various criteria that are proposed as being in some way
desirable, but the desirability is actually speculative. We already
know, for example, that the intuitively-satisfying Condorcet
Criterion can fail miserably once we realize the matter of preference
strength. (Though, in fact, it would perform reasonably well under
most circumstances because *usually* preference strengths, on
average, are reasonably equal, considered over a large population.
But the circumstances where this may not be true are not necessarily
rare, either.... ICC is an example of the extreme in reduction of
preference strength.)

>Having a voter or faction of voters better informed is not a problem
>for the formal "equality of voting power" in an election.

I disagree. If a voter has prior knowledge of the exact election
outcome without his vote, that is a special privilege. I agree that
being "better informed" about the likely votes of others is not a
violation. In this case we have something much more than that, we
have prior knowledge of the outcome of a process which was, in the
design of the criteria, considered to be random.

It is as if the designer of this proof discovered a way in which time
travel *was* possible,by allowing the voter to go *back* and change
his vote. Warren didn't make this argument, Benham did.

Benham designed a method deliberately constructed, in certain
aspects, to make prior knowledge a factor, while leaving all voters
on the same footing. That is, all of them have the opportunity to
change their vote. This can make AFB failure possible even if the
method would not otherwise fail. However, the attempt of the proof
was to show that *any* method satisfying all the assumptions would
fail, under some election conditions -- which doesn't mean
methodological conditions, it means the state of the candidate set
and of the electorate and how they vote. What Benham has shown
(reasonably well, it seems) is that a method can be designed which
can fail, by providing special provisions in the method that cause it to fail.

Those who favor ranked methods will simply reply -- Then don't allow
votes to change after the random selection!
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