[EM] "good method" ? was "IRV ballot pile count (proof of closed form)"

[Abd ul-Rahman Lomax]

At 01:59 PM 2/14/2010, Kathy Dopp wrote:
> >> > From: Chris Benham <cbenhamau at yahoo.com.au>
> >> > 35:A
> >> > 32:B>C
> >> > 33:C,
> >> >
> >> > by which I mean
> >> >
> >> > 35:A>B=C
> >> > 32:B>C>A
> >> > 33:C>A=B.
> >
> > Kathy doesn't seem to recognize this, or maybe she does, but the two
> > statements are equivalent. By not ranking B and C, the voter
> > equal-ranks them bottom. That is the exact effect of the vote.
>
>I seem to be one of the few people on this list who recognizes that I
>don't read voters' minds and cannot convert one vote-type to another
>for voters.

Kathy, there was no reading of voter's minds. What was expressed was 
the votes themselves. Not the voter's internal, unexpressed preferences.


>For example, in the above example, by:
>
> >> > 35:A
>
>Some voters if they chose to rank further might have meant:
>
>A>B=C
>or they might have wanted:
>
>A>C>B
>or
>A>B>C

Whether they "wanted" that or not, they did not vote that. It's like 
Plurality, with three candidates, A, B, and C:

35:A

is exactly the same as

35:A>B=C.

Which, of course could reflect an internal preference profile that is 
just that, or it could reflect on where the voter prefers B to C or 
the reverse.



>and the same for
> >> > 33:C
>
>We disagree on whether or not you and other members' interpretations
>of how voters would alter their votes are self-evident or not.

I did speculate on *possible* alterations, but nobody presented this 
as a "self-evident" interpretation of this. What was stated was just 
that in a three-candidate election, 35:A is the same vote as 35:A>B=C.

There is an exception to this. Suppose the method is two-rank 
Bucklin. The voter votes first rank, A. Second rank, B and C. This is 
indeed a literal vote of A>B=C, and it can have a different effect, 
depending on conditions, than the bullet vote A. That's because it 
could be read as an implicit approval of both B and C, thus a runoff 
election might be avoided.

But that was not the context being discussed, and it was about how a 
Condorcet method is -- I asserted -- a plurality method, that, unless 
there is some kind of majority requirement -- to be defined, to be 
sure -- it can elect without the explicit approval of a majority of voters.

And all single-ballot methods that don't coerce voters are "plurality 
methods," in this sense.

>   Poll
>100 voters, I doubt that the mind-reading abilities of persons on this
>list will hold uniformly for all of the voters you poll.

Except, Kathy, there was no mind-reading. There were only two 
alternate ways of stating the same vote that simply expressed it with 
complete expressed preference profiles. Truncation is, in general, 
equal-ranking-bottom. It's possible to design methods where it means 
something different (for example, consider Approval voting, where the 
voter votes Yes or No on each candidate, but then doesn't vote on 
this question for some. It's an interesting method, in fact, but 
that's beyond the scope of this. It corresponds to the Average Range 
that is proposed by many Range advocates. I consider it to be 
probably politically impossible at this time, but it would be 
interesting to study. It *is* how multiple conflicting ballot 
questions are decided. The basis for majority approval is different 
for each question.

>We may disagree with the counting method that is applied when
> >> > 35:A
> >> > 32:B>C
> >> > 33:C
>
>occurs, but it seems very clear that the Condorcet winner in this case
>is C, as you seem to agree with me in this case.

Yes. The A voters express no preference between B and C. A is the 
plurality winner. But only 35 voters support A, 65 oppose A by their votes.

A beats B, 35:32.
but C beats A, 65:35. That's because the B voters express their 
preference for C over A.

B, on the other hand, gets no support from the C voters over A. The C 
voter *means* A=B, bottom ranked equal.

In most voting methods, not voting for a candidate is quite 
equivalent to ranking the candidate bottom or voting *against* the candidate.

This example was *not* an example of majority failure in a Condorcet 
method, which points out how, unless there is some approval cutoff 
specified, giving a candidate any rank above bottom can become a vote 
*for* the candidate (in any pairwise race with a lower-ranked candidate).

Now, with approval cutoff expressed, one can do a different kind of 
analysis. Suppose there is a dummy candidate called Z. Ranking a 
candidate above Z means that one approves the candidate. The 
practical meaning of this is clear if a runoff is needed if a 
candidate doesn't gain a majority.

So the B votes in the example above might be A>Z>B>C or A>B>Z>C. It's 
also possible that it could be Z>A>B>C, meaning I'd prefer to see a 
runoff in any case, I really don't like any of these candidates and 
maybe we could get some write-in campaign together or I can do more 
investigation and change my mind.... Or it could be A>B>C>Z, meaning, 
I'd prefer any of the three to holding a runoff, there isn't that 
much difference between them, and the worst of them is still good enough.

>However, that is not the case when we make assumptions for how the
>voters would change their votes (see below)

But there was no such assumption. There was only the normal 
interpretation of the votes.

> >>I agree with Chris (below), If you require every winner to "have a
> >>majority *over*" ever other candidate, then there is no system that
> >>would give you any winners.  Clearly above, C has 65 votes and B only
> >>has 35 votes, at least in scenario #1 above.
> >
> > Actually, with the votes above, C has a plurality over every 
> other candidate:
> > A:B 35:32, A
> > B:C 32:33, C
> > C:A 65:35, C
> >
> > C wins every pairwise election, C is the Condorcet winner. "Condorcet
>
>Yes. As I think I said in my first email. But A is the winner if you
>alter the votes as per the email I was responding to.

You did not quote that.


> > winner" does not at all require a majority in every pairwise
> > election, and, in fact, that was part of my point. It's possible that
> > the Condorcet winner only has a plurality in all of them. But in this
> > particular case, a majority of voters have chosen to cast a vote that
> > can be read as a vote for the Condorcet winner. All we have to do is
> > take the B>C votes as such. They are clearly votes against A. So
> > two-thirds of the voters have voted against A. A's out.
> >
> > 65 voters have voted for C, but only 35 have voted for B.
> >
> > "Vote for" means "cast a vote that can be used to elect."
> >
> >>Guessing as to what voters really mean, by assuming scenario #2 from
> >>scenario #1 -- you may have read the minds of all those voters who you
> >>believe all think exactly alike in each category, incorrectly.
> >>However, in scenario #2, I think A is the correct winner.
> >
> > They are the same scenario, in fact. I think you misread this, Kathy.
> > Equal-bottom is the same as not expressing the candidate's rank at all.
> >
> >>I think election methods enthusiasts too often think they can read
> >>voters' minds and translate votes between between two different
> >>scenarios for voters.
> >
> > Perhaps you mean that the B=C part of A>B=C means that the voter
> > really did mean to actually equate them. Okay, let's look again. I
> > assumed in a previous mail, Range utilities of
> >
> > 35: A, 3 / B, 1 / C, 0
> > 32: B, 3 / C, 2 / A, 0
> > 33: C, 3 / A, 1 / B, 0
>
>In this case, A is the Condorcet winner 68 persons preferring A over
>B, 67 preferring B over A, 65 preferring C over A, 67 preferring B
>over C, etc. not C as in your first, not-equivalent, example.

In other words, here I did assume that you were taking a vote of 
A>B=C as meaning real expressed equality. Which cannot be, in fact, 
read from the original votes. So I looked at the situation with that 
interpretation. But what is quoted above is the previous 'guess' at 
what the voters might mean from the votes, how they *might* be 
interpreted. Not mind-reading, but simply suggesting an 
interpretation that is consistent with the votes, and not claiming 
that this ruled out other interpretations.

The reason for the range analysis is that raw preferences do not tell 
us nearly as much as a real preference profile that includes relative 
preference strengths. If we start with a real preference profile, we 
can then, to some degree, predict votes or voting patterns.

Basically, if we have preferences of A:3, B:2, C:0, the voter is more 
likely to vote A>B than if the preferences are A:3, B:1, C:0, in 
which case the voter may vote A>B or just A. If the voter actually 
equates B and C, then the voter is indeed likely to truncate, just 
voting for A, and the vote A>B or A>C doesn't really make much sense.

In the example above, A is not the Condorcet winner, for C beats A by 
65:35. The Condorcet winner is the "beats-all" winner. In the 
expanded example, there is no Condorcet winner, there is a cycle, 
because B beats C as well, and A beats B.

>So it appears that Range voting does not find the Condorcet winner 
>in this case.

In this case, there is no Condorcet winner so there is no Condorcet failure.

However, a Condorcet method might come up with A as a winner, but 
that's "condorcet method" as distinct from "condorcet winner." 
Condorcet methods have means of resolving cycles, and if a win with 
the maximum number of voters is considered more important than a loss 
with fewer voters, then this method could resolve with A.

> > Range totals:
> >
> > A: 138
> > B: 131
> > C: 163

C wins.

Now, I assumed that if the voters really felt that B was a worthwhile 
candidate, they would not have bullet voted for A, but would have 
added a ranked preference for B. And this would correspond to the 
vote of A:3, B:2, C, 0. In other words, the preference of B over C is 
strong, relatively. So I wrote a value of 1 for the rating as an 
upper bound. But, of course, the vote could equally be considered to 
favor C over B with the same arguments. What falls out from this is 
that the inferred preference strength between B and C, as seen by the 
A voters, is a maximimum differential of 1/3 vote.

I'm looking at a ranked voting scenario and attempting to extract 
some underlying utility data. It's speculative. It's not 
mind-reading, not for any individual voter, for sure, but rather 
looking at what utilities would make sense of the votes. There are 
obviously alternate interpretations.

For example, if the B voters actually had preferences of B:3, C:1, 
A:0, but did not think that B had much of a chance (which would be 
inaccurate unless they had *very* accurate poll data), they might 
have assigned more importance to the C:A race and voted as they did. 
(Whereas I inferred from this vote that their preference profile was 3, 2, 0.

Voting systems that do not allow the expression of preference 
strength in some way are doomed to ambiguity. Range ballots allow 
direct expression, and this could be so useful for studying election 
phenomena that I'd like to see Range ballots used even if, say, 
Condorcet analysis or Bucklin analysis or even IRV analysis is used.

(It would have to be IRV with equal ranking allowed, which is 
theoretically a little better, but which still suffers from problems 
like center squeeze.)

Basically, a Range ballot can be used for preference analysis, 
whereas a preferential ballot, ordinarily, cannot be used for Range 
analysis without making the kinds of assumptions that you were 
protesting, it seems. I think the assumptions I made were 
*reasonable*, but certainly not definitive.

> > This allows the A voters to still have a preference between B and
> > [C], but at a lower level. If, in fact, they had no preference, the
> > result becomes simply a lower sum of ratings for B. And if the C
> > votes really did mean that A and B were equal, the result becomes
> > fewer votes for A.
> >
> > C still wins.

In this case, as well, C is the Condorcet winner from the shown 
preferences. Remember, in the original election C was the winner 
because of truncation, so it was a bit misleading to then talk about 
the "Condorcet winner" from the imputed utilities. Those expressed 
preferences and bounded them, but they were not voted.

And I showed that whether or not the preferences were actually equal 
or existed at the lower possible level, the result was still C, from 
Range analysis. That is, I showed this if I didn't make some big 
mistake, always a possibility.

I went into some details.... no more original content it below.

> > If the method were Bucklin, then, again, C would win, easily, with
> > 65/100 voters approving of C, if the votes were as writ.
> >
> > With Bucklin, would more of the B voters truncate? Maybe. Maybe not.
> > Depends, doesn't it?
> >
> > It depends on how strong their preferences are. If it's Bucklin and
> > everyone truncates, A wins, by a narrow margin. Same as with
> > Plurality. But if a majority is required, it would go to a runoff.
> >
> > Do we know, then, who would win? No, we do not. We do not have enough
> > information! If the B votes really do show a higher approval of B
> > voters for C, then C might win, but in that case some of them would
> > probably also approve of C in Bucklin.... and then C could still win
> > in the first round.
> >
> > The scenario I worry about here is that B is really the Condorcet
> > winner, and the Range winner, and the B voters were merely more
> > willing to disclose lower preferences. IRV will choose A and C for
> > the instant runoff, and top-two runoff for a real runoff, which is
> > fine, except for that contingency.
> >
> > I do believe that Bucklin would handle this well, in general. Some
> > voters will add lower preferences, enough to show, even if there is
> > majority failure requiring a runoff, what the best two candidates
> > would be for the runoff. Again, if write-ins are allowed in the
> > runoff, and the wrong two candidates get there, and there is real
> > preference strength behind that error, the voters can fix it.
> >
> > They will be in a better position to do so if the ballot is actually
> > a range ballot, they will have a better idea of the chances of a
> > write-in campaign in the runoff.