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

[Abd ul-Rahman Lomax]

At 12:46 AM 1/26/2007, Dave Ketchum wrote:
>My initial reaction to your words:  If not about public elections, 
>then why bother?

It is theoretical work. It has practical applications, *maybe*. Was 
Arrow's theorem about "public elections." No, it is about elections, 
period. Public, NGO, private.

>Public elections are such an important part of the purpose of this 
>group that anything that does not apply to them should be labeled accordingly.

This group is *full* of highly theoretical work.

>Sure seemed like Warren intended it to be inclusive except, to me 
>his discussion methods do not quite fit.

He's presenting a formal proof. He's a mathematician. He's not 
writing a piece for popular consumption. There is a great deal of 
such work on this list. Have you been following the apportionment 
discussions which are mostly between Smith and Ossipoff? It is highly 
technical and theoretical.

The work has practical applications, yes, but it is at another level, 
the level of theory.

>Certainly theory and other types of elections are worthy topics - 
>provided their applicability is understood.

No. Theory is of value entirely on its own. Application of theory to 
practice is a completely different matter. It actually requires, 
often, different expertise. It's the difference between science and 
engineering. Science is generally about research and exploration, 
engineering is about practical application.

Often theory exists for a *very* long time before practical 
applications are found.

Of course, theory of elections has obvious practical application, but 
it is not obvious that this particular issue does, beyond a certain 
political point which might be made with it. The fact is that it is 
not at all difficult to show a major defect in election methods based 
on pure ranks without consideration of preference strength, and this 
defect is addressed by Range. That does not, by itself, mean that 
Range is superior, because Range might suffer from *other* defects. 
However, some of the criteria by which Range is commonly judged as 
defective are "contaminated" with the assumption of ranked ballots 
and "intuition" based on thinking for several centuries only about them.

If you have a group of people, and they need to pick from two 
choices, and slightly more than half of them have a very slight 
preference for A over B, but the other half would be very unhappy 
with A, but love B, it is obvious that a group which cares about the 
maximum satisfaction of *everyone* would pick B. That choice would 
leave everyone satisfied, choosing A would leave half of the 
population, so to speak, out in the cold.

But the Majority Criterion and the Condorcet Criterion both would 
require A to win. This is so blinkin' obvious that it is amazing that 
it has not attracted much attention. Range methods solve this 
problem. But do they introduce other problems? That is a major question.

Range and Approval do not suffer from Favorite Betrayal. It never 
hurts you to do your best to elect your favorite with these methods. 
As to what additional votes you cast, that may be subject to some 
degree of strategy, particularly with coarse Range (and Approval is 
as coarse as Range could get.)

What about ICC? That is a matter currently under debate. It depends 
quite a bit on definitions. There are Range *strategies* which, if 
followed in a fixed manner, are vulnerable to ICC. But the method 
itself is not, not by any reasonable interpretation of the intuition 
behind ICC.

The intuition is that the introduction of a candidate who is 
identical to another candidate, in the eyes of all voters, should not 
change the election outcome, that is, if the cloned candidate would 
lose, so would the clone. And if the cloned candidate would win, then 
either they will tie (strict clone) or one of them will win (clone 
error less than resolution of Range method, or with preference 
strength less than one rank with ranked methods).

The alleged vulnerability to clones of Range is based on ballot 
normalization. The simplest view of normalization is that voters will 
vote the maximum and minimum ratings, for at least one candidate 
each, in the election. If voters are rating candidates in comparison 
with each other, then, the argument goes, they may give a differing 
rating of one vs. the other. This couldn't happen with a strict clone 
except as noise. I.e., it is equally likely that this variation would 
affect one candidate vs another. This is perhaps why probabilities 
are mentioned in these proofs.

But with clones defined according to the usage in ranked elections, 
they may differ from each other, that is, voters may have preferences 
between them. If this preference shows up in the Range ratings, even 
one point of difference with one voter could, under some 
circumstances, swing the election to or from the clone set. Even 
though the ranks on all ballots are identical. (A Range ballot is 
converted to a ranked ballot by ordering the candidates in order of 
rating. If equal ratings are allowed in the ranked method, then it is 
that simple. If not, then an obvious method of conversion is to rank 
the identically-rated candidates randomly. If the probability of A>B 
coming out of this is equal to the probability of B>A, or similarly 
with more than two equally rated candidates, then it is fair. It is, 
in fact, the coin toss at the end if there is a tie, only it is a 
series of coin tosses, perhaps.

So does Range pass ICC or not? The "ranked" definition of clone was 
written by people who were not considering Range. They were only 
thinking, I would bet, about ranked methods. They first thought of 
cloning as identical, I would presume. (and that is the basis of the 
common-sense definition.) Then, since they were dealing with ranks, 
and there *could* be variations between the clones, i.e., voter 
preferences among them, when considering ICC, they loosened the 
definition to allow the candidates to be other than identical.

But, when used with a method that really allows "ranks" finer than 
the candidate set, it no longer is an appropriate use of the term 
"clone," which, in general usage, means an *exact* copy. (Range 
allows ranks up to the smaller of two numbers: the number of 
candidates, or the resolution of the implementation. So Range has the 
ICC problem -- if we think of it as a problem -- only because it 
collects and uses finer data than pure ranked methods. Basically, for 
a clone to shift a Range outcome, the clone must *not* be identical.

In Range, if two candidates are identically *rated* by all voters, 
then dropping one of them could have no effect on the outcome of the 
election (unless that candidate was the winner, in which case the 
victory would necessarily shift to the other candidate -- unless 
there were more clones.

It is utterly obvious that ICC is a purely theoretical criterion, if 
applied to Range, unless the electorate is very small. However, the 
thinking is that, if ICC is not satisfied, then a political party 
could manipulate the outcome by manipulating the number of 
candidates. There are methods which are seriously vulnerable to this. 
In particular, methods which award value to a candidate by how many 
other candidates are defeated by the candidate, are vulnerable. By 
adding more "defeated" candidates, the outcome can be shifted toward 
the preferred candidate among the clone set.

*This* has practical application. If we look at the practical 
application with Range, however, Range is a method where the rating 
of one candidate is not affected by the presence of other candidates. 
Except when ballots are normalized. If a new candidate is introduced 
who is either Maximum Worst, or Maximum Best, then the ratings of 
other candidates would shift, so the thinking goes. In fact, it might 
not, at least not at the bottom end.

Got it now?

>>No rerun. One of the assumptions, I think, is that the method is 
>>deterministic. That is, it *will* choose a winner, but random 
>>choice is only allowed if there is a tie (or more than one tie).
>
>
>But now we are back to:  responding to a tie by chance should assume 
>equal probability of comparable results.

Yes. In fact, I think that may have been explicitly stated.