[EM] Problem solved (for pure rank ballots): ICC & AFB incompatible (essentially)
Abd ul-Rahman Lomax
abd at lomaxdesign.com
Fri Jan 26 08:53:45 PST 2007
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.
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