On May 7, 2010, at 12:19 AM, Jameson Quinn wrote:
In particular, for Schulze voting, here's the pitch: "The basic idea
is to elect the person who wins against all others. If there's no
such person, you try to eliminate the minimum number of ballots
until there is. But you don't want to have to bring the ballots to a
central location and then try every combination of ballots to
eliminate. So there's a process that is designed to almost always
give the same answer, but can be done using a local count..." (now,
if people ask, you can describe the method.)
I know, the beatpath is not always the same as eliminating ballots,
if you have more than 3 candidates in the Smith set and [some other
improbable criteria which are too involved to state here]. But for
me, personally, I am more likely to support the Schulze method now
that I understand it as a summable approximation of minimal-ballot-
elimination. And for those who support it more than I do, I think
that pitching it as such is honest and useful.
The concept of ballot elimination is a bit complex since there could
be different kind of ballots and some kind of ballots (that we would
like to eliminate) might not exist. I guess you were saying that if we
had all the original ballots available (not only the summable pairwise
comparison matrix) then we could see how many voters we need to ignore
to get a Condorcet winner.
Another closely related concept is how many additional voters we would
need to make someone a Condorcet winner. The benefit of this approach
is that it is simpler and now we are not bound to the actual votes. It
is enough to just use simply additional bullet votes if we want to
lift one of the candidates above the others (later preferences have no
influence on the pairwise comparisons between the to be winner and
other candidates).
With the additional votes concept the answer to what is the optimal
method from this point of view is also obvious. It is minmax(margins).
I don't know how much the result would change if we would use the
concept of eliminating some (existing) votes instead. There are also
other criteria like the clone criterion. Minmax(margins) fails that
criterion (in some special situations). This means that the lowest
number of additional votes criterion and the clone independence
criterion are mutually exclusive.
One could discuss which rule should apply in those special cases when
both criteria can not be met. In order to determine exactly when we
have true clones in our hands we would need to have the original
votes, and also the preference strengths to know if the candidates are
closely related or not. (Actually also near clones should be treated
as clones since we can not expect that all voters treat those
candidates as clones.) The pairwise matrix contains only partial
information. If we make a method 100% clone proof using the matrix
information only we can not limit to the clone cases only but we are
bound to influence the result also in other cases. One pairwise matrix
can be obtained both from votes that have clones and from votes that
do not have clones. It is for example possible that no voter ranks
together those candidates that we must now deem to be (potential)
clones since there is a possibility that in some other vote set they
could be clones. The Schulze method uses path heuristics to eliminate
all cases where clones could exist and influence the end result.
(Are there other strong reasons behind the use of paths? In real life
the existence of the long beat paths maybe doesn't refer to any
natural key target.)
I'll ask myself second time which rule should apply in those special
cases when both criteria can not be met.
In minmax(margins) one could have three strongly looped clones that
together as a group have 51% support, and one more candidate with 49%
support. If one knows that all those circular preferences are weak in
the sense that most sincere opinions (with utility information) of the
clone supporters look like C1=99 C2=98 C3=97 A=1 then minmax(margins)
maybe makes a mistake and one of the clones should be elected instead
of A (whose worst loss was much smaller in terms of pairwise
preference counts than the worst loss of any of the clones). On the
other hand those clones could be severely fighting against each others
and the strong circular opinions and resulting strong opposition could
hamper the work of those clones (or "clone looking bitter enemies of
one wing of the political spectrum") if elected, in which case A (that
is anyway few votes short of being a Condorcer winner) might be a good
winner.
In another possible situation the same pairwise matrix has been
generated from votes where all voters rank one of the C candidates
first and one last and thus never all together. In this case the
argument that the candidate that needs only few additional votes to
become a Condorcet winner should win gets stronger since there is no
clone argument present. Any clone proof method that uses the pairwise
matrix to make the decisions must pick one of the (non-clone) C
candidates in this case.
Another characteristic feature of the Schulze method is the use of
winning votes. My understanding is that the history behind winning
votes is mostly based on strategic voting related concerns. Unless use
of winning votes is considered ideal for sincere votes, this decision
means some deviation from electing ideal winners wit sincere votes.
Yet another possible factor that may influence this discussion on what
the basic idea behind Schulze method (and other methods) is is the
concept of implicit approval cutoff after the ranked candidates. Some
criteria and discussion on what the ideal winner is do refer to the
assumption that voters have indicated that they support/approve those
candidates that they have ranked and do not support candidates that
they have not ranked. (Depending on the ballot type and number of
candidates and interpretation that could mean truncation or candidates
ranked equal last.) This interpretation of the votes is thus not
purely ranking based but includes also additional information. One
problem of this approach is that if voters behave this way then will
not express their preferences on the preference between those
candidates that they do not like, and that could mean high level of
truncation and bullet voting. Implicit approval argumentation usually
appears together with arguments on why winning votes are natural or
how they work.
My understanding of the history of developments behind the Schulze
method is that in addition to Condorcet compatibility one has aimed at
summable matrix, 100% independence of clones, defence against some
strategic voting patterns (=> winning votes), deterministic decisions
(best candidate elected, no lotteries, except when exact ties). I
don't think the interest to aim at minimum number of ballots that must
be eliminated (or added) has been a key target, at least not the
leading one. The strategic concerns must have been strong if one
considers winning votes not to be optimal with sincere votes.
In addition I tend to think that Smith set compatibility (that is
related to clones) and maybe some interest to serialize the group
opinions (related to Smith) have played some role (not necessarily a
good target). The evolution of the Condorcet methods has gone from
simpler methods to more complex ones, with the intent to patch some of
the identified problems. In such evolution process it is possible that
some fixes may unintentionally cause more damage than they fix
problems. For example in the area of strategies it is typical that a
modification that helps in some set-up will make the vulnerabilities
worse in some other set-up (e.g. winning votes). Similar balance
related problems may appear also in performance with sincere votes
(e.g. additional votes vs. clones).
Maybe Markus Schulze and others that have worked with and studied
Ranked Pairs, River etc. can give some more light on the historical
and current motivation.
Juho
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