On May 8, 2010, at 12:43 AM, Kristofer Munsterhjelm wrote:
Juho wrote:
On May 7, 2010, at 7:11 PM, Kristofer Munsterhjelm wrote:
Schulze's primary argument is that the use of paths let one make a
method that is very close to Minmax, yet is cloneproof and elects
from Smith. Thus, if one thinks the Minmax yardstick is a good
one, yet that Minmax's clone susceptibility means one has to
diverge from it in certain cases, Schulze is a good method.
Yes, Schulze has some such properties. If both criteria are
considered important, then one should just estimate which method is
closer to ideal. Minmax may ignore clones that have strong losses
to each others (it puts more weight on the distance to being a
Condorcet winner). Path based methods may defend "clones" also when
there are no clones (and a candidate that meets neither criterion
might win).
Schulze also satisfies some "internal non-contradiction" criteria
that I like, such as reversal symmetry. It seems reasonable that a
method should handle "likes" and "hates" equally but opposite.
However, it would also be reasonable, at first observation, that a
voter can never be worse off by showing up, but that (Participation)
is very strict and almost no methods pass it.
Perhaps there is an element of aesthetics to those criteria
(monotonicity, reversal symmetry, and also monotonicity). This would
fall within the legitimacy meta-criterion, I think; voters would
suspect something fishy is going on if raising a candidate makes him
lose, if polling for "who do you dislike" doesn't return the loser
of "who do you like", etc.
From minmax point of view reversal symmetry is not an obvious
requirement if there are cycles. If three candidates are in a cycle
that is a bad thing. That is a bad thing irrespective of if one looks
at the original or reversed votes. That would thus drop the cyclic
candidates lower in both cases.
A world where voters may give the same answer to question "who should
win" and "who should not win" doesn't appear very natural at first
sight (from transitive opinions based point of view) but that seems to
be possible in minmax. And the explanation is that cyclic defeats are
considered to be as bad as any other defeats (i.e. cyclic clones or
cyclic non-clones will not be treated as if there were no defeats
among them), and the cycle exists when looking from either direction.
In a way there is no agreement to do anything with the cycled
candidates.
As for your second part, there is naturally a tradeoff between
strong paths and short paths. Schulze considers paths equally no
matter their length, but the question is sensible. Methods that
focus on short paths are more like Copeland (which focuses on
"paths" of a single step), and methods that elect from the
uncovered set would have short paths from the winners to the
candidates not in the uncovered set.
I see the "one step philosophy" as answering to question "if we
would elect x, would the society be happy with x or would it be
interested in changing candidate x to someone else" (not on
questions on if the society would be interested in multiple
sequential changes). The philosophy of Copeland's method would make
sense in principle. I guess the minmax philosophy can be said to
focus only on the strength of the losses and not on the number of
them because of the clone related problems that Copeland has. The
number of losses also has no meaning if the intention is to check
how close each candidate is to being a Condorcet winner.
I can see three philosophies/approaches, and associated simple
methods (neither cloneproof):
- The minmax approach: What matters is the worst outcome. Minmax is
its simple method. Schulze fits here because of the strength of a
beatpath being equal to the weakest link.
Yes, the philosophical difference is first step vs. multiple steps.
- The Copeland approach: What matters is the number of short paths.
Second-order Copeland goes here as well.
- The least reversal approach: What matters is the sum of victories
or defeats. Condorcet least-reversal fits here, as does the
"attacker's version" where the candidate with the greatest victory
sum wins. These methods seem to get low Bayesian Regret.
Ranked Pairs, I'm not sure. Its idea is more complex, perhaps
embodied by the immunity against majority complaints criterion,
which goes like: If voters that support Y (and Y beats the winner X)
complain that Y should have won, not X, then those who support X can
point out that X beats Y at least as strongly through an indirect
path, no matter who Y is.
It's possible to get Ranked Pairs closer to the Copeland approach,
as by Short Ranked Pairs (http://www.mail-archive.com/[email protected]/msg04266.html
).
The simplest explanation to Ranked Pairs might be that strongest
opinions must be respected and weaker ones can be violated. Just like
many other rules this seems very rational.
Actually I think the problems mostly come from the fact that people by
nature see preferences as transitive preferences and therefore the
impact of cyclic group opinions is confusing and many of their
properties seem irrational. Many of the simple rules take one concept
from the world of transitive opinions and then try to force that on
the cyclic opinions. What would be needed is a change from Newtonian
(transitive) rules to relative rules that are valid also in the cyclic
cases.
I have no idea where River fits into this - or Kemeny, for that
matter.
Kemeny seems to focus on the full raking of the candidates (not only
on the winner) and finding the least distorting way to do this. If one
wants to define a full ranking of the candidates then this approach is
very natural. (But often the main target of single-winner methods is
to just pick the ideal winner and full "transitivization" of the group
opinion is not needed.)
Juho
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