On 09/16/2012 02:35 PM, Juho Laatu wrote:
On 16.9.2012, at 9.57, Kristofer Munsterhjelm wrote:
(More precisely, the relative scores (number of plumpers required)
become terms of type score_x - score_(x+1), which, along with SUM
x=1..n score_x (just the number of voters), can be used to solve
for the unknowns score_1...score_n. These scores are then
normalized on 0..1.)
It seems to work, but I'm not using it outside of the fitness
function because I have no assurance that, say, even for a monotone
method, raising A won't decrease A's score relative to the others.
It might be the case that A's score will decrease even if A's rank
doesn't change. Obviously, it won't work for methods that fail
mono-add-plump.
What should "candidate's score" indicate in single-winner methods? In
single-winner methods the ranking of other candidates than the winner
is voluntary. You could in principle pick any measure that you want
("distance to victory" or "quality of the candidate" or something
else). But of course most methods do provide also a ranking as a
byproduct (in addition to naming the winner). That ranking tends to
follow the same philosophy as the philosophy in selecting the winner.
As already noted, the mono-add-plump philosophy is close to the
minmax(margins) philosophy, also with respect to ranking the other
candidates.
What should "candidate's score" indicate? Inasfar as the method's winner
is the one the method considers best according to the input given, and
the social ordering is a list of alternatives in order of suitability
(according to the logic of the method), a score should be a finer
graduation of the social ordering. That is, the winner tells you what
candidate is the best choice, the social ordering tells you which
candidates are closer to being winners, and the rating or score tells
you by how much.
If the method aims to satisfy certain criteria while finding good
winners, it should do so with respect to finding the winner, and also
with respect to the ranking and the score. A method that is monotone
should have scores that respond monotonically to the raising of
candidates, too.
I note that some methods like Kemeny seem to produce the winner as a
byproduct of finding the optimal ranking. Also expression "breaking a
loop" refers to an interest to make the potentially cyclic socielty
preferences linear by force. In principle that is of course
unnecessary. The opinions are cyclic, and could be left as they are.
That does not however rule out the option of giving the candidates
scores that indicate some order of preference (that may not be the
preference order of the society).
I think most methods can be made to produce a social ranking. Some
methods do this on its own, like Kemeny. For others, you just extend the
logic by which the method in question determines the winner. For
instance, disregarding ties, in Schulze, the winner is the candidate
whom nobody indirectly beats. The second place finisher would then be
the candidate only indirectly beaten by the winner, and so on.
Turning rankings into ratings the "proper" way highly depends on
the method in question, and can get very complex. Just look at this
variant of Schulze: http://arxiv.org/abs/0912.2190 .
They seem to aim at respecting multiple criteria. Many such criteria
could maybe be used as a basis for scoring the canidates. Already
their first key criterion, the Condorcet-Smith principle is in
conflict with the mono-add-plump score (there can be candidates with
low mono-add-plump score outside the Smith set).
My favourite approach to scoring and picking the winner is not to
have a discrete set of criteria (that we try to respect, and violate
some other criteria when doing so) but to pick one philosophy that
determines who is the best winner, and also how good winners the
other candidates would be. The chosen philosophy determines also the
primary scoring approach, but does not exclude having also other
scorings for other purposes (e.g. if the "ease of winning" differs
from the "quality of the candidate").
If you do that, you get into a problem when comparing methods, however.
Every method can be connected to an optimality measure that it
optimizes. That measure might be simple or it might be very complex, but
still, there's a relation between the method and something that it
attempts to optimize. Discussing methods could then easily end up on
cross purposes where one person says: "but I think minmax is the obvious
natural thing to optimize", and another says "but I think mean score is
the obvious natural thing to optimize", and nobody gets anywhere.
At least with criteria, we have some way of comparing methods. We can
say that this method behaves weirdly in that if some people increase the
ranking of a candidate, the candidate might lose, whereas that method
does not; or that this method can deprive candidates of a win if
completely alike candidates appear, whereas that method does not.
Or perhaps it's more appropriate to say that if we want to compare
methods by some optimization function or philosophy, we should have some
way of anchoring that in reality. One may say "I think Borda count is
the obvious natural thing to optimize", but if we could somehow find out
how good candidates optimizing for Borda would elect, that would let us
compare the philosophies. Yet to do so, we'd either have to have lots of
counterfactual elections or a very good idea of what kind of candidates
exist and how they'd be ranked/rated so that it may be simulated,
because we can't easily determine how good society would be "if" we
picked candidate X instead of Y.
(Well, we might in very limited situations: for example, one could take
the pairwise matrix for a chess tournament once it's halfway through and
use that to find the winner, then determine how often each election
method gets it right. However, it's not obvious if "accuracy at
predicting chess champions" is related to "being able to pick good
presidents", say.)
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