On May 5, 2010, at 10:28 AM, Kristofer Munsterhjelm wrote:
Juho wrote:
Let's say that there are three different criteria that an election
method should meet. They are related to three different strategic
voting related problems. As in the world of security also election
methods are as vulnerable to strategic voting as their weakest
link. In this situation there may be a solution that almost meets
all the three criteria but does not meet any of them fully. That
solution is often (but not always) the best solution.
So, there are patterns where a good method should meet only a
strict subset of the requirements that some competing methods meet.
One may thus improve a method by making it incompatible to some
(good) criterion that it earlier met. The point is that
compatibility with various criteria should often not be an on/off
comparison but a richer analysis where also partial and almost
complete compatibilities are counted.
This seems to be an issue of the criteria being too demanding.
Consider, for instance, independence from covered candidates. I
think Forest Simmons said (in private correspondence) that IFCC
implies nonmonotonicity.
If one considers monotonicity important, but also would like to have
the method be independent of as many types of non-winning candidates
as possible, the natural next thing to do is to ask: how strong can
we make this independence criterion and still satisfy monotonicity?
Yes, either the criteria are too strict or alternatively the laws of
nature don't allow us to build perfect methods but we must always
violate some important criteria. In the pairwise comparison based
methods we must also take into account the fact that we do not measure
the strength of personal preferences, and therefore the decisions that
we make are often simplified assumptions (i.e. one set of votes may
refers to a wide range of different possible sincere preference
strengths). The pairwise matrix for example can not be used to
determine which candidates are clones (or almost clones) (same matrix
can be derived from ballots with clones or from some very different
set of votes). Performance with sincere votes is one step easier to
handle and define than performance when also strategic voting is
included.
In some earlier discussion I proposed concepts like IIA-IAC (where IAC
means "in the absence of cyclic preferences"). That approach gives us
in one rough lower bound that for example in the case of IIA could be
set as a requirement. After that one could either try to fine-tune the
criterion to more exactly describe the level that we can reach, or
alternatively we could take a practical approach and just estimate how
often and how serious problems the "irrelevant" candidates can cause
when there are cycles. It may be that often the optimal level of
meeting IIA is not at some level that can be clearly defined as a new
criterion but at some more vague level where the resistance to these
problems just happens to be so small that we need not care about that
problem any more.
Also monotonicity is a criterion that could be met only "in most
cases". For example in multi-winner STV the non-monotonic features
typically fall below the noise level and do not cause any complaints
(or are not even noticed or can not be be proven). (There is however
no reason to deviate from monotonicity if one is not forced to so for
other reasons.)
Similarly for your own situation; say that Smith compliance breaks
something else you want. Then you can try to find out what that
thing that you want is, and afterward attempt to find the closest
thing to Smith compliance that doesn't break it.
My use of the Smith set criterion (at least in some environments /
societies / needs if not all) is such that Smith-IAC is obvious (same
as Condorcet criterion). When there are cycles then I try to see what
I'd like to happen with sincere votes. Most Condorcet based approaches
try to minimize the opposition against the elected winner. In 99% of
the cases (vote sets) it makes still sense to elect one of the Smith
set members. But there are cases (<1%) where opposition against the
Smith set members is (according to some definitions) is higher than
against some canididate outside the Smith set. In that case and with
these criteria (on who should be elected with sincere votes) it makes
sense to elect outside the Smith set. The border line and definition
of those cases is very clear if we use the amount of additional
support needed to overcome the opposition to measure the level of
opposition. We thus have a clear definition of the modified (less
strict) Smith set criterion (but it is easier to define that criterion
as the number of required additional votes than to modify the Smith
criterion although the new criterion is close to the Smith set
criterion).
Although that border line is very well defined there are still some
questions left. Many different sets of sincere opinions and actual
ballots may lead to one pairwise comparison matrix. In some situations
the Smith set members could be clones (maybe actual clones from the
same party with minor differences in preferences, maybe just
candidates that look like clones but are actually highly competitive
and against each others). Or one may find a mutual majority that would
have strategic interest not to elect the candidate outside the Smith
set. Or the cyclic preferences could be a result of some strategy. But
on average and as a general rule and for practical use the default
rule of measuring the strength of opposition after the election (from
the pairwise comparison matrix) could be used. (Except that in real
life these situations are probably so rare that most Condorcet methods
lead to the same results anyway (with full rankings at least).)
(For the record, I think Smith compliance is a good thing, and that
perhaps one should go even further - thus my interest in uncovered
set methods, IPDA, etc.)
Is that a rule for all societies / elections? Or a rule for most
societies / elections?
Are you saying that there are no such elections where it would be in
the interest of the society to elect the ("most agreeable") candidate
with least opposition (to change the winner to one of the others)? Or
maybe only that it would be better to limit the winners to the Smith
set (and further) if if one must elect one method as a method that
will be recommended as the best general purpose single-winner method
for all societies and in all (or most) single-winner elections of the
world.
Juho
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