[EM] Simmons's Method

[Anthony Simmons]

Forest,

You've posted so much in the past few days that I'm unable to
respond in kind.  There's a lot there to think about.

One thing I wonder about is the idea of coming up with a set
of criteria that any decent system can satisfy.  I understand
the appeal of having a way of recognizing the best possible
method, and then deriving it from the criteria.  But it
doesn't look encouraging, does it?

> But here is a related problem of IRV for which examples
> are easy to supply: When IRV is applied to the reversed
> rankings to figure out the worst candidate, if that
> candidate is eliminated instead of the one with the fewest
> first place preferences (in the search for the best
> candidate), then continuing IRV as usual results in a
> different (and better?) choice.

Aha!  This business about getting a different winner
depending on who is eliminated first is a major difficulty
for IRV, since, as you say, the choice of who to eliminate
first is a pretty crude one.

A quick check shows what happens if you have a three-cornered
race with a preference loop.  A > B, B > C, and C > A.
Anyone could win, depending on who is the initial loser.
Basically, whoever wins the first round wins the election.
Multiple iterations don't change anything.  So in the case of
a three-way, (on first glance, anyway), the rule is:  If
there's a CW, he/she/it wins, otherwise use IRV, Simmons's
Method,  or whatever the base method is.  So the puzzle
becomes how to choose the optimal method for when there's no
CW.

> Weak version of IIA: The winner shouldn't change if the
> worst candidate is thrown out.  This seems like a
> reasonable requirement for a decent method.

But is "worst candidate" well defined?  Perhaps only if
"worst candidate" doesn't change when the winner is thrown
out?  But the winner isn't defined until "worst candidate" is
defined ...

I think a problem here is that each step is a local one, but
depends on knowing the answer to a global answer, while the
global answer depends on the local steps.  Iterative to the
bone.

Tony