[EM] eliminations methods like IRV

[Forest Simmons]

Can anything be salvaged from IRV?  I think so: it's an ill wind indeed
that blows no good at all.

One idea implicit in IRV is this:  Keep eliminating the worst candidates
from the rankings until the best choice among the remaining candidates is
obvious.

The idea is appealing. I like it, and I suppose it is part of the reason
that IRV (which claims to implement it faithfully) has garnered so much
support.

The trouble with implementing the "elimination of the worst" part of the
idea, is that determining the "worst" is mathematically equivalent to
determining the "best."  In fact, if you reverse all of the preferences in
the rankings, best and worst switch places. 

This symmetry may be considered one source of IRV's problems; when you
start eliminating candidates from the rankings in the early stages, if you
eliminate candidates other than the worst, you risk eliminating some of
the best, if not the best.

In IRV, you eliminate the candidate with the least number of top
preference votes. Obviously, the candidate with the least number of first
place votes is not the worst; if it were, then by symmetry we could say
that the candidate with the least number of last place votes is the best.
We could disband this whole EM list and go home.

Of course, IRV supporters can say it is not really necessary to know the
worst, just as long as you are not eliminating the best.  In this regard,
is it possible for IRV to pick the same candidate as both the best and the
worst?  In other words, is there a pair of examples which are identical
except for the reversal of preference directions, that both have the same
winner when IRV is applied?

I doubt that IRV is that bad.

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.

If this modification does indeed improve on IRV, then it suggests an
iterative scheme for improving elimination methods. To get the eleventh
improvement, use the tenth improvement (applied to the reversed order
preferences) to do the eliminations.

Note that in this context IRV is just the result of the first iteration
applied to the crude method that says choose the candidate with the least
number of last place preferences.

We shouldn't expect too much from one iterative of a iterative scheme,
especially when starting from a very crude approximation.

Under what circumstances would such an iterative scheme converge to a
solution?  When the method converged would the solution depend on the base
method?

If the base method is Condorcet, and there is a Condorcet winner, then
that winner is chosen immediately without iteration, since Condorcet
applied to the reversed preferences simply reverses the (partial) order of
the results. In other words, Condorcet is invariant under this
transformation.  Is it the only method invariant under this
transformation?  In other words, when the iteration converges, does it
necessarily converge to a Condorcet Winner? 


(More Later)


Forest