Patching Up Condorcet In Multi-Winner Elections

[David Catchpole]

MULTIPLE CONDORCET "WINNER GROUPS"
This e-mail is the follow up to the one I put in before, outlining how one
could guarantee the "Irrelevance of Alternatives" in a multi-winner
election if an option existed to do so. As with single member Condorcet,
there is the possibility that such an option does not exist; however, with
more than one winner, there is an added threat- that more than one
"winner group" of candidates exists which satisfies my IA interpretation-
	*That any contest between that "winner group" and one candidate
outside it would result in the "winner group" being successful*

Solving this problem, we note that any candidates outside any "winner
group" will not have any influence on the outcome, so we ignore them. The
problem then is to decide between "winner group" options. My preferred
option is (numbers in brackets indicate loops)(urgh)-

(1) your typical at-large household garden STV amongst those
candidates who are part of "winner groups" (but that's MY STV, of course),
but without the threat to dishomogeneity by exclusion (my objection to
going out of our way to exclude candidates will come up later in the
piece). If not all quotas are met (revised Droop quotas, so suck on eggs,
silly US barbarians), or the resultant group isn't a "winner group", one
simply conducts the n elections where one of these is missing in the
"melee" each time (2), and finds if that works to find a single "winner group"
which wins every time it is in those elections; if it finds "winner
groups", but not just one, one ignores those candidates outside them and
starts at (1) again. If there are again no "winner groups" from the n
elections which leave out one "winner", one tries the n*(n-1)/2 elections
which leave out two "winners" and keeps on trucking as (2), letting the
size of the "electorates" slip by one every time. If the dimunition of
"electorates" gets to the point where they contain only the number to be
elected plus one, or one is stuck in a rut because the pool of "winner
groups" fails to get smaller, then you finally get to the point where we
might try overt exclusion (or um-thingy-can't-remember-his/her-name:
raise threshold or weight preferences).

LIVING WITHOUT CONDORCET- EXCLUSION AND HOMOGENEITY
Unfortunately, there are cases where either no "winner group" exists, as
is common with Condorcet, or it is difficult for us to decide between
those groups. In this case it may be prudent to exclude a candidate
independent of their "winner group"/s. This is dangerous, as it may lead
to a paradox- voters might hurt candidate's chances by giving them higher
preferences. As people experienced in electoral issues, I'm sure you're
aware of this threat. It is a quite serious one with both plurality
exclusion (smallest highest preferences) and Black exclusion (highest
lowest preferences). One system of exclusion which I think holds promise
is effectively to hold an STV election of n-1 people (with Droop quota-
ha!) which leaves one person out. I haven't had the time or skill to try
to find if there are any contradictions of homogeneity from this system,
but I would appreciate any responses.

I would appreciate any criticism.
---------------------------------

David Catchpole