Condorcet and multi-winner elections

[David Catchpole]

THE CONCEPT OF IRRELEVANT ALTERNATIVES AND CONDORCET
This concept I believe was first detailed by Kenneth J. Arrow in his book
"Social Choice and Individual Values". My understanding is as follows-
	*Any withdrawal of any candidate or group of candidates from an
election in which they would not win if they stood should not influence
the outcome of that election.*

Basically, the vote shouldn't have the potential to be split.

One can prove that in a single-winner election, if such a situation
exists, the winner is a Condorcet winner (and thus that such a
situation exists only where a Condorcet winner exists). Picture this: for
IA to be satisfied in an n-candidate election with one winner, any
(n-1)-candidate elections against that winner must be won by that winner,
and any (n-2)-elections... and so on to the level of 2-candidate elections
where the winner contests the others one-by-one (and it's a basic norm
for us to assume that the winner in such contests has a simple majority).
This will by definition be a Condorcet winner. The moral of the story is
that there is only no potential for splitting the vote where a Condorcet
winner will be elected.

TAKING CONDORCET INTO MULTI-WINNER ELECTIONS
I contend that taking IA into multi-member elections will also give us a
set of Condorcet winners, and that if such a set is empty then IA is not
satisfied. picture, as before, an n-candidate election for w winners. For
IA to be satisfied, the outcome of any contest involving the winners and
any combination or number from the losers should elect those winners, and
again this boils down to the winners winning against any other
one candidate in, obviously, (w+1)-candidate elections.

A challenge then is designing selection in these (w+1)-candidate
elections. I'm going to offend you silly pro-Hare Quota Americans by
advocating STV/Quota Preferential with a revised Droop quota (more on this
in a later submission). So, say there's an election of 3 representatives
from a pool of 5 candidates. A 3-winner group which was in the set of
IA-satisfying outcomes will be the winners of the STV vote in any election
in which the three "winning" candidates and one other competed.


EXPECT LATER-
Electoral rules for this system where a set of "Condorcet winning groups"
does not exist or when the set holds more than one group, and a discussion
of exclusion and the concept of Homogeneity, and

A defense of Droop.

I would love to get criticism from anyone.
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David Catchpole