Hello Jobst, James,
On Aug 10, 2005, at 14:02, Jobst Heitzig wrote:
James:
2. Does the Condorcet criterion plus the independence of
clones criterion imply the Smith criterion?
Rule a: Picks the Condorcet Winner if it exists, otherwise determine
which candidates are defeated *most* often and pick one of them at
random. Fulfils Condorcet Criterion and Independence of Clones, but
certainly not the Smith criterion.
Does this algorithm meet the Independence of Clones criterion? I'm not
sure if I got the definitions right, but here is an example that shows
what I was thinking. The idea is that the most defeated candidate is in
the d-e-f clone set, but if one of the a-b-c clone set candidates is
removed, we have a Condorcet winner in the a-b-c clone set.
11: abcdef
11: bcaefd
11: cabfde
10: defcba
10: efdbac
10: fdeacb
Here is also another counterexample algorithm candidate.
Find the largest clone sets (genuine subsets of the set of candidates).
Replace the clone subsets in the ballots each with one symbolic
candidate representing the whole clone set. Find the MinMax winner. If
the winner is a symbolic candidate representing a clone set, take the
original ballots, eliminate all other candidates than the members of
this clone set and start again from the beginning.
This modified MinMax should still be Condorcet compliant and
non-Smith-compliant. And the modifications should make it independent
of clones.
BR, Juho
((P.S. Some time ago I brought up the idea of using "voluntary and
manually generated clone wannabe sets" (=parties) on this list (April
28th). The algorithm above finds the clone sets automatically but it is
quite rough. It would be quite possible to enhance it so that the
border line between clone sets and "close to clone sets" would be
smooth (strengths of defeats would be weighed based on the
strength/weakness of clone relationship between the candidates).))
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