On Sun, Dec 08, 2002 at 07:57:09AM -0500, Raul Miller wrote: > I am uncomfortable this for the axiom that the option ranked last must > lose. It's just too arbitrary. For example, consider also a ballot with > only one option (not that our current system allows this). The resulting > statement is rather akward to accept as being true without proof. Assume you have a non-trivial election, ie with multiple options. A given majority vote some particular option, X, last. That means there are N/2+1 ballots of the form "abcde...X", "edcba...X", etc, where X is always the last option. In this case _any_ of options a, b, c, d, e, ... are preferred over X by that majority, and further, in the traditional Condorcet sense, there can't be any cycles involving X (ie, A beats X, X beats B, B beats A, etc), since that would requires N/2+1 ballots to have the form "...X...Y" -- but there are only N/2-1 ballots left. Indeed, there can't be _any_ options that beat X in the traditional Condorcet sense. Another way of looking at the undemocratic example is to say that the original vote was "B versus D", and that a few people who wanted to stymie the outcome (50:10 in favour of B) introduced a new option "A". In the example given, A fails, but in so doing, also knocks B out of the running. Another way of looking at it is a restatement of the "Condorcet loser criterion", or a modification of the "local independence from irrelevant alternatives criterion" to deal with supermajority requirements. (see http://www.condorcet.org/emr/criteria.shtml) Cheers, aj -- Anthony Towns <aj@humbug.org.au> <http://azure.humbug.org.au/~aj/> I don't speak for anyone save myself. GPG signed mail preferred. ``If you don't do it now, you'll be one year older when you do.''
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