On May 4, 2010, at 6:17 AM, C.Benham wrote:
I think the idea that the CW should always be elected but it is sometimes ok to elect from outside the Smith set is a bit philosophically weird, and not easy to sell.
I think electing outside the Smith set is a healthy idea :-). I agree that it is not the easiest to sell (if someone first brings the Smith set argument in).
If group opinions would be transitive / linear as we expect the opinions of individual voters to be, then one could argue that the cyclic opinions in the Smith set must be "fixed" and in the resulting transitive order it would not make sense to elect anyone else but the first in that order. And that candidate could be only someone from the Smith set.
However, opinions of groups are not always transitive but may contain sincere cycles. The "cycle fixing" approach that I described above removes all the cycles from the opinions and when doing so it ignores and hides the defeats within the Smith set. There are rare cases where the defeats of all the members of the Smith set are stronger than the defeats of some candidate outside the Smith set. In such cases it makes sense to elect that candidate outside the Smith set if the intention of the election is to elect a candidate that would have lowest opposition against her (as Condorcet methods typically do). No good method should have a tendency to elect outside the Smith set, but good methods may well be prepared to elect outside the Smith set in the rare cases where some of those candidates is considered to be a better choice (e.g. with less opposition) than any of the Smith set members.
Human beings may visualize the defeat graph as a structure where the Smith set can be drawn at the top and other candidates below that set. That drawing / imagining technique is based on the hidden assumption of linear preference order of the candidates. The Smith set members are also generally not clones that could be logically replaced with one big bubble (= a new imaginary candidate that would represent all the clones). The cyclic relationships within the Smith set are hidden or maybe shown as strange / illogical curved or backwards pointing arrows. The world of potentially cyclic world of group preferences has been distorted. There is no natural two dimensional geometric way to express the cyclic preferences. The preference order or values describing the level of opposition of each candidate could be expressed in a one dimensional space, but one might not draw the Smith set members together and in the first positions.
The explanation behind electing always the Condorcet winner but not necessarily always from the Smith set is that the Condorcet winner is not defeated by anyone but all the the Smith set members are, and they may be beaten badly when compared to some candidate outside the Smith set.
Juho ---- Election-Methods mailing list - see http://electorama.com/em for list info
