[EM] MinMax(AWP)

[Juho]

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