[EM] Saari's Basic Argument

[Alex Small]

Steve Barney said:
> In this case, a second pref must be given less than 1/1,000,001 of the
> weight  of a first pref, if A should win. That's a stretch.

So, what is the appropriate weight for a second choice?  Half the weight
of first?  Equal to first?  Maybe a third?  How about 1/Pi?

Possible solutions include:

1)  Let the voters decide--Use Cardinal Ratings, of which Approval is a
variant (albeit a variant with little flexibility).
2)  Make it contingent on the non-existence of a strong first
choice--Majority Choice Approval.
3)  Refuse to specify a numerical weight, and simply insist that it carry
less weight than first choice and more weight then third choice.  IRV and
Condorcet both do that in some sense.
4)  Impose some arbitrary weight and insist that all voters give that
weight to their #2 choice.  There is no real way to pick that weight.  We
can try to use symmetry arguments as Saari does, but we find that it all
depends on the order in which we examine the symmetries.  If we use his
order we get a rather perverse system (Borda).  If we use the opposite
order we get Condorcet, with a rather bizarre completion system.

Anyway, although Saari is insightful in his analysis, his recommendations
make me wonder if he isn't smirking and waiting for the world to realize
he was just joking ;)  I mean, how can anybody recommend Borda with a
straight face for any public election?



Alex


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