Finding the probable best candidate

[Forest Simmons]

On Sat, 23 Feb 2002, Steve Barney wrote:

> Forest:
>
> Saari's approach doesn't seem to be to "maximize" this or that fairness
> criterion at all. It is not an "axiomatic" approach. Instead, he strictly
> focuses on the ballots themselves, and looks for symmetries, such as reversals
> and circular triplets, which should cancel out and produce complete ties. He
> scrutinizes all positional and pairwise procedures on those grounds, and
> iterative extensions of those methods.
>
> SB

Borda Seeded Bubble Sort yields a tie if and only if Borda yields a tie.

Borda Seeded Bubble Sort does not iterate Borda like Borda elimination
does, nor does it remove any candidate from the process until the bottom
seeded candidate has had its go, so the theorem Saari quoted does not
apply.

The title of this EM thread is "finding the probable best candidate."
"Best" implies optimization of something.  Perhaps minimum number of
uncanceled ties would be Saari's objective. But that still begs the
original question of what makes a candidate "best".

None of your postings have addressed the fundamental problems of Borda
that Rob and I raised.

Borda has two insuperable faults for public elections: (1) clone
sensitivity (2) it encourages highly insincere rankings.

Bubble Sorted Borda largely overcomes both of these problems while giving
ties whenever Borda does.

Assume for a moment that among methods based on pure preference ballots,
Borda is most likely to pick the best candidate when the voters vote
sincerely.

This just says that Borda would be the method to use when the information
available was in the form of sincerely voted preference ballots.

This assumption would do nothing to recommend Borda for actual public
elections, since sincere ballots are extremely unlikely when they are
scored according to Borda.

As I said before, Saari's work is great for showing how valuable Borda is
for applications where the preference information cannot be falsified. But
as we have just seen, even professional judges in Olympic Figure skating
cannot be trusted to vote sincerely.

Saari's book, "The Geometry of Voting," was one of the books that
helped convinced me that Approval was superior to Borda.

Throughout the book, Saari makes implicit use of Cardinal Ratings as a
standard of comparison. That convinced me that CR was at least as good as
Borda.  But CR, like Borda, encourages voters to vote insincerely (given
accurate or inaccurate polling information) but not to the extent of
requiring any order reversals. And Approval gives the same result as CR,
so Approval is as good as CR, while not requiring the fancy ballots.  All
in all, Approval is superior to Borda.

By the way, Approval does better than Borda on symmetries.

Let's take one of your examples:

50 A>B>C
50 C>B>A

Most likely the apparent symmetry is not real; since the preference
ballots do not show intensity of preference, we have no way of knowing if
the B preferences are distributed symmetrically about the middle utility
value. Here are some of the possibilities consistent with these preference
ballots:

50 A>B>>C
50 C>B>>A

OR

50 A>>B>C
50 C>B>>A

OR

25 A>B>>C
25 A>>B>C
50 C>B>>A


Approval will distinguish almost all of the myriad possibilities, but
Borda will not.

This one that Approval does not distinguish

25 A>B>>C
25 A>>B>C
25 C>B>>A
25 C>>B>A

is the one that Borda would break into if B were distributed with precise
symmetry about the middle.

Forest