Majority Criteria

[Mike Ossipoff]

As I said, mathematicians, including Bruce, have defined
the Majority Criterion in such a way that it's met by 
pretty much every method (maybe not Borda).

On the other hand academic mathematicians have also defined
that criterion in such a way that _no_ method meets it:

A method meets the Majority Criterion if & only if any
alternative that is favorite to a majority will always win.

No method meets that criterion, because there's no assurance
that people will vote their favorite alone in 1st place--
especially with a method like Copeland or Regular-Champion,
where there's often strategic incentive to not vote one's
favorite alone in 1st place.

So, as with the academics' "Condorcet Criterion" which would
be more useful as a standard than as a criterion, we could
define the Majority Standard by saying that a method does
well by the majority standard if it does a good job oe
electing an alternative that's favorite to a majority.

Because Condorcet, not Copeland, is the method that doesn't
give defensive strategic incentive to not vote one's favorite
alone in 1st place, it's Condorcet, not Copeland that does
well by the Majority standard.

***

Or we could define a Majority Criterion which, unlike the
academic ones, is met by some methods but not others
(that's the most useful kind of criterion):

A method meets the Majority Criterion if a favorite
to a majority will always win if everyone votes it alone
in 1st place unless they have strategic incentive not to.

I admit that "strategic incentive not to" is imprecise wording,
but you know what it means (For instance, whom are you voting
for in November?). Besides, the wording could be made more
precise if necessary.

Plainly the method that doesn't give incentive not to vote
one's favorite over everything else is the method that 
meets this Majority Criterion. That's Condorcet's method.
Actually I don't know that any method _strictly_ meets
that criterion, because, not giving strategic incentive
to not vote one's favorite alone in 1st place is 
a special-case way of saying the part of the Defensive
Strategy Criterion that Condorcet meets for all practical
purposes, but not strictly.

Still, Condorcet comes a lot closer to meeting that criterion
than does Copeland. Still, since it can't be said to be
_strictly_ met, maybe it would he better to speak of the
Majority standard discussed earlier, instead of the Majority
Criterion.

***

But, for something precisely stated that's strictly met by
some methods but not others, here's what I now call the
Generalized Majority Criterion:

A method meets the Generalized Majority Criterion if & only if
it will never elect an alternative with a majority agasint
it unless every alternative in the Smith set has a majority
against it.

***

If one wanted to really be exacting, one would leave out the
words "...in the smtih set...". Then only plain Condorcet
would meet the criterion. But because of the importance of
the Smith set in making methods meet certain criteria, to
avoid criticism, and because Smith//Condorcet meets the
standards & criteria I've named as well as plain Condorcet
does, the above wording, containing the words "...in the
Smith set..." seems a desirable, and not completely arbitrary,
compromise.

The fact that only plain Condorcet can say that it never
elects an alternative with a majority against it unless
every alternative has a majority against it, that counts
as an advantage that plain Condorcet has over Smith//Condorcet
& Schwartz//Condorcet. But the abililty of those latter 2
methods to avoid criticism (whether the criticism is warranted
or not), and the fact that those 2 methods have the other
properties that I've described and meet the othler critreria
& standards that I've named, suggests the desirability of
including the words "...in the Smith set" in the
definition of the Generalized Majority Criterion, as I
did when I stated it above.

That compromise probably seems inelegant to a mathematician,
and someone could even try to abuse it by replacing
"Smith set" with the set of winners of their favorite method.
But for the reasons I've given, and because the Smith Criterion
(and other criteria implied by it) is important to some people,
it seems reasonable to expand the definition of the Generalized
Majority Criterion so that it isn't automatifcally failed by
methods meeting the Smith Criterion.


Mike





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