[EM] Smith//PC , GSFC, & SDSC

[MIKE OSSIPOFF]

Markus asked how Smith//PC can fail GSFC & SDSC.

Let me just tell how it can happen--outlines of examples. Let me
start with SDSC:

[In these example outlines I use "that majority" to refer to the
majority who, in SDSC, prefers A to B, and who, in GSFC, votes
someone in the sincere Smith set over B]

Say a majority prefer A to B and that some of them also like C, D,
& E better than A. Say that their sincere preferences among C, D, &
E are such that (maybe including other people's votes) C, D, & E
would be in a cycle of majority defeats if that majority voted
sincerely.

Now, say the B voters vote C, D, & E over A, so that the B voters'
ballots, plus those of the aforementioned majority, if they vote
sincerely, will make A be majority-defeated by C, D, & E.

That majority's way of making sure B can't win is giving him a
majority defeat and making sure someone else doesn't have one.

If we replaced {C, D, E} with just C, then it would be easy. That
majority need only refuse to rank B. Then B doesn't have a majority
against anyone, so that when they all vote A over B, B has a majority
defeat and C can't have one. So  B can't win.

But that won't work if we have {C,D,E) instead of C. The members of
that majority who prefer C to A have to vote that preference, otherwise
they're insincerely ranking equal. So they can't keep A from being
majority-beaten. And the members of that majority can't decline to
express their preferences among C, D, & E, because some of that
majority like C, D, & E better than A, and in order to vote A over
B, they must rank A. And so, in order to not insincerely rank equal,
the must rank C, D, & E over A. They can't decline to express their 
preferences among C, D, & E.

That means that C, D, & E are majority beaten, and the members of that
majority can't prevent that without insincerely ranking equal.

In that example outline, I've specified votes of people outside that
majority, and preferences of members of that majority, as called for
by the criterion. That's what's called for, because the criterion
speaks of a way of voting for that majority, and requires that it
not violate their sincere preferences in a certain way. Since the other
people's votes aren't specified by the criterion, the example writer
can specify them any way he wants to.  The criterion requires that
that majority prefer A to B, but says nothing about their other
preferences, which means that the example-writer can specify those
other preferences of theirs any way he wants to.

GSFC:

5 candidates: A1,A2,A3,B, & C.

A1,A2, & A3 are in a cycle of majority defeats, and they're the
sincere Smith set. A majority prefers a member of {A1,A2,A3) to B,
and no one falisifes a preference.

Now, if, instead of {A1,A2,A3}, we just had a 1-candidate sincere
Smith set, A, then Smith//PC would pass the criterion in that example.
Because A can't have a majority defeat, since he's SCW and no one
falsifies a preference. Since B has a majority defeat, and
A doesn't, then A can't win. Of course if B is in the voted Smith set,
then so must A be, since A beats B.

Why doesn't that work when A is replaced with {A1,A2,A3}? Because
they're in a cycle of majority defeats. Their majority defeats could
be greater than that of B. B's majority defeat could be weaker than
any other defeat in the election, and B could win by having the weakest
maximum defeat.

Of course if B is in the sincere Smith set, qualifying him to win
if his greatest defeat is the least among that set, then A1, A2, &
A3 are also in the sincere Smith set, since they beat B.

Any BC complying method meets both criteria, because that majority
has the power to ensure that B hasn't a majority defeat against anyone
(by not ranking B). Since they can give B a majority defeat, and ensure
that B doesn't majority-beat anyone, that means that B has a defeat
that isn't the weakest defeat in a cycle.

BC says "Never choose a winner who has a defeat that isn't the weakest
defeat in a cycle". So any BC complying method meets SDSC & GSFC.

Note that B isn't in the sincere Smith set, but that doesn't mean that
he can't be in the voted Smith set. It would be easy to write an
example in which he's in the voted Smith set, and therefore wins if
his greatest defeat is less than those of the other members of that
set.

This criterion of course says something about voters' sincere
preferences when it specifies that {A1,A2,A3} is the sincere Smith
set. Other than that, it merely specifies that a majority vote
someone in the sincere Smith set, {A1,A2,A3} over B. Everything else
is unspecified by the criterion and thererore can be specified by
the example-writer any way he wants to.

If these example outlines don't convince you that Smith//PC
fails GSFC & SDSC, then say so and I'll post actual numerical
examples. But it's obvious that such examples could be written. There's
nothing impossible about the example outlines that I've written here.
Markus, if you ask for numerical examples, would you tell me why
you believe that these example outlines aren't possible?


Mike Ossipoff


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