[EM] Why wv is nonfalsifying. Definitions.

[MIKE OSSIPOFF]

Before I start--A few days ago I posted an example in which
the margins methods' only equilibria were ones in which order-reversal
was being used. They were equilibria in which the C voters were using
defensive order-reversal to protect B, the sincere CW, against
offensive order-reversal by the A voters. That example showed that
the margins methods are falsifying.

Winning-votes:

By the premise of these definitions, there's a sincere CW, and
there's no indifference between the CW & the other candidates.

That means that, for each candidate other than the CW, there's
a majority who prefer the CW to that other candidate.

Suppose that everyone sincerely ranks everyone down to the CW,
but doesn't rank anyone whom they like less than the CW.

Of course the CW is the winner. Suppose some people who like X
better than the CW order-reverse by ranking Y over the CW.

A majority of the voters prefer the CW to X, and so they rank the
CW, but not X. That means that X has a majority against him.

That majority of the voters don't rank X, and so it's impossible for
X to have a majority against anyone, since a majority aren't ranking
him over anyone.

If the order-reversal caused some candidate, Y, to beat the CW,
and X has a beatpath to Y, then conceivably X could win with
some circular tie solutions.

But with BeatpathWinner, the CW has a majority-strength beatpath
(a 1-defeat beatpath) to X, and X can't have that strong a beatpath
against anyone, since he can't beat anyone by majority.  So X can't
win in BeatpathWinner, since he has at least one beatpath win
against him, from the CW.

RP never drops or declines to keep a defeat unless it's the weakest
defeat in a cycle, according the procedure that defines RP.

The defeat of X, by the CW, is stronger than any defeat by X, and
is therefore never the weakest defeat in any cycle containing X.

So the CW>X defeat never gets dropped or passed-up for keepting.
X doesn't win in RP.

I showed in early 2000 that any defeat among the members of the
Schwartz set is in a cycle. If the defeat from A to B isn't in
a cycle, then there's  no beatpath from B to A. By the beatpath
definition of the Schwartz set, then, B isn't in the Schwartz set.
So any defeat that's among members of the Schwartz set must be
in a cycle.

I showed some time ago the equivalence of the beatpath definition
and the unbeaten set definition of the Schwartz set.

So SSD never drops a defeat unless it's the weakest defeat in a cycle.
If it wasn't initially the weakest defeat in a cycle, then dropping
other defeats won't make it the weakest defeat in a cycle, since
dropping defeats can't create a stronger beatpath that wasn't already
there.

So X can't win in SSD for the same reason that X can't win in RP.

In PC, X has a majority defeat, and the CW never has a majority
defeat, and so X can't win in PC, since it always has a stronger
defeat than the CW does.

X can't win by order-reversal. If truncation could make some Y
beat the CW, it would be a weaker defeat, and wouldn't improve
X's chance of winning over order-reversal.

Obviously extending one's ranking past the winner, to someone
less-liked can't improve a voter's outcome.

No change in strategy--order-reversal, truncation, extension of
a ranking--can improve anyone's outcome.

I have to finish up right now, but if I've left anything out,
I'll add it in a subsequent posting.

Mike Ossipoff


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