[EM] Why CSSD & PC are monotonic

[MIKE OSSIPOFF]

There could be an error here, but it appears to me that CSSD & PC are both
monotonic.

CSSD is Cloneproof SSD, which is equivalent to BeatpathWinner.

In BeatpathWinner, if I raise candidate X's position in my ranking, without 
changing the
order of the others, I might increase X's defeat of someone, or I might 
decrease his
defeat by someone, but that's the only way that I affect any defeats.

If I decrease his defeat by someone, then I'm decreasing one or more 
beatpaths to him.
If I increase his defeat of someone, then I'm increasing one or more 
beatpaths from
him. Those are the only ways that I'm affecting beatpath strengths.

Comparing X to some other candidate Y, then, I might be increasing X's 
strongest
beatpath to Y, or weakening Y's strongest beatpath to X. In no way could 
that make Y
have a beatpath win against X that Y didn't already have.

So, by upranking X, without changing the orders of the other candidates in 
my ranking,
I can't make X not win, if he'd previously won.

With PC (Minimax), by upranking X without changing the order of the other 
candidates,
I might decrease defeats of X, or I might be increasing defeats of other 
candidates.
That won't increase X's strongest defeat, and it won't lessen someone else's 
strongest
defeat. So if X's strongest defeat was less than anyone else's strongest 
defeat, before
I upranked X, then it must still be.

I don't call those proofs,  and their hasty wording might not be as explicit 
as it should.
But you know what I mean, and it could be worded more explicitly. Aren't 
those
arguments valid  & convincing?

Any general definition of Monotonicity will be expected to result in a 
requirement that
upranking someone without changing the order of the other candidates mustn't 
make
him lose. Even though we don't have a general definition yet, it will have 
to be like that,
in order to be accepted.

MIke Ossipoff


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