[EM] Demonstrations that BeatpathWinner meets WDSC & SFC

[MIKE OSSIPOFF]

WDSC's premise says that a majority of the voters prefer X to Y.  WDSC's 
requirement says that they must have a way to ensure that Y loses, without 
reversing a preference.

If that majority rank X over Y, then Y has a majority defeat. A majority 
defeat is a 1-defeat majority beatpath. Therefore there's a 
majority-strength beatpath from X to Y.

Can there be a majority-strength beatpath from Y to X? Well, say that that 
majority don't rank Y. That means that they aren't ranking Y over anyone. 
That means that X can't pairwise-beat anyone by majority. Since every 
beatpath must start out with a defeat, that means that there can't be a 
majority beatpath from Y to anyone, including X.

And that means that there's a majority-strength beatpath from X to Y, but 
not from Y to X. The strongest beatpath from X to Y is stronger than the 
strongest beatpath from Y to X. That means that Y can't win, because someone 
(X) has a beatpath win against Y.

That majority were able to make Y lose merely by ranking X over Y, and not 
ranking Y over anyone. They didn't have to reverse a preference in order to 
make Y lose.

[end of demonstration that BeatpathWinner meets WDSC]

While we're at it, how about why BeatpathWinner meets SFC:

SFC compliance is tested only in examples in which there's a CW, because its 
premise mentions a CW.

The premise of SFC says that no one falsifies a preference, and that a 
majority prefer the CW to Y and vote sincerely. The requirement says that Y 
shouldn't win.

By my definition of sincere voting, if voters are allowed to rank as many 
candidates as they want to, a sincere ranking must vote all of the voter's 
preferences.

So the stipulation that a majority prefer the CW to Y and vote sincerely 
means that that majority rank the CW over Y.

As before, that gives Y a majority defeat, which is a 1-defeat majority 
beatpath from the CW to Y.

Can Y have a majority defeat to the CW?

In the previous demonstration, I mentioned that a majority beatpath 
obviously must start with a majority defeat, since all of its defeats must 
be majority defeats. Likewise, a majority beatpath must end with a majority 
defeat. For a beatpath from Y to the CW to end with a majority defeat, the 
CW must have a majority defeat.

Can the CW have a majority defeat if no one falsifies a preference? If for 
any candidate Z other than the CW, it must be that more prefer the CW to Z 
than prefer Z to the CW. That's the definition of a CW. That means that 
there can by no majority preferring someone to the CW. And, if no one 
falsifies a preference, then there can be no majority voting someone over 
the CW. That means that the CW can't have a majority defeat. Since a 
majority beatpath must end with a majority defeat, that also means that 
there can be no majority beatpath from anyone to the CW.

Since there's a majority beatpath from the CW to Y, and there can't be a 
majority beatpath from lY to the CW (because there can't be a majority 
beatpath from anyone to the CW), then the CW has a beatpath win against Y. 
That disqualifies Y from winning, by the definition of BeatpathWinner.

So, as required by SFC's requirement, when SFC's premise conditions are met, 
and the method is BeatpathWinner, candidate Y can't win. BeatpathWinner 
meets SFC.

[end of demonstration that BeatpathWinner meets  SFC]

In my earliest advocacy of wv, I pointed out that Worst's majority defeat 
will be recorded, preserved, and always counted. I pointed out that that 
means that majority rule will be honored, and that lesser-of-2-evils voters 
will be reassured that their pairwise vote against Worst will be recorded 
and preserved, as will its effect in reducing Y's winnability, as described 
by WDSC and SFC.  I pointed out that those things aren't so in margins. 
Those things affect Worst's ability to win in BeatpathWinner just as surely 
as they do in PC or Smith//PC, as shown by the demonstration above.
Those are general facts about wv, arguments in general for wv.

As I said in an earlier reply: Though voting systems are important, I'm only 
doing this because I like to, with no goals regarding results. That's 
because what someone else does or doesn't do isn't my department. But it's 
fun to find out what kind of opposing arguments I get. The lengths to which 
some people, like Markus, will go are remarkable, and often hilarious, and 
provide much entertainment.

Mike Ossipoff

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