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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