[EM] Why RP(wv) & CSSD meet GSFC

[MIKE OSSIPOFF]

Stephane wrote:


>Strategy-Free Criterion (SFC)
>
>If an Ideal Democratic Winner (IDW) exists, and if a majority prefers
the IDW to another
>candidate, then the other candidate should not win if that majority
votes sincerely and no
>other voter falsifies any preferences.

Both "majority" definitions fit.
B is the IDW.
A is the other candidate.

I reply:

We use the .5+ majority definition. IDW is another way of saying
sincere CW.

Staphane continues:

If I understand well, you mean
"No group should be able to defeat a Condorcet winner simply by
truncating its preferences"

I reply:

Yes, but it's more than that. It applies to all methods. For instance
IRV fails SFC even if no one truncates or votes insincerely, as
does Plurality (except that, of course, every Plurality ballot
could be called an involuntary truncation). But, with pairwise-count
methods, SFC effectively means what you said above. Truncation can't
steal the election from a CW whom a majority prefer to the truncators'
candidate. Most imporantly, though, SFC &/or GSFC compliance means
that, in the absence of preference falsification, everyone can
vote sincerely, with no penalty or possibe reason to regret it.
No one need consider strategy when voting. I consider that the
biggest & most important gain possible from rank methods.


I'd said:

>Sincere preferences:
>
>100: ABC
>49: BAC
>75: CBA

Stephan replied:

>I see how Ranked Pair with winning-votes will
ensure B to still win
100: A
49: BAC
75: CBA

It works for this example, do you have a proof describing why it would
always work?

I reply:

The demonstration that RP(wv) meets SFC is similar to the demonstration
that it is conditionally completely expressive.

Let me show that RP(wv) & BeatpathWinner/CSSD meet GSFC. Anything
that meets GSFC meets SFC. Since you asked about RP(wv), I'll do
that one first.

Say that no one falsifies a preference, and that X is in the
sincere Smith set, and that Y is not, and that a majority prefer
X to Y and vote sincerely.

(The sincere Smith set is the set of candidates such that every candidate in 
that set is preferred to every candidate outside the
set by more voters than vice-versa. In other words, under sincere
complete ranking, every member of that set would beat every nonmember).

If the defeat of Y by X isn't in a cycle, then of course it will
not be passed-up for keeping by RP. To make Y win, it would be
necessary to make a circular tie of which Y is a member.

Since there's no falsification, that must be done by truncation.
Some voters could truncate to let some Z outside the Smith set
beat someone in the Smith set. Maybe Y beats Z.

Say that a cycle like that is successfully made. By GSFC's
premise, a majority of the voters, .5+, prefer X to Y & vote
sincerely. In a rank method that means that they rank X over Y,
since rank balloting allows that.

So the XY defeat is a majority defeat. But there can be no majority
defeat in which a nonmember of the Smith set beats a member, because
more people prefer each member to each nonmember than vice-versa,
and no one falsifies. The cycle inluding Y and the Smith set members
must include a defeat of a Smiths set member by a nonmember, for
obvious reasons.

So, the XY defeat can never be the weakest defeat in a cycle,
because the defeat by which a nonmember of the Smith set beats a member
can never be a majority defeat.

Since the XY defeat can never be the weakest defeat in a cycle, it
can never be passed-up for keeping in RP, by RP's rules. To be
passed-up, a defeat must cycle with already-kept defeats, stronger
ones.

So the XY defeat is kept in RP, and Y can't win.

BeatpathWinner/CSSD:

For the reasons I've just described, no member of the Smith set
can have a majority defeat from outside the Smith set, and the XY
defeat is a majority defeat.

Y can't win without a beatpath to X, because X has a beatpath to Y.

Can Y's beatpath to X be as strong as X's beatpath to Y? X's beatpath
to Y is a 1-defeat beatpath, a majority beatpath consisting of 1
majority defeat.

Any beatpath from Y to X must, at some point, include a defeat by
a nonmember of the Smith set to a member of the Smith set, since
X is in the Smith set and Y isn't.

But no defeat of a Smith-member by a nonmember can be a majority
defeat. That means that no beatpath from Y to X can be a majority
beatpath. X has a beatpath win against Y. Y can't win.

RP(wv) and BeatpathWinner(wv)/CSSD both meet GSFC, which means that
they both also meet SFC.

Mike Ossipoff





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