[EM] RP elects A & B in Forrest's example

[MIKE OSSIPOFF]

Forrest--

In the example that you posted, Ranked-Pairs chooses A & B.

Likewise, BeatpathWinner/CSSD, ordinary SSD, SD, and PC all choose
A & B.

Norm once objected that RP has a mid-count tiebreaking problem when
it encounters 2 or more equally strongest unconsidered defeats.
But no random tiebreaking is needed. There's a natural deterministic
way to deal with the equal defeats.

Of course the first thing to do might be to look at the margins
when several defeats have the same winning-votes. But in your example
those defeats are tied by both measures, since everyone votes a
complete ranking.

But the equal defeats are still easily dealt with. Just as we
keep a single strongest unconsidered defeat if it doesn't cycle with
already-kept defeats, we could similarly keep whichever of the
equally strongest unkept defeats don't cycle with defeats kept before
those equal ones are considered.

In your example that returns zero winners. But there's another
argument against it: It means that a defeat is nullified only if
it's in a cycle with defeats that are all stronger. Plainly a defeat
should be considered nullified if it's in a cycle consisting of defeats
that are at least as strong as it is.

What that means for an RP equal-defeats rule is:

If there are several defeats that are equally the strongest unconsidered 
defeats, call them the "tie-defeats". Say that each
of the tie-defeats that doesn't cycle with previously-kept defeats
is "qualified". Keep every tie-defeat that isn't in a cycle consisting
only of itself and some combination of kept defeats and qualified
defeats.

It's regrettable that the need for that wording spoils the great
brevity of RP. But it isn't something that has to be brought up
initially when proposing RP publicly. It's important mainly in small 
committees.

For public proposals, the definition without the tie-procedure
could be presented first, getting justified credit for its brevity.
Then the tie-procedure could be mentioned afterwards, so as to not
hit the person with it all at once.

Returning to the matter of what various methods do in your example:

Methods that choose A:

Plurality, IRV, Runoff, original CS with rankings and Euclidian distance, 
and the new method.

I've found that original CS with rankings & Euclidian distance
is vulnerable to truncation so as to not meet SFC.

Methods that choose B:

Borda, Bucklin, Approval, original CS with rankings and city-block
distances.

(With Approval I assumed that utilities of ranked candidates are
equally-spaced, that the election is 0-info, and that voters use
above-mean strategy, though it probably isn't really quite optimal
with so few voters. With Bucklin I assumed that people rank at least
as many candidates as they'd vote for in Approval).

I'm not qualified to discuss how the new CS method should be done,
because I haven't studied linear algebra. But of course I suggest that
the way of doing it be chosen so as to achieve the best properties.

Mike Ossipoff








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