Clarifying the subcycle rule & its effect

[Mike Ossipoff]

I've recently posted new definitions, like those of the subcycle rule,
the Generalized Majority Criterion, & the Lesser-of-2-Evils Criteria.
Sometimes I've improved or clarified these while writing them, and it
would be best if I clearly state my final definitions for these things.
Especially since Bruce has claimed that I haven't precisely defined
any criteria.

***

The subcycle rule:

Only after all cycles are solved is a final winner chosen by Condorcet's
method (when this rule is being used). Cycles are solved as follows:
If a cycle contains a cycle as one of its elements (a subcycle of the
cycle), that subcycle is solved before solving the cycle. A cycle or
subcycle is solved by determing which of its members has a better
Condorcet score than the other members of the cycle; the other members
are the eliminated, and that member with the best Condorcet score
occupies the place formerly occupied by the cycle.

[This isn't inconsistent with Smith//Condorcet or plain Condorcet]

***

When the subcycle rule is worded in that way, it ensures that
both Smith//Condorcet & plain Condorcet will strictly meet both
of the Lesser-of-2-Evils Criteria (without it, they strictly meet
only Lesser-of-2-Evils Criterion #1, LO2E1).

That's because, since a group, M, of voters consisting of a majority
have ranked the alternatives in set S1 over the alternatives in S2,
all the S2 alternatives have a majority against them. If the M voters
refuse to include S2 alternatives in their rankings, then it isn't
possible for any S2 alternative to have a majority against a non-S2
alternative. That means that the only way for a non-S2 alternative
to have a majority against it is for another S2 alternative to have
a majority against it. And that means that the only for there to not
be any alternative without a majority against it would be for there
to be a cycle among the non-S2 alternatives, such that every one of
them has a majority against it.

But the subcycle rule will get rid of any such cycle, leaving only
its winner. So there will be at least 1 alternative outside of S2
that doesn't have a majority against it, and therefore no alternative
in S2 can possibly win, if the M voters refuse to include S2 alternatives
in their rankings, and if the subcycle rule is used, and, of course,
if the M voters have all ranked the alternatives in some set S1 over
the S2 alternatives.

***

Incidentally though, as I said, it would be so difficult for 
supporters of S2 alternatives to create a cycle among the non-S2
alternatives, and such a long-shot for them to count on there
being one, that that's why I say that plain Condorcet & Smith//Condorcet
meet LO2E2 for all practical purposes. Especially since, with 
Smith//Condorcet, S2 order-reversers would have to create a circular
tie in which S2 alternatives are members, AND somehow engineer a
cycle among the non-S2 alternatives such that every alternative
outside S2 has a majority against it. Difficult, since the S2
voters aren't a majority, and would need much predictive knowledge
and improbable luck to succeed. This would require their engineering
a subcycle as an element of the larger cycle that is the Smith set.
That's asking really a lot of them. Forget it, Condorcet meets LO2E2
for all practical purposes, even without the subcycle rule.

***

Mike





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