[EM] Rouse reply II

[MIKE OSSIPOFF]

Michael Rouse wrote:

Otherwise, find the Smith set (which I usually see
                 defined as the smallest set where none of the candidates 
outside the set
                 beat any of the candidates within the set in a pairwise 
contest).

I reply:

The Smith set is the smallest set such that every candidate in
the set pairwise-beats every candidate outside the set.

Rouse's definition is the definition of the smallest unbeaten
set. Something along those lines is desirable. But there can be
2 or more disjoint unbeaten sets; the smallest unbeaten set isn't
necessarily inside the other unbeaten sets. So it seems arbitrary
to choose from a particular unbeateaten set merely because it's
the smallest one.

But a set has been defined that achieves what "smallest unbeaten
set" is intended to achieve:

The Schwartz set:

1. An unbeaten set is a set of candidates none of whom are beaten by
   anyone outside the set.
2. An innermost unbeaten set is an unbeaten set that doesn't contain
   a smaller unbeaten set.
3. The Schwartz set is the set of candidates who are in innermost
   unbeaten sets. (In other words, the Schwartz set is the union
   of the innermost unbeaten sets).

[end of definition]

In public elections, where there are no pairwise ties or equal
defeats, the Schwartz & Smith sets are indentical. Of course the
Smith set is briefer to define, and so it should be the one used
when definiing public proposals.

For small committees, you might want to substitute the Schwartz set
for the Smith set, if the electorate are tolerant of the longer
definition.

SSD speaks of the Schwartz set, because when SSD drops defeats,
that's like new pairwise ties, and so the Schwartz & Smith sets are
no longer the same. And the Schwartz set is the one that suits the
requirements that are served by SSD.

Mike Ossipoff

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