# [EM] Benham and Woodall for small electorates. Schwartz Woodall.

Michael Ossipoff email9648742 at gmail.com
Tue Mar 19 10:35:34 PDT 2013

```To allow for pairwise-ties, possible in small electorates (small
elections and polls), I define Benham in terms of unbeaten-ness:

Do IRV till there's an uneliminated candidate who isn't
pairwisee-beaten by any other uneliminated candidate. Elect hir.

[end of Benham definition]

Woodall can be similarly improved, by specifying the Scwhartz set
instead of the Smith set. Of course in public elections, where
pairwise-ties are rare, the Smith and Schwartz sets are the same.

Schwartz Woodall:

Do IRV till there's only one uneliminated member of the initial Schwartz set.

[end of Schwartz Woodall definition]

Definitions of the Schwartz set:

Definition 1, Cycle definition:

The Schwartz set is the set of candidates who don't have a pairwise
defeat that isn't part of a cycle.

A cycle is cyclical chain of defeats: X>Y>Z...>X.

Definition 2, Unbeaten-Set Definition:

1. An unbeaten set is a set of candidates none of whom are
pairwise-beaten by any candidate 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.

The cycle definition is briefer, but the unbeaten set has its own
compellingness. But of course being in a cycle reduces the
meaningfulness of a defeat.

Michael Ossipoff

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