# [EM] Scott--BeatpathWinner & CSSD

MIKE OSSIPOFF nkklrp at hotmail.com
Thu Jun 2 21:55:23 PDT 2005

```On Thu, 2005-06-02 at 05:40 +0000, MIKE OSSIPOFF wrote:
>BeatpathWinner and CSSD, though they're equivalent, are completely
>different algorithms

You replied:

what

BeatpathfWinner and CSSD are two different count rules, implemented via
different algorithms.

But they always give the same results as eachother, so they're said to be
equivalent.

Let me define them here:

First of all, the preliminary definitions that I posted a few days ago for
wv methods apply here. CSSD, SSD, and BeatpathWinner can be considered to be
circular tiebreakers, as referred to in those preliminary definitions.

CSSD uses the Schwartz set:

Schwartz set:

1. An unbeaten set is a set of candidates none of whom are beaten by anyone
outside that set.

2. An innremost 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.

[end of Schwartz set definition]

Cloneproof Schwartz Sequential Dropping (CSSD):

The "current Schwartz set" is the Schwartz set based only on undropped
defeats.

1. Determine which candidates are in the current Schwartz set.

2. If there are no defeats among those candidates, then they win and the
count ends.

3. Otherwise, drop the weakest defeat among those candidates. Go to 1.

[end of CSSD definition]

Before I get to BeatpathWinner, let me define SSD, which is similar to CSSD,
but more proposable, because it stops when there's an unbeaten candidate, a
more natural stopping rule:

1. If one or more candidates are not beaten, they win and the count ends.

2. Otherwise, determine which candidates are in the current Schwartz set.

3. Drop the weakest defeat among those candidates. Go to 1.

[end of SSD definition]

BeatpathWinner:

1. There is a beatpath from X to Y if either X beats Y, or if X beats
someone who has a beatpath to Y.

2. A beatpath from X to Y is a sequence of defeats that makes it possible to
truly say that there is a beatpath from X to Y.

3. The strength of a beatpath is defined as the strength of its weakest
defeat.

4. If the strongest beatpath from X to Y is stronger thana the strongest
beatpath from Y to X, then X has a beatpath win against Y.

5. A candidate wins if no other candidate has a beatpath win against
him/her.

[end of BeatpathWinner definition]

As I said, BeatpathWinner and CSSD always give the same result as eachother,
and so they're said to be equivalent.

The BeatpathWinner algorithm is much simpler and briefer, but the CSSD
algorithm seems more naturally and obviously motivated and justified.

I often refer to BeatpathWinner and CSSD as one method, by calling it
BeatpathWinner/CSSD. I've been recommending it for committees,
organizations, and meetings.

I've been recommending SSD for public elections. In public elections, where
there are no pairwise ties, SSD and CSSD give the same results. In small
committees, SSD isn't clone-independent, and CSSD is. But in public
elections they both are.

For public proposals, SSD is my favorite of the best wv methods, because its
definition doesn't refer to cycles or beatpaths, directly or indirectly.
There has been good reason to believe that that makes SSD more acceptable to
people.

More recently I"ve heard tha MMPO meets FBC (with AERLO, it meets Strong
FBC). Also, MMPO is much briefer and simpler to define and propose, because
it doesn't need the list of preliminary definitions that the wv methods
need. So MMPO has become my favorite public proposal.

I hadn't considered whether MMPO is a better proposal for committees,
organizations and meetings too, but it probably is. Especially since the
most valuable thing about electoral reform in committees, organizations and
meetings is that it provides experience and precedent for public proposals.

Mike Ossipoff

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