[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:


I reply:

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 

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 

[end of Schwartz set definition]

Cloneproof Schwartz Sequential Dropping (CSSD):

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

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]


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 

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 

[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 

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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