# SSD and CSSD Condorcet

## Basic Condorcet

The Basic Condorcet method of ambiguity resolution is the original method proposed by Condorcet himself. It can be stated as follows: drop the weakest (smallest magnitude) defeat, repeating if necessary until one of the candidates is unbeaten.

Although it is superior to Plurality and Instant Runoff Voting, Basic Condorcet suffers from some technical deficiencies compared to the SSD method to be discussed below and is not recommended for use in actual public elections - unless its simplicity makes the difference between public acceptance or lack thereof.

## Schwartz Sequential Dropping (SSD)

The Schwartz Sequential Dropping (SSD) method has a "plain" version and the "cloneproof" version.

The cloneproof version gives no group or party any advantage or disadvantage for having additional candidates that are essentially "clones" of each other. Except for the case of ties, the two versions give the same result. When the number of voters is small, ties are likely and the cloneproof version is needed.**The cloneproof version is slightly more complicated than the plain version, but it works well regardless of the number of voters, so it will serve as our standard. **The cloneproof SSD procedure can be stated as follows:

- Determine the Schwartz set.
- The Schwartz set is the innermost unbeaten set, or the smallest set of candidates such that any candidate outside the set beats none.
- If no defeats exist among the Schwartz set, then its members are the winners (plural only in the case of a tie, which must be resolved by another method).
- Otherwise, drop the weakest defeat among the Schwartz set, determine the new Schwartz set, and repeat the procedure.

Take this ordered list of defeats as example:

1. D/B (60) 2. B/C (50) 3. A/B (40) 4. C/A (30) 5. C/D (25) 6. D/A (20)

The SSD method works as follows for this example:

- Initially the entire set of candidates constitutes the Schwartz set because it doesn't contain a smaller unbeaten set.
- The weakest defeat is D/A, so it is dropped.
- The innermost unbeaten set is now still the set of all the candidates, and the smallest defeat is now C/D, so it is dropped
- Candidate D is now unbeaten and wins.

Note that Basic Condorcet and SSD produced the same winner in this case.

The cloneproof SSD method of resolving cyclic ambiguities has been found to be equivalent to another method called Beatpath Winner, which can be stated as follows:

- X has a beatpath to Y if X beats Y or if X beats another candidate that has a beatpath to Y.
- A sequence of defeats that makes it possible to say that X has a beatpath to Y is called a beatpath from X to Y.
- The strength of a beatpath is measured by the strength of its weakest defeat.
- X has a beatpath win against Y if the strongest beatpath from X to Y is stronger than the strongest beatpath from Y to X.
- The winner (or winners in the case of a tie) is the candidate against whom no candidate has a beatpath win.

## "Plain" Schwartz Sequential Dropping (SSD)

We start with a definition of the Schwartz set:

- An unbeaten set is a set of candidates of whom none is beaten by anyone outside that set.
- An innermost unbeaten set is an unbeaten set that doesn't contain a smaller unbeaten set.
- The Schwartz set is the set of candidates who are in innermost unbeaten sets.

Now, the rules for SSD:

- If there's a candidate who isn't beaten by any other candidate, then that candidate wins.
- Otherwise, calculate the Schwartz set, based only on undropped defeats.
- Drop the weakest defeat among the candidates of that set. Go to 1.

SSD is another favourite Condorcet variation among experts.

## Cloneproof Schwartz Sequential Dropping (CSSD)

**In public elections where the number of voters is large, the chance of pairwise ties is small. In the absence of pairwise ties, SSD and CSSD give the same result. However, SSD can have problems when the number of voters is small, as in small committees.** Then CSSD is the better choice, although it is slightly more complicated. *CSSD can also be used in public elections, of course.*

CSSD is the same as SSD, except for the stopping rule. Whereas SSD stops the count when a candidate is unbeaten, CSSD stops the count only when there are no defeats among the candidates of the current Schwartz set.

Here are the CSSD rules:

- Calculate the Schwartz set based only on undropped defeats.
- If there are no defeats among the members of that set then they (plural in the case of a tie) win and the count ends.
- Otherwise, drop the weakest defeat among the candidates of that set. Go to 1.

## Beatpath Winner

The brief Beatpath Winner algorithm most easily implements CSSD. Beatpath Winner is equivalent to CSSD in the sense that they always find the same winner(s). Beatpath Winner can be implemented in an elegantly brief algorithm. If that is an important consideration, then Beatpath Winner should be considered. Otherwise SSD would probably be a more acceptable public proposal. A beatpath Winner proposal could be justified via the CSSD definition.

Here are the definitions and rules for the Beatpath Winner algorithm:

- X has a beatpath to Y if X beats Y or if X beats another candidate that has a beatpath to Y.
- A sequence of defeats that makes it possible to say that X has a beatpath to Y is called a beatpath from X to Y.
- The strength of a beatpath is measured by the strength of its weakest defeat.
- X has a beatpath win against Y if the strongest beatpath from X to Y is stronger than the strongest beatpath from Y to X.
- The winner (or winners in the case of a tie) is the candidate against whom no candidate has a beatpath wins.

Again, barring pairwise ties or equal defeats, there will be only one winner. Condorcet has many other variations. **Except for Basic Condorcet, all of these variations have no significant differences in merit. The choice should be based entirely on which one would be more acceptable to the public or the enacting legislature.**