# [EM] Fw: IBCM, Tideman, Schulze

Markus Schulze schulze at sol.physik.tu-berlin.de
Mon Jul 24 09:40:48 PDT 2000

```Dear Mike,

you wrote (23 July 2000):
> Markus wrote (23 July 2000):
> > Mike wrote (23 July 2000):
> > > That SSD definition has a natural & obvious motivation
> > > & justification that Schulze doesn't have. Both your
> > > definitions use beatpaths.
> >
> > But the SSD definition uses Schwartz sets. And the
> > definition of Schwartz sets uses beat paths.
>
> The Schwartz set can be defined in terms of innermost
> unbeaten sets. The Schwartz set is the set of candidates
> who are in innermost unbeaten sets. But my SSD definition
> yesterday avoided mentioning the Schwartz set, and only
> spoke of innermost unbeaten sets.

You wrote (23 July 2000):
> That isn't simple. This is simple:
>
> 1. An unbeaten set is a set of candidates none of whom are
>    beaten by anyone outside that set.
> 2. An innermost unbeaten set is an unbeaten set that doesn't
>    contain a smaller unbeaten set.
> 3. Drop the weakest defeat that is among an innermost unbeaten
>    set. Repeat till there's an unbeaten candidate.

It is dangerous to believe that you can make an election method
simpler simply by using the term "innermost unbeaten set" instead
of "Schwartz set" and simply by avoiding explaining properly how
this set is actually calculated.

***

In so far as SSD and Schulze differ only when there are pairwise
ties, it is necessary to consider examples with pairwise ties to
decide which method is better. But when there are pairwise ties
then SSD violates independence from clones.

Example:

Suppose that the Senate uses SSD to elect its President
pro tempore. Suppose that 50 Senators are Democrats and
50 Senators are Republicans. Suppose that the Democrats
nominate three candidates A1, A2 and A3 and that the
Republicans nominate only one candidate B. Then a possible
situation looks as follows:

40 Senators vote A1 > A2 > A3 > B.
35 Senators vote B > A2 > A3 > A1.
15 Senators vote B > A3 > A1 > A2.
10 Senators vote A3 > A1 > A2 > B.

The pairwise matrix looks as follows:

A1:B = 50:50
A2:B = 50:50
A3:B = 50:50
A1:A2 = 65:35
A1:A3 = 40:60
A2:A3 = 75:25

SSD elects candidate B decisively.

On the other side (A1,A2,A3) is a set of clones. And when this
set of clones is substituted with a single makro candidate A
then the situation above looks as follows:

50 Senators vote A > B.
50 Senators vote B > A.

Therefore, independence from clones says that candidate B
must be elected with a probability of 50% and that one of the
candidates A1,A2,A3 must be elected with a probability of 50%.

***

You wrote (23 July 2000):
> It's _obvious_ that the members of an innermost unbeaten set
> are uniquely deserving of winning.

This statement demonstrates that Steve erred when he suggested
that you don't consider the Schwartz set to be important (26 Feb
2000). When you say that "it's _obvious_ that the members of an
innermost unbeaten set are uniquely deserving of winning" then
this includes that you consider the Schwartz criterion to be
_very_ important.

Markus Schulze
schulze at sol.physik.tu-berlin.de
schulze at math.tu-berlin.de
markusschulze at planet-interkom.de

```