# RE: Condorcet Voting and Supermajorities (Re: [CONSTITUTIONAL AMENDMENT] Disambiguation of 4.1.5)

```Buddha Buck wrote:

> The Smith Set is defined as the smallest set of options that are not
> defeated by any option outside the Smith Set.

This definition of the Smith set is incorrect.  Suppose we have:

A>B, B>C, A=C, A>D, B>D, C>D. ('>' means 'dominates', or beats pairwise)

Then by your definition, the Smith set would consist of only {A}, whereas in
fact the Smith set contains three members: {A,B,C}.  In other words, any
candidate which is tied with a member of the Smith set is also a member of
that set.  The corrected definition would be something like:

"The Smith Set is defined as the smallest set of options that defeat
('dominate') all options outside the set."

I mention this, because a number of messages posted recently to debian-vote
contain this subtle error, and if an incorrect definition was used in a
constitutional amendment, it could affect the interpretation of results in
certain elections.  Your incorrect definition has a name, too -- it's called
the 'Schwartz Set'.  In large-scale public elections, where pairties are
very rare, the Smith and Schwartz sets are the same.  However, for committee
voting, or decisions made by other small groups (like Debian), pairties can
easily occur during vote counting.  In the above example, if the winners are
restricted to being members of the Schwartz set, then there is a single
winner; if the Smith set is used, then a tiebreaker procedure (STV?) would
need to be invoked to determine the winner, which might not be A.

Note that I am *NOT* saying the Schwartz set is preferable to Smith, or that
either one should necessarily be used as part of any voting procedure Debian
might adopt (I'd recommend against it, in fact -- it is better to just use a
method which satisfies the Smith criterion, than use Smith as a method).
I'm merely pointing out how easy it is to make mistakes like this, so it is
important to word any proposed constitutional amendments *very* carefully.

--
Norm Petry

```