# [EM] Majority winner set

Markus Schulze schulze at sol.physik.tu-berlin.de
Mon Nov 20 02:03:37 PST 2000

```Dear Craig,

beat-path GMC says:

"X >> Y" means that an absolute majority of the voters
strictly prefers candidate X to candidate Y.
"There is a majority beat path from X to Y" means that
(1) X >> Y or
(2) there is a set of candidates C[1],...,C[n] with
X >> C[1] >> ... >> C[n] >> Y.

If there is a majority beat path from candidate A to
candidate B and no majority beat path from candidate B
to candidate A, then candidate B must not be elected.

A "Sincere Absolute Smith Winner" (SASW) is a candidate
who is a Smith winner when every voter votes sincerely
and votes a complete ranking of all candidates. Then -in
the formulation above- (when truncation does occur and
order-reversal does not occur) it is possible that
candidate A is a SASW and candidate B is no SASW but it is
not possible that candidate A is no SASW and candidate B
is a SASW. Therefore -if either candidate A or candidate
B has to be elected- rather candidate A than candidate B
should be elected. In other words: Beat-path GMC says
that when only truncation occurs the used election method
shouldn't unnecessarily choose a candidate who is not a
SASW.

It seems to me that the majority winner set and beat-path
GMC have the same intention.

Markus Schulze

```