[EM] Identifying ranked ballot sets

[John]

[Note:  I'm not on-list; please CC me on replies]

Somebody suggested a two-ballot set to me with three candidates, and
it turns out you can mutate this set and achieve the same pairwise
votes.

BAC
CAB

      LOSE
     A  B  C
W A  _  1  1
I B  1  _  1
N C  1  1  _

You'll note this set also derives from BCA ACB.

This expands to larger sets:

BACD = BDCA
DCAB = ACDB

      LOSE
     A  B  C  D
W A  _  1  1  1
I B  1  _  1  1
N C  1  1  _  1
  D  1  1  1  _

If we clear the first votes:

      LOSE
     A  B  C  D
W A  _  .  .  .
I B  .  _  .  .
N C  1  1  _  1
  D  1  1  1  _

(B)DCA = (B)CDA
(A)CDB = (A)DCB

I'm trying to identify a concise representation of a set of ranked
ballots.  At this time, I believe this may be:

1.  The pairwise race matrix (Win-Lose)
2.  The matrix of appearances at each ordinal rank

For BACD and DCAB, #2 is as such:

   1  2  3  4
A 0  1  1  0
B 1  0  0  1
C 0  1  1  0
D 1  0  0  1

It probably doesn't matter if you rank the truncated ballots by
counting all unranked as ties at that rank or by not counting them at
that rank.