Ranking Array Math

[DEMOREP1 at aol.com]

Using relative rank order voting (1, 2, 3, etc.) produces an array (or
matrix). In the below the winner in a pairing is at the top and the loser is
at the left. Thus AB means the number of ballots that rank A over B. CT=
Column Totals, RT= Row Totals
(View table in 9 point Monaco)
X  A   B   C   D   E   F   G    RT
A  X   BA  CA  DA  EA  FA  GA   ART
B  AB  X   CB  DB  EB  FB  GB   BRT
C  AC  BC  X   DC  EC  FC  GC   CRT
D  AD  BD  CD  X   ED  FD  GD   DRT
E  AE  BE  CE  DE  X   FE  GE   ERT
F  AF  BF  CF  DF  EF  X   GF   FRT
G  AG  BG  CG  DG  EG  FG  X    GRT

Thus, votes for a candidate are in the (vertical) columns and votes against a
candidate are in the (horizontal) rows.

Note that the simple two candidate case (A versus B) is in the upper left
corner.

Various vote ranking methods use the numbers in different ways to determine a
single winner.
In the plain Condorcet method the two opposing amounts are subtracted from
each other to see who wins the pairing. Example-  BE minus EB-- if positive,
B Wins-- if negative, E wins.
The plain Condorcet method does all the pairings and sees if one candidate
wins against each other candidate (i.e.- the column votes minus the row votes
for all pairs with such candidate would all be positive). If there is such a
candidate, then he/she is the Condorcet winner.

With multiple candidates, there may not be such a winner. Thus, the need for
a tie breaker. 
Which tie breaker to use is definitely not obvious and is arbitrary.

One tie breaker looks at the votes against in each row in cases wherever a
candidate loses a pairing. Example- A loses to F-- The FA amount is noted. 
The highest defeat amounts are noted for each candidate involved. The
candidate whose worst defeat amount is smallest among such worst defeats is
deemed the tie breaker winner.

The matrix also has use in (a) the multiple winner executive/ judicial case
and (b) the proportional representation case. Each single candidate at the
top would be, in effect, 2 or more candidates paired against 2 or more other
candidates at the left. Each whole cell would be divided into a mini matrix
with the votes for/against each individual candidate involved. 

Partial example- 5 candidates, 2 to be elected
Each pair of 2 candidates would be matched against the three pairs among the
3 other candidates. In each whole cell, there would be a 2 by 2 mini-matrix.
If the same 2 candidates win in all of their combinations then they would be
Condorcet winners. If not (as is likely), a tie breaker is needed.
More later regarding the use of *absolute* votes in tie breakers.