[EM] ranked pair method that resolves beath path ties.

[Ross Hyman]

When beat path produces a tie, this method can produce a single winner unless the tie is genuine.  It is the same method I presented earlier except for the addition of the Removing step, which resolves the ties. 

Candidates are classed in two categories: Winners and
Losers.  Initially, all candidates are Winners.  Every candidate has
an associated Set of candidates that includes itself and those candidates that
have defeated it.  Every candidate initially has a set composed of itself
and no other candidates.  Winners are those candidates who have no Winners
in their set aside from themselves.



The pairs are ranked in order.  All pairs
are ranked in the form A>B indicating more voters rank A above B than rank B
above A.  Pairs with equal votes for A
above B and B above A are not ranked.  For
winning votes ranking, A>B is ranked higher than C>D if more voters
ranked A above B than ranked C above D. 
If the same number of voters ranked A above B as ranked C above D then
A>B is ranked higher than C>D if more voters ranked D above C than ranked
B above A.  If the same number of voters
ranked A above B as ranked C above D and the same number ranked D above C as
ranked B above A then these pairs are equally ranked.

 

Affirm each group of equally ranked pairs in order, from
highest to lowest.   The count can
be ended before all pairs have been affirmed if only one Winner remains.

 

Affirming is composed of three steps: Combining sets,
Removing candidates from sets, and Reclassifying candidates.    

 

Affirming Step 1: Combining

When A > B is affirmed, the set for candidate A is added
to every set that includes candidate B (not just candidate B’s set).  The Combining step is performed for all pairs
of the same rank before moving on to the Removing step.

 

Affirming Step 2: Removing

Each pair of winning candidates that are in each others'
sets are deleted from those sets. Example if C is in D's set and D is in C's
set and both C and D are winners, then delete C from D's set and delete D from
C's set.).  All such pairs of candidates
are removed before moving on to the Reclassifying step.  The inclusion of this step resolves ties that
are not resolved by the beat path method.

 

Affirming Step 3: Reclassifying

All Winners that now have Winning candidates other than
themselves in their set are reclassified as Losers.

 

Example

7 A > B > C > D

6 C > D

5 D > B > A > C





A         B          C         D

A         0          7          12        7

B          5          0          12        7

C         6          6          0          13

D         11        11        5          0



C> D 13,5

A>C and B>C 12,6

D>A and D>B 11, 7

A>B  7,5



affirm C>D

A(W): A(W)

B(W): B(W)

C(W): C(W)

D(L):C(W), D(L)

D was reclassified as a Loser since C(W) is in its set. 



affirm  A>C and B > C

A(W): A(W)

B(W): B(W)

C(L): A(W), B(W), C(L)

D(L): A(W), B(W), C(L), D(L)

C was reclassified as a Loser since A(W) and B(W) are in its set. 



affirm D > A and D > B

A(W): A(W), B(W)*, C(L), D(L)

B(W): A(W)*, B(W), C(L), D(L)

C(L): A(W), B(W), C(L), D(L)

D(L): A(W), B(W), C(L), D(L)

A and B are both winners. A is in B's set and B is in A's set. So A is deleted
from B's set and B is deleted from A's set.

A(W): A(W), C(L), D(L)

B(W): B(W), C(L), D(L)

C(L): A(W), B(W), C(L), D(L)

D(L): A(W), B(W), C(L), D(L)



affirm A > B

A(W):A(W), C(L), D(L)

B(L): A(W), B(L), C(L), D(L)

C(L): A(W), B(L), C(L), D(L)

D(L): A(W), B(L), C(L), D(L)

B was reclassified as a Loser since A(W) is in its set. 





A wins.  With beat
path, A and B are tied.


 

 

 

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