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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