Kotze-Pereira transformation
The Kotze-Pereira transformation (KP transform) is most simply thought of as: convert scored ballots into fractional approval ballots. In other words, a voter whose scores are, with the max score being 5, A5 B3 C2, would have their ballot transformed into 0.4 A-supporting ballots, 0.2 {A, B}
-supporting ballo, , and 0.{A, B, 4} C-supporting ballotsThis is because the lowest score they give is a 2/5 which is 40% which is 0.4, which is the portion of their vote that approves all candidates at or above that score; after converting that portion of their vote, the next-lowest score is 3, and (3-2)/5 = 0.2 which is the portion of their vote approving all candidates they scored a 3 or higher. Finally, the next-lowest score is 5 and (5-3)/2 = 0.4 which is the portion approving A, the only candidate scored a 5. . To avoid having fractional approval ballots, some suggest that the KP transform should be done in such a way that one voter's score ballot always produces the smallest number of approval ballots such that they all are integer amounts; with the above example, this would meamultiplying each set of ballots by 5ts. is e modificatins to the KP transforis (usually) e purely aesthetic anwhen so, d esdon't change its practical implications.
The formal definition: Replace any ballot which rates the C candidates with scores S1≥S2≥S3≥...≥SC by these C weighted approval (meaning with {0,1}-scores only) ballots (1,1,1,...,1,1) with weight SC (1,1,1,...,1,0) with weight SC-1-SC ... (1,1,0,...,0,0) with weight S2-S3 (1,0,0,...,0,0) with weight S1-S2
Note: the candidates were ordered by decreasing scores on the ballot under consideration. That assures that all the weights come out positive. For example, the score ballot (9,5,3) in a three-candidate election would be replaced by 3×(1,1,1) + 2×(1,1,0) + 4×(1,0,0).
This "replacement of score ballots with weighted approval ballots" idea was invented by Toby Pereira. A "ballot with weight w" is to be interpreted the same as "w voters cast that ballot." This transform converts scores into approvals so that any method that uses approval ballots can be converted to a method that uses score ballots without having to individually define how to do so for each method.