I've got a new fpA-fpC-compliant method and this time I am very satisfied with its behaviour (after checking some test examples that some other methods failed, though I guess it needs some robust checks on monotonicity). Instead of just pairwise comparing the candidates, it compares the "alliances" of these candidates in order to create a special pairwise comparison matrix (in a more burial-resistant manner). The mechanism tends to break Condorcet cycles, even though there are some rare cases where it doesn't clear *all* the ambiguity and a completion method has to be used. The "agreement" rule was created specifically to make the method cloneproof (but it also had to be strict enough so that a) the burial resistance was not weakened, b) the method didn't stop converging, i.e. achieving stable solutions). It's worth noting that - in a given iteration - a covered candidate can't have any allies against the covering candidate, so a covering relation should cause a visibly strong defeat. *Convergent Allied Comparison (CAC)* Definitions: · (A>B)-ally - a candidate that pairwise beats B, doesn't pairwise beat A and is not in agreement with B against A · (A,X)-agreement against B - occurs when A and X are in the same pairwise relation with B (win, lose or tie), number of A>=X>B votes is greater than number of A>=B>X votes and number of X>=A>B votes is greater than number of X>=B>A votes · (A>B)-alliance - a set that consists of A and all (A>B)-allies · score(A>B) = [number of votes that prefer some member of the (A>B)-alliance to every member of the (B>A)-alliance] + [number of votes that prefer A to every member of the (B>A)-alliance if A pairwise beats B, otherwise 0] Procedure: · Calculate the scores and use them to create a pairwise comparison matrix - then use this matrix to redefine which candidates are pairwise beaten by whom (which in turn can affect the alliances and the scores). Repeat until the comparison matrix does not change with a new iteration - this is the final CAC matrix. · If there exists a candidate that is unbeaten according to the CAC matrix, choose this candidate. Otherwise apply a defeat-dropping Condorcet method (River, RP/MAM or Schulze) to the CAC matrix. Example: 9: A>B>C 6: B>C>A 8: C>A>B There's no agreement between any candidates. (A>C)-alliance is {A,B}, (B>A)-alliance is {B,C}, and (C>B)-alliance is {A,C}. Scores in the first iteration are as follows: | A | B | C A | - | 9+9=18 | 9+6=15 B | 6+8=14 | - | 6+6=12 C | 8+8=16 | 9+8=17 | - So we already see a strict order (C>A>B) in this matrix, but as the beating relations (and, with them, the alliances and the scores) have changed, let's see the next iteration to make sure that the solution is stable. Now the only multi-member alliance is the (C>B)-alliance: {A,C}. | A | B | C A | - | 9+8+9+8=34 | 9 B | 6 | - | 6 C | 6+8+6+8=28 | 9+8+8=25 | - The pairwise relations remain unchanged, so this is the final CAC matrix, with C candidate confirmed as the winner. - Filip Ejlak

On 5/19/23 14:40, Filip Ejlak wrote: > I've got a new fpA-fpC-compliant method and this time I am very > satisfied with its behaviour (after checking some test examples that > some other methods failed, though I guess it needs some robust checks on > monotonicity). Instead of just pairwise comparing the candidates, it > compares the "alliances" of these candidates in order to create a > special pairwise comparison matrix (in a more burial-resistant manner). > The mechanism tends to break Condorcet cycles, even though there are > some rare cases where it doesn't clear /all/ the ambiguity and a > completion method has to be used. > > The "agreement" rule was created specifically to make the method > cloneproof (but it also had to be strict enough so that a) the burial > resistance was not weakened, b) the method didn't stop converging, i.e. > achieving stable solutions). > > It's worth noting that - in a given iteration - a covered candidate > can't have any allies against the covering candidate, so a covering > relation should cause a visibly strong defeat. Great! That does look rather complex; I'll have to investigate further. I guess I really should make those automatic criterion checkers, but it's really god that you are picking up fpA-fpC and trying to improve it :-) -km