After putting together some ideas from recent conversations on approval based chain building and MMPO based approval ... here's a burial resistant method that popped out: Let CO be the candidate whose max pairwise opposition is smaller than any other candidate's MPO. [Think CO for Cut Off as in approval cutoff candidate] If this MMPO candidate is undefeated, elect it, otherwise ... Initialize a chain with the candidate X with the greatest pairwise opposition to CO, that is with the most winning votes against the cutoff CO. [This X has the greatest.MMPO based approval] If X is uncovered, add to the chain the most approved candidate that covers X. [That is, the candidate covering X with the greatest pairwise opposition to CO] In general, as long as the (head of) the chain is uncovered, keep augmenting the chain with the most approved candidate that covers the (head of) the chain. Finally, elect the last candidate that was added to the chain ... the one that covers all of the other members of the chain. In the examples below all three candidates are uncovered, because in a cycle of three candidates each has a short beatpath to each of the others. Example 1. 46 A>B 44 B>C(Sincere B>A) 5. C>A 5 C>B Candidate B is the MMPO approval cutoff CO. The chain is initialized with candidate A, the candidate with the greatest pairwise opposition to B. As the only member of the chain, candidate A wins. Example 2. 49 A 26 B>C 25 C(sincere C>B) Candidates B and C are tied for MMPO candidate ... but the tie is broken in favor of B, because B beats C pairwise. So B is the approval cutoff candidate. Candidate A is the approval winner, that is the one with the most PO against the cutoff B. And A is uncovered, so A wins. Example 3. a A>B b B>C c C>A Where c is the smallest faction size, but large enough to keep both a and b below half of the total number of ballots. The respective max PO's of A, B, and C are b+c, a+c, and a+b, which can be written respectively as n-a, n-b, and n-c, where n is the total number of ballots. The largest of these MPO's is n-c, since C is the smallest faction size, so the MinMaxPO candidate must be either A or B, depending which of A or B has the larger faction. Case 1: a>b>c Candidate A is the cutoff. Then C has the greatest approval since C's PO against A is c+b which is greater than b, which is B's PO against A. So C wins ... which means that if the cycle was created by the largest faction burying the smallest faction, the buried sincere CW won anyway. Case 2. b>a>c. Candidate B is the cutoff. Candidate A has the most winning votes against this approval cutoff, so A wins as the one and only member of the approval chain. If the cycle was created by the largest faction (B) burying its sincere second choice C below its last choice A, then its ploy backfired since its anti-favorite A got elected. Note that IRV agrees with this method except on example 3 in the case where a>b>c. In that case IRV rewards the burial of C by the A faction. fws

Each example would elect precisely the same candidate if the rule were simply ... lacking an undefeated candidate, elect the candidate with the most (winning) votes against the MMPO candidate. This simple method might be a good entry level public proposal. You could even change it into an elimination method to satisfy the expectation for one-by-one eliminations ... something like what Martin Harper did for Approval to turn it into a vote transfer method like IRV. He said ... For each ballot B let X(B) be the candidate that has the greatest total approval of any candidate that is approved on ballot B. Transfer all of B's votes to X(B). Then elect the candidate with the most transferred votes. The tongue in cheek punch line is that the winner of this method is simply the candidate with the greatest total approval. But the technique is actually (potentially) useful for resolving equal first votes without resorting to fractional votes ... of those candidates voted equal first on ballot B, give B's vote to the one voted equal first on the most ballots. At least that was my suggestion for someone trying to figure out how to handle IRV vote transfers (without fractional votes) when voters are allowed equal ranking of candidates. Another application is an idea for Approval DSV ... iteratively at each stage put the approval cutoff just below X(B) determined by Martin Harper's method from the previous stage approval values. This DSV method implements Michael Ossipoff's approval cutoff advice ... put your approval cutoff immediately below the candidate you would vote for in a Plurality election. Ossipoff's rule shows that Approval strategy is no harder than Plurality strategy. fws On Wed, Aug 9, 2023, 3:13 PM Forest Simmons <forest.simmons21@gmail.com> wrote: > After putting together some ideas from recent conversations on approval > based chain building and MMPO based approval ... here's a burial resistant > method that popped out: > > Let CO be the candidate whose max pairwise opposition is smaller than any > other candidate's MPO. > > [Think CO for Cut Off as in approval cutoff candidate] > > If this MMPO candidate is undefeated, elect it, otherwise ... > > Initialize a chain with the candidate X with the greatest pairwise > opposition to CO, that is with the most winning votes against the cutoff CO. > > [This X has the greatest.MMPO based approval] > > If X is uncovered, add to the chain the most approved candidate that > covers X. > > [That is, the candidate covering X with the greatest pairwise opposition > to CO] > > In general, as long as the (head of) the chain is uncovered, keep > augmenting the chain with the most approved candidate that covers the (head > of) the chain. > > Finally, elect the last candidate that was added to the chain ... the one > that covers all of the other members of the chain. > > In the examples below all three candidates are uncovered, because in a > cycle of three candidates each has a short beatpath to each of the others. > > Example 1. > > 46 A>B > 44 B>C(Sincere B>A) > 5. C>A > 5 C>B > > Candidate B is the MMPO approval cutoff CO. The chain is initialized with > candidate A, the candidate with the greatest pairwise opposition to B. > > As the only member of the chain, candidate A wins. > > Example 2. > > 49 A > 26 B>C > 25 C(sincere C>B) > > Candidates B and C are tied for MMPO candidate ... but the tie is broken > in favor of B, because B beats C pairwise. > > So B is the approval cutoff candidate. > > Candidate A is the approval winner, that is the one with the most PO > against the cutoff B. > > And A is uncovered, so A wins. > > Example 3. > > a A>B > b B>C > c C>A > > Where c is the smallest faction size, but large enough to keep both a and > b below half of the total number of ballots. > > The respective max PO's of A, B, and C are b+c, a+c, and a+b, which can be > written respectively as n-a, n-b, and n-c, where n is the total number of > ballots. > > The largest of these MPO's is n-c, since C is the smallest faction size, > so the MinMaxPO candidate must be either A or B, depending which of A or B > has the larger faction. > > Case 1: a>b>c > Candidate A is the cutoff. Then C has the greatest approval since C's PO > against A is c+b which is greater than b, which is B's PO against A. So C > wins ... which means that if the cycle was created by the largest faction > burying the smallest faction, the buried sincere CW won anyway. > > Case 2. b>a>c. > Candidate B is the cutoff. Candidate A has the most winning votes against > this approval cutoff, so A wins as the one and only member of the approval > chain. > If the cycle was created by the largest faction (B) burying its sincere > second choice C below its last choice A, then its ploy backfired since its > anti-favorite A got elected. > > Note that IRV agrees with this method except on example 3 in the case > where a>b>c. In that case IRV rewards the burial of C by the A faction. > > fws >