Just as rankings allow you to order preferences without specifying a numerical strength of preference, so ranked preferences allow one to order the preference strengths without quantifying those strengths numerically ... for example the notation A>B>>>C>>D>>>>E makes clear that the strongest preference shown is D>>>>E and the weakest is A>B, but the notation does not imply that the stronger of these is four times as strong as the weaker. Ranked rankings allow us to simulate multi-round Approval elimination, i.e. to implement Instant Implicit Approval Loser Elimination (IIALE): While more than two alternatives remain eliminate the one that is currently (implicitly) approved on the fewest ballots. An alternative X is currently (implicitly) approved on a given ballot if on that ballot there is another alternative Y such that X is ranked before Y and there is no stronger (implied) rank relation remaining on the ballot than the one between X and Y. By convention no other (impliied) rank relation is stronger than the truncation relation, "X trunc Y", meaning Y is truncated but X is not. Example: 49 C 26 A>B [or even A >>> B] 25 B A eliminated in the first round ... B wins. 49 C 26 A>>B>C 25 B B eliminated in first round ... C wins. 49 C 26 A > B >> C 25 B A eliminated, B wins. Another possible method would calculate the Martin Harper Lottery at each stage, and eliminate the alternatives tied for least probability. Example: 51 A>C 49 B>C Under IIALE alternative B is eliminated, then C is eliminated. Under the Harper Lottery elimination, both A and B are eliminated in the first round. Any other ideas for using ranked ranks? Remark: If the current definition of Universal Domain were expanded to allow use of the ordinal information (about preference strength) provided by ranked rankings, then .... ? FWS

> On 10/13/2021 9:09 PM Forest Simmons <forest.simmons21@gmail.com> wrote: > > > Just as rankings allow you to order preferences without specifying a numerical strength of preference, so ranked preferences allow one to order the preference strengths without quantifying those strengths numerically ... for example the notation > A>B>>>C>>D>>>>E > > > makes clear that the strongest preference shown is D>>>>E and the weakest is A>B, but the notation does not imply that the stronger of these is four times as strong as the weaker. > but it implies that it's stronger than B>>>C which is stronger than C>>D which is stronger than A>B . Sure, maybe we can assign the strength of A>B as -inf, that of C>>D to be zero, B>>>C to be sqrt(pi) and D>>>>E to be +inf, but that wouldn't be particularly meaningful. It **is** some quantitative information. Not just preferential. -- r b-j . _ . _ . _ . _ rbj@audioimagination.com "Imagination is more important than knowledge." . . .

Just as Borda is an attempt to quantify normal rankings, i.e. convert them into ratings, so also we could attempt to convert ranked rankings into ratings. If we did it correctly it would yield a clone free method, unlike Borda. It would be a natural generalization of Dyadic Approval, if you remember that from fifteen years ago. El jue., 14 de oct. de 2021 1:24 a. m., robert bristow-johnson < rbj@audioimagination.com> escribió: > > > > On 10/13/2021 9:09 PM Forest Simmons <forest.simmons21@gmail.com> wrote: > > > > > > Just as rankings allow you to order preferences without specifying a > numerical strength of preference, so ranked preferences allow one to order > the preference strengths without quantifying those strengths numerically > ... for example the notation > > A>B>>>C>>D>>>>E > > > > > > makes clear that the strongest preference shown is D>>>>E and the > weakest is A>B, but the notation does not imply that the stronger of these > is four times as strong as the weaker. > > > > but it implies that it's stronger than B>>>C which is stronger than C>>D > which is stronger than A>B . Sure, maybe we can assign the strength of A>B > as -inf, that of C>>D to be zero, B>>>C to be sqrt(pi) and D>>>>E to be > +inf, but that wouldn't be particularly meaningful. It **is** some > quantitative information. Not just preferential. > > -- > > r b-j . _ . _ . _ . _ rbj@audioimagination.com > > "Imagination is more important than knowledge." > > . > . > . > ---- > Election-Methods mailing list - see https://electorama.com/em for list > info >