Let's start with a list L of candidates listed from top to bottom in order of their score totals ... high scores listed above low scores Initialize a second list L' with the the top member of L at its head. Then incorporate the next member of L into L', above or below its initial member depending on whether or not the new member of L' is scored above its initial member on more ballots than not. So far the list L' is the Score Then Automatic Runoff "STAR" finish order. In general, while some member X of L can be incorporated into L' without defeating any member of L' listed above it and without being defeated by any member of L' listed below it, so incorporate the highest such X from L into the list L'. Then elect the candidate that ends up at the top of L'. Example: Suppose the score order list L is S1 S2 S3 and that the pairwise defeats are S3 defeats S2 defeats S1defeats S3. Then L' is initialized as S1, and S2 is incorporated above S1. So far the list L' is S2 S1 Can S3 be incorporated? Not at the head, because that would entail defeat by a lower listed candidate S1. Not at the middle or bottom because then it would defeat a candidate (S2) listed above it. So the completed list L' is S2 S1 The head S2 is elected from the top of the list L'. Since the method is independent from Smith dominated alternatives, and since Smith is highly unlikely to have more than three members, we can easily boil it down to three cases (barring unlikely ties): I. Smith has one member: Then that member will be the head of L'. II. The cyclic order counters the score order as in the above example. As we have just seen, the Smith member with the 2nd highest score is elected in this case. III. The cyclic defeat order does not counter the score order. In this case it is easy to see that the first two members of L' are S1 and S2, and that S3 cannot be incorporated. So the head of L' is the score winner S1. So we see that in public elections this (Banks!) method reduces to the simple rule "Elect the lowest score candidate that defeats every higher score candidate," which is another formulation of Score,Benham as well as Ranked Pairs, River, and BeatPath,etc when defeat strength is measured by winning score. Which of these equivalent formulations will be easiest to sell? I mention Banks only to add prestige to the method, and to show how to generalize it to specialized, non-public elections that are likely to have more interesting Smith structure (twisted prism, etc). -Forest

On 28.06.2022 01:21, Forest Simmons wrote: > So we see that in public elections this (Banks!) method reduces to the > simple rule "Elect the lowest score candidate that defeats every higher > score candidate," which is another formulation of Score,Benham as well > as Ranked Pairs, River, and BeatPath,etc when defeat strength is > measured by winning score. > > Which of these equivalent formulations will be easiest to sell? > > I mention Banks only to add prestige to the method, and to show how to > generalize it to specialized, non-public elections that are likely to > have more interesting Smith structure (twisted prism, etc). Nice! I would lean towards the Benham phrasing, but that's probably because I've been considering how to tersely describe Benham lately: "A candidate defeats another if the former is ranked (rated) ahead of the latter on more ballots than vice versa. A candidate is undefeated if nobody (still in the running) defeats him. While there is no undefeated candidate, eliminate the Plurality (Range) loser. Elect the remaining undefeated candidate." Of course, with the first two sentences staying the same, you could replace the latter two sentences with: "Elect the lowest score candidate that defeats every higher score candidate", which is shorter but not as close to the mechanical procedure. If the jurisdiction is currently using Condorcet, or is the STAR method is intended as a first step to Condorcet proper, then "just do RP but with this defeat strength measure" might be more natural. I guess you lose monotonicity because it's hard to get all of Banks, monotonicity, and polynomial runtime. -km

El mar., 28 de jun. de 2022 5:29 a. m., Kristofer Munsterhjelm < km_elmet@t-online.de> escribió: > On 28.06.2022 01:21, Forest Simmons wrote: > > > So we see that in public elections this (Banks!) method reduces to the > > simple rule "Elect the lowest score candidate that defeats every higher > > score candidate," which is another formulation of Score,Benham as well > > as Ranked Pairs, River, and BeatPath,etc when defeat strength is > > measured by winning score. > > > > Which of these equivalent formulations will be easiest to sell? > > > > I mention Banks only to add prestige to the method, and to show how to > > generalize it to specialized, non-public elections that are likely to > > have more interesting Smith structure (twisted prism, etc). > > Nice! > > I would lean towards the Benham phrasing, but that's probably because > I've been considering how to tersely describe Benham lately: > > "A candidate defeats another if the former is ranked (rated) ahead of > the latter on more ballots than vice versa. > > A candidate is undefeated if nobody (still in the running) defeats him. > > While there is no undefeated candidate, eliminate the Plurality (Range) > loser. > > Elect the remaining undefeated candidate." > Excellent! > > Of course, with the first two sentences staying the same, you could > replace the latter two sentences with: > > "Elect the lowest score candidate that defeats every higher score > candidate", > > which is shorter but not as close to the mechanical procedure. > Your Benham formulation is much better psychologically. in particular, some voters could get hung up on "... elect the lowest score ..." ...wondering why lowest? Also, (1) people are used to eliminations ... "How can it be a real voting method without eliminations?" And (2) they are more attuned to step-by-step procedures than characterizations. > > > If the jurisdiction is currently using Condorcet, or is the STAR method > is intended as a first step to Condorcet proper, then "just do RP but > with this defeat strength measure" might be more natural. > In any case, after defining the method ... as an aside ... "it turns out that another way to find the exact same winner is Ranked Pairs, as long as defeat strength is gauged by winning score rather than winning votes or margins." RP(winning score): Go down the list of candidates one-by-one in order of decreasing score. When you get to candidate X, lock in all of the defeats where X is the victor, unless that would contradict a previously locked in defeat. ETC. > > I guess you lose monotonicity Actually RP(winning score) is perfectly monotone ... and so all of its equivalent variants. The Banks efficient method is not equivalent, but does elect the same candidate when Smith has fewer than four members ... as does STAR when restricted to Smith (when Smith has fewer than four members). because it's hard to get all of Banks, > monotonicity, and polynomial runtime. > -km > Another interesting method to consider is RP(losing score) where the defeat weakness is the score of the defeated candidate ... i.e. the strength is the reciprocal of the score of the pairwise loser. This version of RP always elects the same candidate as Sequential Pairwise Elimination where the agenda is the score order from lowest to highest. Furthermore, in the case of #Smith<4, this method always elects the highest score Smith member. If I remember correctly, Score(Smith) had the highest VSE of all the methods tested. And I bet the VSE of Score SPE would agree to several decimal places, since even in the rare cases of #Smith>3, it is more likely than not that Score(Smith) winner would also be the Score SPE winner, etc. I'm not suggesting abandoning RP(winning score) for RP(losing acore); it wouldn't be worth giving up your Benham formulation of RP(ws). I haven't checked yet, but I'm guessing that RP(score margins) is another description of Score Sorted Margins, which would be a great proposal, but not as good psychologically as your Benham proposal. -Forest >