Score Sorted Margins
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@jack-waugh I explained it in https://www.votingtheory.org/forum/topic/213/under-what-scenarios-will-the-smith-set-differ-from-copeland-set/5
First, find both the total scores for each candidate, and the pairwise matrix for each candidate.
Candidates are seeded by their total score. For each adjacent pair of candidates in the ranking, we check whether their order in the ranking agrees with their pairwise matchup. If each pair agrees, then stop and take this as the final ranking. If at least one pair does not agree, select the pair with the smallest difference in score that disagrees and switch the candidates. Repeat until you end up with a ranking that has no disagreements between adjacent candidates.Score Sorted Margins[100]; STAR[90]; Score[81]; Approval[59]; IRV[18]; FPTP[0]
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I take it that "seeded by" means you will tentatively rank them according to their total scores as in Score Voting. So then you are going to look at the adjacent pairs. If any adjacent pair, A ranked higher than B, has B preferred to A by more count of voters, look at all the adjacent pairs that share that discrepancy, and whichever one exhibits the smallest difference in Score scores, switch them. This changes the set of adjacent pairs. Repeat until there are no such discrepancies. Have I paraphrased the algo correctly?
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@jack-waugh Yes.
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I had never heard of this, but my first impression is it is really hard to wrap one's head around.
It sounds like an "iterate until it settles on an equilibrium" sort of thing. At first blush, I'm not 100% convinced it will always find an equilibrium. Has it been proven to?
There's also the issue that it uses score ballots, which I'm stating to become convinced is a major deal since almost all voting machines and infrastructure support ranked ballots, but not score ballots.
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@rob said in Score Sorted Margins:
It sounds like an "iterate until it settles on an equilibrium" sort of thing. At first blush, I'm not 100% convinced it will always find an equilibrium. Has it been proven to?
Yes, it can be proven to always find an equilibrium.
Note that only adjacent candidates can be swapped in the ranking, and that when adjacent candidates are swapped, the only relation that changes is the one between those two candidates.
Thus if we start at a ranking in which A>B, and move to a ranking in which B>A, then at some step some point, A and B must have been adjacent, and then they must have swapped positions, implying that B beats A pairwise. So to move back to a ranking in which A>B, A would have to beat B pairwise. But A and B cannot both beat each other pairwise.
Therefore we cannot move from a newer ranking to an older one.
I'll grant that this system is probably too complex to be implemented in practice, but it's my favorite theoretically.
Score Sorted Margins[100]; STAR[90]; Score[81]; Approval[59]; IRV[18]; FPTP[0]
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@marylander That makes sense.
It seems to be Condorcet compliant. I think. It would be very, very strange to see a Condorcet winner that doesn't make it to the top of the list in this method.
If it is indeed Condorcet compliant, is there are reason you prefer this to any other Condorcet method with cardinal ballots?
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Unless I'm misinterpreting one description or another, this system is Condorcet compliant, and to go a step further, it is Copeland compliant. It looks as though the idea is that if there is a Copeland tie, the system will resolve it in the way that does the least violence to the Score outcome. I'm not sure whether this technique for quantifying that violence can produce a different result than resolving the ties by ordering by overall score, like Reverse STAR.
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Here is a variation for ranking ballots that permit equal-ranking (and so would be an RCV-c system):
- From each ballot, derive an "implied set of ranks". If the ballot ranks every candidate, the implied set of ranks equals the explicit set of ranks. Otherwise, add a rank at the bottom and fill it with the remaining candidates.
- For each possible count of implied ranks, other than unity (which is an abstention and can just be thrown out), form a mapping via a logistic function, from each rank to a score. The sharpness of the logistic function is characterized by the mapping [6, 5, 4, 3, 2, 1, 0] => [100, 99, 90, 50, 10, 1, 0].
- Use the mappings and rankings to derive a score for each candidate from each voter.
- Proceed as before.
Some may say, why a logistic mapping and not a linear mapping? I ask them, why a linear mapping? I think fineness near the top will provide needed support to lesser-evil candidates in case true favorites don't win. Of course, any mapping from ranking to rating is imperfect and presumptuous, but I'm suggesting this as one of the least bad from the viewpoint of defeating two-party dominance (2PD) and as a sop to RCV advocates. They don't seem to appreciate the benefits of collecting actual scores from voters. And my example to try to show the importance of that is to compare the meaning and intent of Score votes Gore 0, Bush 1, Nader 100 vs. Gore 0, Bush 99, Nader 100. These would affect the pairwise scores of the candidates very differently in Score, but in RCV with whatever tallying, would collapse to the same vote Nader > Bush > Gore.
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