Score Difference Stratified Condorcet
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Let each voter v score candidate c as v(c). For each pair of candidates (c1,c2), consider the distribution of absolute score differences as v varies uniformly over the voters:
|v(c1)-v(c2)|~f(c1,c2).
To determine the marginal victory of c1 over c2, count -p, 0, or +p depending on the sign of the score difference (-1, 0, +1) and the percentile p of the ballot’s absolute score difference relative to f(c1,c2).
Elect the “Condorcet winner” from this marginal victory graph if one exists. Otherwise, elect the ordinary Condorcet winner if one exists. Otherwise, elect the score winner.
This method attempts to strike a balance between strict ordinal preference and cardinal preference—it becomes possible for a sufficiently passionate and sufficiently large minority to overrule a sufficiently dispassionate and sufficiently small majority.
It represents a more general class of methods that operate in a similar way, with perhaps a different activation function for score difference percentiles.
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@cfrank said in Score Difference Stratified Condorcet:
Let each voter v score candidate c as v(c). For each pair of candidates (c1,c2), consider the distribution of absolute score differences as v varies uniformly over the voters:
|v(c1)-v(c2)|~f(c1,c2).I believe I suggested something similar to this on both ElectoWiki and the EM list, by taking
|v(c1)-v(c2)|as the margin to get a cardinal-augmented Condorcet method. -
@lime if we take the raw score difference, doesn’t that just become score voting? Or maybe I misunderstand your meaning. I think c1>c2 if and only if c1 has a higher score than c2. In score voting, c1 and c2 will each accrue from each ballot the average of their scores on that ballot, and the sum of the score differentials will be the only deciding factor between them.
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@cfrank said in Score Difference Stratified Condorcet:
@lime if we take the raw score difference, doesn’t that just become score voting? Or maybe I misunderstand your meaning. I think c1>c2 if and only if c1 has a higher score than c2. In score voting, c1 and c2 will each accrue from each ballot the average of their scores on that ballot, and the sum of the score differentials will be the only deciding factor between them.
Sorry, I forgot to mention the important bit, which is using the median instead of the sum or average
If you take the median difference, this is positive only when most voters prefer A > B.
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@lime but if you take the median and positivity means majority rule, then it’s just an ordinary Condorcet method, right?
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@cfrank said in Score Difference Stratified Condorcet:
@lime but if you take the median and positivity means majority rule, then it’s just an ordinary Condorcet method, right?
Yes, but with defeat strength being measured using cardinal scores.
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@lime I see, so this would be used in absence of a Condorcet winner like for a ranked pairs resolution?
I was trying to think about burial but I don’t think my method addresses it quite as I conceived. My reasoning was that, by replacing absolute score differences with their more robust percentiles, burial (and bullet voting) strategies will suffer from severely diminishing returns compared with less risky and more honest ballots. For example, burying a second-favorite below a turkey to support a first favorite probably won’t significantly improve the score percentile provided by that voter to the first favorite’s runoff with the second favorite, but will significantly improve the chances of the turkey winning. This makes dishonest burial more severely punished and risky, meaning that fewer rational voters will choose to do it. Also, the effects of the fraction that do will be significantly reduced, since they will not only be fewer in number, but the magnitude of their indicated score differences will be majorly reeled back upon being replaced by their percentiles relative to the more honest bulk.
At the same time, the method is not restricted to Condorcet compliance, since, for example, it is possible for a [1-sqrt(1/2)]~0.2928… fraction minority of voters to overrule a sqrt(1/2)~0.707… fraction majority as long as the whole minority has the top quantile of absolute score differences and all of them have the same sign. That is the smallest possible minority that can overrule a majority in this method. It’s in one sense a generalized, more flexible extension of some of the reasonable measures we already have in the legislative houses, where for example a supermajority (2/3) is required for certain decisions.
Alternatively, each absolute score difference percentile could be measured relative to the distribution of all absolute score differences across all differences. The data set would consist of N*K(K-1)/2 values where N is the number of voters and K is the number of candidates.