I am rewriting the overall “XXX: a xxx proportionality metric” section because I’ve thought of a more-interpretable metric. So, where it used to be “Representational fairness: an overall proportionality metric”, now it will be “Vote wastage: a combined proportionality metric”. Here’s the old version, before I erase it:
Since we’ve structured RQ_d as an “efficiency” — 100% at best, 0% at worst — we can take each voter’s “quality-weighted voter power” (QWVP) to be the sum of their responsibility for electing each candidate, times their RQ_1 for that candidate. Ideally, this would be 1 for each voter; so we can define the overall “quality-weighted proportionality” (QWP) of an outcome as the average of squared differences between voters’ QWVP and 1, shifted and scaled so that no difference gives a QWP of 100% and uniform zeros gives a QWP of 0. (Note that in principle, a dictatorship could score substantially less than 0, depending on the number of voters).
(To do: better notation and LaTeX)
Since realistic voting methods will usually have at least 1 Droop quota of wasted votes (or, in the case of Hare-quota-based methods, just over half a Hare quota of double-powered votes and just under half of wasted votes; which amounts to much the same thing in QWP terms), the highest QWP you could reasonably expect for a voting method would be S/(S+1).
(show the math for the QWP of the IRV example above. Key point: the D>C>A>B voters have zero responsibility for electing A, so all they do is lower the average RQ of the B>C>A>D and C>B>A>D voters)
Note that this QWP metric, in combining the ideas of overall equality and representation quality, is no longer perfectly optimizing either of those aspects in itself. That is to say, in some cases it will push methods to sacrifice proportionality, in search of better representation, in a way that would tend to hurt the MSE from God’s perspective. I think those cases are likely to be rare enough, especially for voting methods that weren’t specifically designed to optimize to this metric, that I’m OK with this slight mis-alignment. That is to say: I think the true ideal quality ordering would be closer to a lexical sort with priority on proportionality (“optimize for proportionality, then optimize RQ only insofar as it doesn’t harm proportionality`); but I think most seriously-proposed, practically-feasible voting methods are far enough from the Pareto frontier that “optimize the product of the two” is fine as an approximation of that ideal goal.
One more note: in passing, this rigorous framework for an overarching proportional metric also helps define the simple concept of “wasted vote”; any vote with 0 responsibility for electing any winner. Although “wasted votes” are already commonly discussed in the political science literature, I believe this is actually the first time the idea has been given a general definition, as opposed to ad-hoc definitions for each voting method.
I am rewriting the overall “XXX: a xxx proportionality metric” section because I’ve thought of a more-interpretable metric. So, where it used to be “Representational fairness: an overall proportionality metric”, now it will be “Vote wastage: a combined proportionality metric”. Here’s the old version, before I erase it:
Since we’ve structured RQ_d as an “efficiency” — 100% at best, 0% at worst — we can take each voter’s “quality-weighted voter power” (QWVP) to be the sum of their responsibility for electing each candidate, times their RQ_1 for that candidate. Ideally, this would be 1 for each voter; so we can define the overall “quality-weighted proportionality” (QWP) of an outcome as the average of squared differences between voters’ QWVP and 1, shifted and scaled so that no difference gives a QWP of 100% and uniform zeros gives a QWP of 0. (Note that in principle, a dictatorship could score substantially less than 0, depending on the number of voters).
(To do: better notation and LaTeX)
Since realistic voting methods will usually have at least 1 Droop quota of wasted votes (or, in the case of Hare-quota-based methods, just over half a Hare quota of double-powered votes and just under half of wasted votes; which amounts to much the same thing in QWP terms), the highest QWP you could reasonably expect for a voting method would be S/(S+1).
(show the math for the QWP of the IRV example above. Key point: the D>C>A>B voters have zero responsibility for electing A, so all they do is lower the average RQ of the B>C>A>D and C>B>A>D voters)
Note that this QWP metric, in combining the ideas of overall equality and representation quality, is no longer perfectly optimizing either of those aspects in itself. That is to say, in some cases it will push methods to sacrifice proportionality, in search of better representation, in a way that would tend to hurt the MSE from God’s perspective. I think those cases are likely to be rare enough, especially for voting methods that weren’t specifically designed to optimize to this metric, that I’m OK with this slight mis-alignment. That is to say: I think the true ideal quality ordering would be closer to a lexical sort with priority on proportionality (“optimize for proportionality, then optimize RQ only insofar as it doesn’t harm proportionality`); but I think most seriously-proposed, practically-feasible voting methods are far enough from the Pareto frontier that “optimize the product of the two” is fine as an approximation of that ideal goal.
One more note: in passing, this rigorous framework for an overarching proportional metric also helps define the simple concept of “wasted vote”; any vote with 0 responsibility for electing any winner. Although “wasted votes” are already commonly discussed in the political science literature, I believe this is actually the first time the idea has been given a general definition, as opposed to ad-hoc definitions for each voting method.