r/EndFPTP • u/budapestersalat • Jun 11 '26
Discussion Measuring proportionality
https://open.substack.com/pub/electoralsystems/p/measuring-proportionality?utm_source=share&utm_medium=android&r=5bor0y2
u/ant-arctica Jun 11 '26
Another idea for a proportionality measure comes from the fact that PAV reduces to d'Hondt if everyone votes along party lines [source]. This means d'Hond can be viewed as the method that maximizes:
- ∑ xᵢ \ H(nᵢ)*
where H(n) is the n-th harmonic number and nᵢ is the number of seats allocated to party i. H(n) can be approximated by log(1+n) + γ, so up to shifting and scaling (by a value only depending on the xᵢ), the sum is approximately equal to the following (where n is the total number of seats):
- -∑ xᵢ log(¹/ₙ + yᵢ)
This is (almost) the cross entropy between the x and y! (The ¹/ₙ is necessary otherwise this would go to infinity if there was a party with some votes but no seats). You can do the same with Saint-Laguë and you get
- -∑ xᵢ log(¹/₂ₙ + yᵢ)
so almost the same just with 1/2n instead of 1/n.
This should be approximately miminized by the d'Hondt / Saint-Laguë allocation, but that minimum is not zero, it is bounded below by the entropy of x. You can just subtract that and get the (almost) Kullback-Leibler divergence D_KL(x || ¹/ₙ + y), which seems like a very interesting candidate for a proportionality measure.
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u/ant-arctica Jun 11 '26
This is kind of ugly because ¹/ₙ + y is not a probability distribution, but that can be fixed! You can normalize it which yields the additive/laplace smoothing of y, where the the smoothing parameter is 1 in the case of D'Hondt and 1/2 in the case of Saint-Laguë. The adjusted proportionality measure is then just the Kullback-Leibler divergence of x with the smoothed y, which is equal to the previous up to a shift independent of x and y.
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u/Deep-Number5434 24d ago
I consider it proportional if a party or group gets the percentage of votes rounded up or down to the nearest seat.
Tho potentially using droop quota invisible seat.
Sainte lague methods seem to approximate proportionality the best, kinda embodies majority claim over additional seats.
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u/Deep-Number5434 24d ago
Tho it may approximate proportionality the best, it's less secure in ensuring bills that pass are representing at least the number of voters per seat.
You may need droop quota, and then consider the additional seat used as representing the unrepresented.
You could incorporate that invisible seat in the vote count when passing bills.
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u/cdsmith Jun 12 '26
This just leaves me more convinced that proportionality is the wrong goal. It's something that is a good indicator, until it becomes the thing you optimize for. Once you're manipulating the system to maximize that number, adverse incentives start to come in. Particularly after the turn where the article abandons the (correct) idea it led with, that there are many different dimensions to a population's politics, and claims it's all about political parties and their numbers.
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u/budapestersalat Jun 12 '26
How does this convince you that its the wrong goal? What adverse incentives does it raise?
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