[EM] Proportionality: the small bias effect seems to be real for Harmonic

[Kristofer Munsterhjelm]

On 2024-09-26 23:49, Toby Pereira wrote:
> The other thing with this is that looking solely at proportionality is 
> not necessarily the best thing to do. It needs to be combined with some 
> reasonable degree of monotonicity as well.

I agree that criteria matter too. As an analogy: even if Borda has a 
better VSE than Schulze, I much prefer to have the majority criterion of 
the latter.

> Take the following approval ballots:
> 
> 99: ABC
> 99: ABD
> 1: C
> 1: D
> 
> With two to elect, arguably CD is the most proportional result (and this 
> is what Phragmen philosophy prefers). However, C and D each have just 
> 50% of the support, whereas A and B each have 99%. The 99s in the 
> ballots could be increased arbitrarily and you'd get the same thing. 
> This is why Phragmen-based methods tend to be only weakly monotonic.

What kind of monotonicity does the example show the limits of? It 
doesn't seem to be mono-raise or mono-add-top.

> I've written about optimal proportionality before, but if you are 
> allowed to elect any number of candidates with any amount of weight 
> (rather than having a fixed number with equal weight), then optimal PAV 
> and COWPEA both have some very nice properties, including a strong 
> degree of monotonicity and passing independence of irrelevant ballots. 
> They differ slightly in their "philosophy" so can produce different 
> results, but I see them as possibly the only two contenders for the 
> "ultimate" in optimal PR.

In a ranked setting, if you're allowed to elect any number of candidates 
with any weight, then electing everybody's first preference can be 
strategy-proof (by Duggan-Schwartz). It would also be optimal by 
Chamberlin-Courant - every candidate would be assigned every voter 
voting that candidate first, with weight equal to their proportion. So I 
would imagine "elect any number with any weight" to be a setting where 
it's relatively easy to do well.

> But the bottom line is that there is no known method that keeps all the 
> nice properties you'd want for all deterministic elections where a fixed 
> number of candidates are to be elected with equal weight. Some 
> compromise is likely to always be needed. It's not that surprising since 
> proportionality and monotonicity are essentially orthogonal things, and 
> it's not a priori obvious that they could be perfectly combined in a 
> non-arbitrary and clean way. It seems they can be anyway for optimal PR 
> (with two nice methods - an embarrassment of riches) but not for fixed 
> candidates with equal weight.

How is proportionality and monotonicity incompatible in the fixed 
candidates setting? It would seem to me that if A is on the winning 
council and someone ranks or rates A higher, then it's possible to 
ensure that A still wins without having to compromise proportionality.

-km