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

[Toby Pereira]

 On weak monotonicity, take the following ballots:
99: AC99: BD1: C1:D
And then these ballots:
99: ABC99: ABD1: C1:D
In the first ballot set, CD would win (2 to elect), but the Phragmen score of AB (under e.g. var-Phragmen) would still be close. In the second set, CD would still win. Despite all those extra approvals for A and B, there would be no improvement in the score for AB. I would say that in a strongly monotonic method, each extra approval should count for the candidate being approved, rather than merely not against. These is why Phragmen-based methods are weakly monotonic. All these approvals count for nothing. I think mono-raise and mono-add-top are defined for ranked methods. Under approval voting, approving an extra candidate is in a sense raising them but also adding them to (potentially joint) top. But I don't think those exact definitions matter in this case. But it would also be the same for raising a score of a candidate or adding them to the top score in Harmonic Voting.
When talking about the "ultimate" in PR, I was largely talking about cardinal voting at that point, and something that could even be used on voters' raw utilities. Also it would have to take into count cases where a voter might be indifferent between two candidates while still passing independence of irrelevant ballots (IIB). If a voter has A and B top, then merely adding an equal weight for each in the overall parliament would not pass IIB.
My point wasn't that proportionality and monotonicity were incompatible in the fixed candidates setting, but that getting a clean method that passes all the criteria you want doesn't seem possible. E.g. I don't think there is a deterministic cardinal method that passes the same desirable criteria as COWPEA Lottery or Optimal PAV Lottery.
Toby
    On Thursday 3 October 2024 at 15:43:08 BST, Kristofer Munsterhjelm <km-elmet at munsterhjelm.no> wrote:  
 
 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
  
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