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<div dir="ltr" data-setdir="false">On weak monotonicity, take the following ballots:</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">99: AC</div><div dir="ltr" data-setdir="false">99: BD</div><div dir="ltr" data-setdir="false">1: C</div><div dir="ltr" data-setdir="false">1:D</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">And then these ballots:</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">99: ABC</div><div dir="ltr" data-setdir="false">99: ABD</div><div dir="ltr" data-setdir="false">1: C</div><div dir="ltr" data-setdir="false">1:D</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">In the first ballot set, CD would win (2 to elect), but the <span><span style="color: rgb(38, 40, 42); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">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.</span></span></div><div dir="ltr" data-setdir="false"><span><span style="color: rgb(38, 40, 42); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><br></span></span></div><div dir="ltr" data-setdir="false">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.</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">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.</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">Toby</div><div><br></div>
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On Thursday 3 October 2024 at 15:43:08 BST, Kristofer Munsterhjelm <km-elmet@munsterhjelm.no> wrote:
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<div><div dir="ltr">On 2024-09-26 23:49, Toby Pereira wrote:<br clear="none">> The other thing with this is that looking solely at proportionality is <br clear="none">> not necessarily the best thing to do. It needs to be combined with some <br clear="none">> reasonable degree of monotonicity as well.<br clear="none"><br clear="none">I agree that criteria matter too. As an analogy: even if Borda has a <br clear="none">better VSE than Schulze, I much prefer to have the majority criterion of <br clear="none">the latter.<br clear="none"><br clear="none">> Take the following approval ballots:<br clear="none">> <br clear="none">> 99: ABC<br clear="none">> 99: ABD<br clear="none">> 1: C<br clear="none">> 1: D<br clear="none">> <br clear="none">> With two to elect, arguably CD is the most proportional result (and this <br clear="none">> is what Phragmen philosophy prefers). However, C and D each have just <br clear="none">> 50% of the support, whereas A and B each have 99%. The 99s in the <br clear="none">> ballots could be increased arbitrarily and you'd get the same thing. <br clear="none">> This is why Phragmen-based methods tend to be only weakly monotonic.<br clear="none"><br clear="none">What kind of monotonicity does the example show the limits of? It <br clear="none">doesn't seem to be mono-raise or mono-add-top.<br clear="none"><br clear="none">> I've written about optimal proportionality before, but if you are <br clear="none">> allowed to elect any number of candidates with any amount of weight <br clear="none">> (rather than having a fixed number with equal weight), then optimal PAV <br clear="none">> and COWPEA both have some very nice properties, including a strong <br clear="none">> degree of monotonicity and passing independence of irrelevant ballots. <br clear="none">> They differ slightly in their "philosophy" so can produce different <br clear="none">> results, but I see them as possibly the only two contenders for the <br clear="none">> "ultimate" in optimal PR.<br clear="none"><br clear="none">In a ranked setting, if you're allowed to elect any number of candidates <br clear="none">with any weight, then electing everybody's first preference can be <br clear="none">strategy-proof (by Duggan-Schwartz). It would also be optimal by <br clear="none">Chamberlin-Courant - every candidate would be assigned every voter <br clear="none">voting that candidate first, with weight equal to their proportion. So I <br clear="none">would imagine "elect any number with any weight" to be a setting where <br clear="none">it's relatively easy to do well.<br clear="none"><br clear="none">> But the bottom line is that there is no known method that keeps all the <br clear="none">> nice properties you'd want for all deterministic elections where a fixed <br clear="none">> number of candidates are to be elected with equal weight. Some <br clear="none">> compromise is likely to always be needed. It's not that surprising since <br clear="none">> proportionality and monotonicity are essentially orthogonal things, and <br clear="none">> it's not a priori obvious that they could be perfectly combined in a <br clear="none">> non-arbitrary and clean way. It seems they can be anyway for optimal PR <br clear="none">> (with two nice methods - an embarrassment of riches) but not for fixed <br clear="none">> candidates with equal weight.<br clear="none"><br clear="none">How is proportionality and monotonicity incompatible in the fixed <br clear="none">candidates setting? It would seem to me that if A is on the winning <br clear="none">council and someone ranks or rates A higher, then it's possible to <br clear="none">ensure that A still wins without having to compromise proportionality.<div class="ydpa8fc15cyqt5728325887" id="ydpa8fc15cyqtfd41643"><br clear="none"><br clear="none">-km<br clear="none"></div></div></div>
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