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<div dir="ltr" data-setdir="false">Kevin</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">Having variable loads/costs might seem counterintuitive and unfair, but it's necessary for the <span>Phragmén </span>method to work at all. The default version (seq-<span>Phragmén) minimises the maximum load on the highest loaded voter. Take the following example where A and B are two parties fielding multiple candidates:</span></div><div dir="ltr" data-setdir="false"><span><br></span></div><div dir="ltr" data-setdir="false"><span>2 voters: A</span></div><div dir="ltr" data-setdir="false"><span>1 voter: B</span></div><div dir="ltr" data-setdir="false"><span>1 voter: AB</span></div><div dir="ltr" data-setdir="false"><span><br></span></div><div dir="ltr" data-setdir="false"><span>If the load for each candidate had to be spread equally across their voters, then the only relevant voter is the AB voter. Simply electing A every time would minimise the load on this voter.</span></div><div dir="ltr" data-setdir="false"><span><br></span></div><div dir="ltr" data-setdir="false"><span>There are variants on <span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">Phragmén, including var-<span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">Phragmén, which looks at the variance of the load rather than the max on the single voter with the highest. Var-<span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">Phragmén with enforced equal loads isn't as broken as the default version would be, but it is still broken, and it would fail monotonicity. E.g.</span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><br></span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">1 voter: AB</span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">1 voter: AC</span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><br></span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">With enforced equal loading, BC is a better result than the asymmetrical AB or AC, so the extra vote for A counts against them. Electing sequentially obviously avoids this particular monotonicity failure, but it's actually quite easy to come up with examples that would also fail in the sequential case. But this shows that <span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">Phragmén's measure of balance/proportionality is non-monotonic if equal loads are enforced.</span></span></span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><br></span></span></span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">Allowing unequal loads is just another way of saying that if an approval for a candidate would count against that candidate, then we will ignore it (or count it only in part). But this leaves <span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">Phragmén with a very weak form of monotonicity. E.g.</span></span></span></span></span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><br></span></span></span></span></span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">99 voters: AB</span></span></span></span></span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">99 voters: AC</span></span></span></span></span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">1 voter: B</span></span></span></span></span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">1 voter: C</span></span></span></span></span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><br></span></span></span></span></span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">Under <span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">Phragmén philosophy, BC is a better result than AB or AC. Again, electing sequentially can save this particular case, but not in general.</span></span></span></span></span></span></span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><br></span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">If you wanted to create a tool that evaluated sets of candidates under different metrics, then PAV score (with different divisors), <span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">Phragmén variance (perhaps with equal loads enforced and not) and the standard min-max version (but only with equal loads not enforced) might be good measures to looks at.</span></span></span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><br></span></span></span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;"><span><span style="color: rgb(0, 0, 0); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">Toby</span></span></span></span></span></span></span></span></span></div><div dir="ltr" data-setdir="false"><span><br></span></div><div dir="ltr" data-setdir="false"><span><br></span></div><div><br></div>
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On Monday, 6 May 2024 at 05:11:02 BST, Kevin Venzke <stepjak@yahoo.fr> wrote:
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<div><div dir="ltr">Hi Toby,<br></div><div dir="ltr"><br></div><div dir="ltr">> I posted the below on the Voting Theory Forum, but thought it might be of<br></div><div dir="ltr">> interest to some people on this list as well. The link formatting won't work<br></div><div dir="ltr">> here in the same way, but URLs can simply be copied and pasted. It should<br></div><div dir="ltr">> still read OK, and I'd be more likely to make a mess of it by changing<br></div><div dir="ltr">> everything around.<br></div><div dir="ltr"><br></div><div dir="ltr">> The problem is that there are essentially two orthogonal goals for a method -<br></div><div dir="ltr">> maximising proportionality and also being properly monotonic (as well and<br></div><div dir="ltr">> passing things like [Independence of Irrelevant Ballots]<br></div><div dir="ltr">> (https://electowiki.org/wiki/Independence_of_Irrelevant_Ballots)) - and there<br></div><div dir="ltr">> was never any guarantee that they could be seamlessly combined.<br></div><div dir="ltr"><br></div><div dir="ltr">I do find this interesting and will have to look into all this... I started<br></div><div dir="ltr">working on a PAV calculator but ended up finding the topic to be bigger than I<br></div><div dir="ltr">thought.<br></div><div dir="ltr"><br></div><div dir="ltr">I understand the satisfaction function of Thiele's methods (sequential or just<br></div><div dir="ltr">finding the best set of winners). Discovering seq-Phragmén, I liked the concept<br></div><div dir="ltr">of candidates "costing" a fixed amount to be borne by a candidate's supporters,<br></div><div dir="ltr">but as I understand it the specific load borne by a given voter can vary, in<br></div><div dir="ltr">particular with the effect of "charging" a voter more for the election of a given<br></div><div dir="ltr">candidate precisely because that voter hadn't managed to elect anyone yet. That<br></div><div dir="ltr">doesn't strike me as fair, or reflective of anything. Maybe I misunderstand it.<br></div><div dir="ltr"><br></div><div dir="ltr">One thought I had is maybe the most helpful thing would be a webpage where users<br></div><div dir="ltr">could play around with selecting sets of winners and see evaluations of those<br></div><div dir="ltr">sets by some metrics. This, as opposed to focusing on the results of specific<br></div><div dir="ltr">methods.<br></div><div dir="ltr"><br></div><div dir="ltr">Not sure if you have any thoughts on what an ideal tool would do.<br></div><div dir="ltr"><br></div><div dir="ltr">Kevin<br></div><div dir="ltr">votingmethods.net<br></div></div>
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