An Argument for MES

This is more or less a continuation of my last topic, but I'm splitting it off since I think I finally have a nice way to think about the classic set of proportionality criteria for scored ballots.
As I see it, there are basically two ways to think about proportionality of scored ballots. The first is a proportional distribution of utility, so two quotas of 0.5 is worth the same as one quota of 1.0. The second is basically thinking about quotas in whole numbers of voters treating the scores more or less like preferences, and so in some sense this is a proportional distribution of influence. The two are not necessarily at odds with each other, but it can be difficult to get both at the same time.
I believe I can formalize the first notion with a statement to the effect of (paraphrasing for readability, but can provide details to those interested)
If there is a group of voters S such that S constitutes at least T quotas, and there is a set of T candidates such that the total utility of S on T is X, then the winning committee must give S at least X/2 total utility.
And the condition on a method required to prove this (within the class of quotaspending rules) are
 If the sum of weighted score for any candidate is at least x >= one quota, then the algorithm must not have terminated yet and the maximum amount of ballot weight per utility paid by any voter must not exceed q/x
I believe I can formalize the second notion with a statement to the effect of
If there is a group of voters S such that S constitutes at least T quotas, and there exists an approval threshold b such that S is cohesive on a set of X <= T candidates when the scores are converted to approvals above b (and 0 below b), then the number of candidates that group receives that are scored above b by a member of S must be at least X
It's a little wordy, but it basically says "Justified Representation holds no matter what the approval threshold is." I think the condition required for this one is simply
 If there exists a candidate receiving a positive score from at least one quota of ballot weight, then the algorithm must not have terminated yet and the winner in that round must also receive a positive score from at least one quota of ballot weight
So, a method like SSS satisfies 1. but not 2.; a method like Expanding Approvals will satisfy 2. but not 1.; and a method like Allocated Score will satisfy neither 1. nor 2.
However, MES actually satisfies both! So at least in this framework it is the best of all worlds. And if you look a little closely at these conditions, it seems that they (nearly) exactly characterize MES. I suppose you could choose something rather contrived that is technically different from MES but still fits both, but in terms of elegant rules I don't see another way.

@andydienes said in An Argument for MES:
there are basically two ways to think about proportionality of scored ballots. The first is a proportional distribution of utility, so two quotas of 0.5 is worth the same as one quota of 1.0. The second is basically thinking about quotas in whole numbers of voters treating the scores more or less like preferences, and so in some sense this is a proportional distribution of influence.
There is also a third. That is how much of your ballot/vote power you are willing to give up to see each winner win.
I would say that 1 is needed and obvious. 2 and 3 are incompatible because they are both about how you set a metric on the scores. SSS gives you 1 and 3 but MES gives you 1 and 2.
3 is basically vote unitarity.

@keithedmonds Could you help me formalize that into a statement about the output of a voting rule rather than the process of a voting rule?
Maybe there is a way to phrase this as some kind of "laternoharm"ness criterion.

@andydienes First, lets never associate this with Laternoharm as I hate that property like none other.
Anyway, I have thought about this for a few days and I do not think it is possible. Just because adding constraints on the process is a nonstandard concept in voting theory does not mean it should not be. In fact, it likely implies something has been missed. There may be similar constrains already implicitly applied.
In liberal theory, equality of process has traditionally been considered to equality of outcome(equity)

I wasn't sure where to post this, but dug up this topic. My reason for bringing this up is that despite all the discussion I've seen for the Method of Equal Shares (MES), I've never seen in defined in an easytounderstand way for voting for a committee with a fixed number of positions. It's generally discussed in terms of participatory budgeting where there is a fixed amount of money and not a fixed number of things to be elected. It has its own website here. It's on the voting wiki here and on the Wikipedia here.
It seems the the cost per project is analogous to the quota (e.g. Droop or Hare), but it's perfectly possible that you simply won't elect the right number of candidates under a strict interpretation of MES.
I find it strange that people have taken this participatory budgeting method for use in committee elections, without really discussing this and therefore what the method is that they are talking about.

@tobypereira We introduced MES originally in this paper https://arxiv.org/abs/1911.11747 for committee elections, but allowing rules to select "up to k" candidates rather than "exactly k" candidates. As you say, MES can select fewer than k candidates (including no candidates at all). We have discussed ways to top up the committee to contain exactly k candidates in various places. In the linked original paper, we suggest continuing with Sequential PhragmÃ©n. In the participatory budgeting focussed followup paper https://arxiv.org/abs/2008.13276 we propose (but don't necessarily endorse) "epsiloncompletion" which behaves quite differently. As far as I know, there is not yet a principled approach to decide which completion is the best one.

@dominikpeters Thanks for the information on this. Do you think that MES + completion is a good method for electing a committee of a fixed size or do you think a single method that works without an addon would be better?