Open Question :

Can anybody come up with ANY party-agnostic rated deterministic non-delegated optimal proportional voting method that is not Thiele-based and passes:

  1. Independence of Irrelevant Alternatives

  2. Independence of Irrelevant Ballots (if a ballot rates all candidates equally, that ballot is irrelevant and adding/removing irrelevant ballots should not change who wins and who loses)

  3. (if possible) Monotonicity

The definition of proportionality this open question will be using: Whenever a group of voters gives max support their favored candidates and min support to every other candidate, at least one seat less than the portion of seats in that district corresponding to the portion of seats that that group makes up (or how many of those candidates exist if there exists less then this amount) is expected to be won by those candidates.

I have trouble defining optimal and non-Thiele-based individually, so here is the combined definitions of Independence of Irrelevant Alternatives, optimal, and non-Thiele-based this open question will be using: The voting method must be defined via a quality function that takes only a set of candidates and ratings the voters give to each of the candidates in that set as inputs and outputs a quality value of that election result. The voting method must pick the set of candidates with a size equal to the number of winners that maximizes this quality function. In the approval ballot case of the voting method, the voting method can NOT be simplified to simply picking a set of candidates with a size equal to the number of winners that maximizes a quality function equal to a sum among all voters f(the number of candidates in that set that that voter approved of) where f(x) is a function.

What makes this extremely difficult is the Independence of Irrelevant Ballots, which seems to be a property that only Thiele-based voting methods (or methods that violate IIA by simply discarding all ballots that give all candidates the same rating) have. However IIB seems like a very reasonable criterion (why should a ballot that does not care about which candidates win be used to influence which candidates win???) which is why I am interested in finding other voting methods that pass it.

Just curious, is it possible to have them count as a “Vote of Indifference” where each candidate has the points symbolically added onto their tally after the process has elected them? I’m guessing you also want a method where a ballot treating some candidates equally is irrelevant in changing which of those candidates wins, while the ballot still counts when determining which of the candidates it differentiates between should win.

Yes. I only care about inputs and outputs where the outputs are who wins and loses. If I add an irreverent ballot, quality values of each set of candidates can all increase or decrease together but who wins and who loses should not be altered by such ballots.

I thought of one, but it’s stupid.

Consider the “Loser Independent PR” criterion I posted in the committee doc:

The winner set must be proportional even if some losing candidates were disqualified and/or scores for some losing candidates were reduced. That is, if at least n quotas of ballots approve the same set of candidates, but there is partial disagreement on unelected candidates outside of that set, then at least n candidates in the set must be elected. (If 2 quotas approve ABCD, 2 approve ABCDE, and E is not elected, the standard PR criterion would require 2 of ABCD to be elected, whereas this criterion would require 4 of ABCD to be elected.)

  1. Find the smallest quota size for which it is possible to choose a winner set that passes this criterion. (We know that it must be between 0 and a Droop quota, assuming there are more candidates than seats).

  2. Find the winner set(s) that would pass the Loser Independent PR criterion for this quota size.

  3. Elect that winner set. If there is more than one such set, consider the election outcome to be a tie between those sets.

I’m not actually sure that this passes IIA. Instead: the score of a given winner set is the largest quota size for which it would be required to be elected by the PR criterion if all candidates outside the set were ignored. (We know that this must be between 0 and a Hare quota.) Elect the highest scoring winner set.

Does my optimal extension of vote unitarity not work for this? It is clealry Monroe based but I do not think it is monotonic. The general class of these methods which use what Warren calls the “corner trick” could have other similar models.

Nop. As we have previously discussed on the wolf committee, it fails IIB.

Almost everything that’s proportional fails IIB, so as a starting point try to pass that and then move on to the other properties.

So as an equation of what the quality (Q) of a two winner election result would be in terms of the % of voters that approve of only the first candidate (va) the % of voters that approve of only the second (vb) and the % of voters that approve of both (vab), Qab = max(vab/2, min(va, vb))? Is this what you mean? Because if so, it doesn’t pass IIB because you can add enough voters that approve of all candidates such that the PR criterion will only ever force groups of candidates to be elected because it can do so with a quota that is to high to elect candidates via their individual supporters.

Yup, you are right. Sorry

It might be possible to prove that all Monroe (quota) systems fail IIB. This would be because adding relevant ballots would change the quotas and thus the winners. This would mean there is no such system which meets your criteria.

The determinism clause is also very important or something like this might get you there.

Would using “Winner Independent Proportionality” instead of the standard PR criterion fix this? In the doc, I described it as:

The winner set must be proportional even if scores for some winning candidates were increased. That is, if at least n quotas of ballots approve the same set of candidates, but there is partial disagreement on m elected candidates outside of that set, then at least n-m candidates in the set must be elected. (If 2 quotas approve ABCD, 2 quotas approve ABCDE, and E is elected, the standard PR criterion would require 2 of ABCD to be elected, whereas this criterion would require 3 of ABCD to be elected.)