Can anybody come up with ANY party-agnostic rated deterministic non-delegated optimal proportional voting method that is not Thiele-based and passes:
Independence of Irrelevant Alternatives
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)
(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.