"Duplicated Quotas" and other testing ideas for PR

If you have a set of ballots:

51 A>B
49 C>B

A would be deemed best by any majority-obeying single-winner method. But if we duplicate these ballots, and turn it into a 2-winner election:

51 A>B
49 C>B
51 A2>B2
49.C2>B2

While any majority-obeying single-winner method would say (A, A2) is the right winner set when examining the individual Hare Quotas, when combining the ballots, the decision may change. The other thing is that if the A and A2 voters bullet vote, most PR methods should give the election to them; does that indicate that when generalizing a majority-obeying single-winner system to PR, “Hare-Majorities” should be given special precedence in who to elect if it allows them to prevent a bigger contender from beating them?

If the example was

51 A>B
49 C>B
51 A2>B
49 C2>B

then even though in the individual Hare Quotas, A and A2 would be considered best, when combined, B may be one of the two best candidates regardless of whether your PR method gives precedence to Hare majorities.

Am I supposed to be inferring a political alliance between A and A1, B and B1, and C and C1? In that case, in the example

Why wouldn’t their supporters preference each other? Then if the A voters bullet vote for A and A2, they win 1 seat and C wins the other (assuming Droop proportionality).

Also, there’s nothing special about a “Hare quota majority”, which is basically the same as “Half a Hare quota”. It appears significant in the single winner case because it’s the same as a Droop quota, but if you want to extend the concept of a majority to multiwinner elections, the Droop quota makes more sense for that purpose.

No, the main idea is to try duplicating the quotas, but make them entirely separate, so that we can judge who should win when trying to get the result from several quotas individually vs. combined. The duplicated names were an attempt to try to say, if Candidate A wins when looking at one quota, and we duplicate that quota, should A and the duplicated version of Candidate A (A2) win, or is some other winner set preferable for PR purposes?

The above examples were intended to demonstrate that when you have four factions, none of whom amount to a Droop Quota, and two candidates who can be the 2nd choice of two disjoint Droop Quotas, but whose ability to win or lose depends on whether or not the largest (51 votes) non-Droop Quota factions rank them or truncate, then it may make sense to give those Hare majorities their preference whenever the method can establish that not doing so would give them an incentive to bullet vote. I’m guessing the best heuristic for determining when those situations occur will have some relation to checking if any faction has a Hare majority of preference, and if so, checking whether, if their ballots were set aside from supporting any other candidate, they’d be able to force a victory.

I’m making these remarks mostly in the context of generalizing Condorcet methods to PR in a sequential manner, but they may be of interest for majority-obeying methods in general, and the idea of “Duplicated Quotas” might be useful even when thinking about logical properties or test cases for cardinal PR.