Dear Kristofer, in section 9.2 of my paper "The Schulze Method of Voting", I define independence of clones for proportional ranking methods and I explain, why the Schulze proportional ranking method satisfies this criterion: https://arxiv.org/pdf/1804.02973.pdf Suppose candidate D had the Z-th place in the proportional ranking. Suppose candidate D is replaced by a set of clones D(1), ..., D(n). Then the highest ranked clone D(1) must get the Z-th place of the proportional ranking and there must be no change in the places 1, ..., (Z-1) of the proportional ranking. Example: Suppose the original proportional ranking was A, B, C, D, E, F, G, ... Then the new proportional ranking could be A, B, C, D(1), G, F, E, D(2), ... One might argue that the order of the candidates between D(1) and D(2) shouldn't change either since D(2) is the first candidate who disrupts the original proportional ranking. However, I believe that already the sheer existence of potential winning sets with two candidates from the set of clones can change the order of the candidates between D(1) and D(2). ********************************************* Now, let's apply independence of clones for proportional ranking methods to multi-winner election methods. So let's say that a board of M seats is filled by calculating a proportional ranking first and then choose the first M candidates of this proportional ranking. When candidate D is not elected, then he wasn't one of the M highest ranked candidates. So when D is replaced by a set of clones, the ordering of the M highest ranked candidates must not change. So the winners stay the same. When candidate D is elected, then we don't know whether candidate D was (say) the top-ranked candidate or (say) the M-th ranked candidate. So we cannot say anything about whether the other candidates are still elected. We only know that at least one of the clones D(1), ..., D(n) must be elected. Therefore, I believe that the maximum that we could ask for is the following: ######################### Independence of clones for multi-winner elections: When a non-winner is replaced by a set of clones, then the election result must not change. When a winner is replaced by a set of clones, then at least one of these clones must be elected. ######################### Markus Schulze

On 2025-10-08 20:07, Markus Schulze via Election-Methods wrote: > Dear Kristofer, > > in section 9.2 of my paper "The Schulze Method of Voting", > I define independence of clones for proportional ranking > methods and I explain, why the Schulze proportional ranking > method satisfies this criterion: > > https://arxiv.org/pdf/1804.02973.pdf > > Suppose candidate D had the Z-th place in the proportional > ranking. Suppose candidate D is replaced by a set of clones > D(1), ..., D(n). Then the highest ranked clone D(1) must > get the Z-th place of the proportional ranking and there > must be no change in the places 1, ..., (Z-1) of the > proportional ranking. > > Example: Suppose the original proportional ranking was > A, B, C, D, E, F, G, ... > Then the new proportional ranking could be > A, B, C, D(1), G, F, E, D(2), ... > > One might argue that the order of the candidates between D(1) > and D(2) shouldn't change either since D(2) is the first candidate > who disrupts the original proportional ranking. However, I > believe that already the sheer existence of potential winning > sets with two candidates from the set of clones can change > the order of the candidates between D(1) and D(2). I'm having some difficulty seeing how that could be considered a good thing, at least for general multiwinner methods. Suppose we're electing five candidates, and before cloning the outcome is {A, B, C, D, E}. Then suppose D is cloned. It would seem to be just an IIA failure to then have the outcome change to {A, B, C, D1, G}. Now we pretty much have to live with IIA failures as a class. But single-winner clone independence says that we don't need to have IIA failures when the irrelevant alternatives are clones. (Something like "outcome is {A,B,C} then we clone A and get {A1, A2, B}" is not an IIA failure since A2 is not an irrelevant alternative, I think.) Do you have an example where it makes sense for the elected candidates to change in this way? It might be more reasonable for proportional orderings (though I don't see why either), but proportional orderings are more constrained than general multiwinner methods (e.g. the LCR example). I'd still be interested in a proportional ordering example where this kind of change makes sense, though. -km