[EM] a solution to the by-election problem?

[Ross Hyman]

Hi all,

The by-election problem is the following:
Starting with a Droop compliant solution for the M-winner problem,
exclude one of the M winners. Now determine a new set of M-winners
that includes the M-1 remaining previous winners and is also Droop
compliant. There is no STV method that does this. Either you preserve
Droop compliance but not all the M-1 previous winners are reelected,
or you protect the M-1 previous winners from exclusion and the new Mth
winner is not guaranteed to be Droop compliant.

However, I believe the method I describe in Section V of "A House
Monotone, Coherent, and Droop Proportional Ranked Candidate Voting
Method"  https://arxiv.org/pdf/2506.12318 does this correctly when
setting the M-1 remaining winners as “previously elected." The proof
is in Section VI Claim 9, with N=M.

Can someone check this?

Best,
Ross