Single transferable vote (STV) STV is a multistage elimination rule that works as follows. In each stage, if there is a candidate c whose Plurality score is at least q = V k+1 +1 (the so-called Droop quota), we do the following: (a) include c in the winning committee, (b) delete q votes where c is ranked first, so that each of these votes is ‘transferred’ to the candidate currently ranked right after c, and (c) remove c from all the remaining votes. If each candidate’s Plurality score is less than q, a candidate with the lowest Plurality score is deleted from all votes. We repeat this process until k candidates are selected. Note that parallel-universes tie-breaking requires us to consider all possible ways to select a candidate among those whose score meets or exceeds the quota, to identify q votes to be deleted among those where this candidate is ranked first (note that this determines the ‘transfers’, i.e., how many extra votes will be gained by each surviving candidate), and to pick a candidate to be eliminated among those with the lowest score; a committee wins under STV if it can be obtained by the procedure described above for some sequence of such choices. There are also many other variants of STV; we point the reader to the work of Tideman and Richardson (2000) for details. In particular, the reader can verify that all results in our paper continue to hold if we require that the number of the ‘extra’ votes of a selected candidate c that are transferred to another candidate a is proportional to the number of votes of the form c a ··· at the respective stage.