A solution the Quinn's puzzle, which greatly generalizes LeGrand's, is now available here: http://rangevoting.org/PuzzKingTourn.html --- The "existence of king" theorem stated in part (a) there suggests that it might be interesting for Condorcetists to demand that the winner always must be a "king." For example, the king-variant of Markus Schulze's beatpath method would be to demand that the winner W is the candidate who (i) is a king (ii) beats every rival via the strongest <=2-hop beatpath i.e. choose the king W such that the minimum defeat margin used in all its (<=2)-hop beatpaths, is maximized. There similarly would be king-variants of virtually every Condorcet method ever proposed. These king-variants might have advantages, such as speedier algorithm; and disadvantages, such as losing some of the properties Schulze's (original) beatpath method enjoyed. I have not attempted to check which properties survive/not. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)