Good work, Alex. I think the argument can be simplified so that it will generalize easier, but nobody else has faced up to it like you have.
BTW it seems like every N+3 candidate election has a three candidate election embedded within it as far as each faction is concerned, since each faction has its Favorite, along with the two front runners to worry about. And if even one faction has incentive to dump favorite in even one election, then the FBC is not satisfied by the method being used. I have one further tangential (actually orthogonal) comment below: On Wed, 29 Jan 2003, Alex Small wrote: .... > Consider the boundary between the A region and the B region. Call the > normal to the boundary |Nab>. If the boundary is a fractal, then there is no normal. Rob LeGrand reported that his Cumulative Repeated Approval Balloting simulations yielded graphs with (what appeared to be) fractal boundaries separating the victory regions. On the other hand, CRAB (in its simplest formulation) satisfies the Majority Criterion, and I think we can prove that no method satisfying the Majority Criterion can also satisfy the (Strong) FBC, whether or not the boundaries are smooth. >From a practical point of view, no method that requires expression of a unique favorite will ever be adopted for public elections unless it satisfies the Majority Criterion. So it is sufficient, for public proposal purposes, to show that the Majority Criterion precludes the Strong FBC. Unfortunately, this doesn't help to distinguish between IRV and Condorcet. It does help to explain why it is hard to improve on Majority Choice Approval: If we restrict favorite status to one candidate per ballot, then we gain the (unique majority version of the) Majority Criterion at the expense of the FBC. If we allow more than one candidate per ballot to have favorite status, then we regain the FBC, but lose the ability to detect a unique majority favorite. Furthermore, we give up the chance of strong FBC, since there are definitely cases where one should give Compromise equal billing with Favorite when that is allowed. This predicament is not just a result of our lack of ingenuity, but a fundamental incompatibility between the majority criterion and the FBC. [This is not intended to be a proof, but only to show an instructive application of the result.] Faced with this predicament, most of us prefer the version of MCA that allows more than one favorite, so that more than one candidate can have a fifty percent plus "majority." It seems to me that if we were consistent in our thinking, then in the context of Condorcet methods we should prefer ballots that allow equal ranking of candidates at the top as well as at the bottom (a.k.a. truncation), if not in between as well. That's another reason why I prefer Grade Ballots (for example, grades A through Z) over traditional preference ballots for Condorcet methods. [Construct the pairwise matrices from the Grade ballots, and then use your favorite Condorcet method to score the matrices.] Forest ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
