Yes, my more general short definition could be interpreted in a way so that Borda, and the pairwise-count methods would fail. It might be an impossible situation, caused by the fact that, though what we feel we mean by Monotonicity is simple--voting someone higher shouldn't cause him to lose--we expect something else when we apply it. It's customary to apply Monotonicity in a way that assumes that, if possible, we don't change the order in which we vote the other candidates.
Say you've ranked Smith in 7th place, and Jones in 5th place, and you vote Smith higher by moving him to 5th place. That causes it to be necessary to move Jones. Lacking any rules about where to move him, we could move him to 1st place, and move the previous 1st place candidate down to 7th place. We do that because we have to move Jones, and that's one way to do that. Whatever happens when we move Jones to 1st place can be said to have been caused by upranking Smith, since doing so caused it to be necessary to move Jones. That's the problem that I notice with my brief definition, the one that seems to say what we mean by Monotonicity--till we apply it. But maybe it isn't to un-general to speak of order. After all, what can a ballot do with a candidate, other than tell how good he is, or tell if he's better than someone else? So my long version, and/or whatever you've written along those lines, might be ok after-all. My long version might be ok with the "voting Smith higher" definition that I posted with it. Probably is, though I haven't studied that yet. At worst, my previous definition of voting Smith higher would be needed. My long definition doesn't mention rating or ranking, but does mention order. It mentions how candidates are marked on the ballot, but that's general. I suspect that the problem defining Monotonicity results from what we intend for Monotonicity being different from how we customarily apply it. The definition(s) that speak of order should be the ones used now, since they conform to custom in applying Monotonicity. If eventually someone writes a definition that doesn't have to mention order, and works as expected by custom, then that could then replace the previous definition. Or maybe order just has to be accepted as part of the customary application of Monotonicity. Mike Ossipoff Forest had written: Our attempts were either too strong (eliminating all Condorcet methods, for example) or, too ambiguous, or lacking in generality. We were striving to be so general that we would never have to mention ranking, rating, or order ... just the results of the election method (i.e. the method being tested for monotonicity) when restricted to various combinations of ballots. Such a definition would work on any method whatsoever, based on any possible ballot type, even those types that have not yet been envisioned due to our lack of imagination and ingenuity. We haven't given up, but it's on the back burner for now. When we have time, we should summerize some of the blind alleys and partial results so as to prevent unnecessary duplication of effort. Perhaps some of the list members could then carry it beyond the point where we left off. _________________________________________________________________ MSN Photos is the easiest way to share and print your photos: http://photos.msn.com/support/worldwide.aspx
