I think I have actually seen definitions of monotonicity which already take this into account. You only need to specify that the remaining candidates stay in the same relative order -- no need to mention "avoidable/unavoidable changes" since such changes are always avoidable.
So maybe something like [starting from Mike's proposal]: [begin definition] If, by a particular set of ballots, Smith wins, then modifying some of the ballots so as to vote Smith higher *without changing the relative order* in which those ballots vote the other candidates, then, after that change, Smith shouldn't lose. [end of definition] I have never heard of a definition of monotonicity which attempted to deal with situations where more than one candidate is modified in relation to the remaining candidates. I think you would be opening a can of worms by doing so, & don't know what the value would be. What would you call it, *multiple monotonicity* ? Bart MIKE OSSIPOFF wrote: > > It's just occurred to me that if I move Smith from 5th place to 1st place, > and move my > 2nd through 4th place candidates down one rank position, it could be said > that that > isn't an unavoidable change in how I mark the other candidates, since it > could be > argued that that particular change could be avoided by making it into a > different change, > by reversing some other candidate pair. I probably made the unspoken > assumption that > all the other candidates are kept in the same order, for the unavoidable > change in the > other candidates' marks on that ballot. > > So here's perhaps a better wording of Monotonicity: > > If, by a particular set of ballots, Smith wins, then modifying some of the > ballots so as > to vote Smith higher, if possible without changing the order in which those > ballots vote > the other candidates, if possible without changing how the other candidates > are marked > on those ballots, then, after that change, Smith shouldn't lose. > > [end of definition] > > A similar change can be made in my definition of a fixed way for John to > mark the > other candidates: > > A way for John to mark the other candidates that, if possible, doesn't > change the > order in which John's ballot votes those other candidates when John changes > how > he marks Smith, and which, if possible, doesn't change how it marks those > other > candidates when John changes how he marks Smith. > > [end of definition] > > These definitions are contrived to give the expected results with the 4 > kinds of balloting > that we encounter the most: Approval, CR, rankings, & Plurality. It isn't > based on a > general study that would guarantee that it would give expected results with > all > methods, or even with all proposable methods, and so it probably isn't the > satisfactory > Monotonicity definition that Forest & Richard were looking for. It's > intended more as > a stopgap. > > My previous definition sounded more general, but I don't know if it could be > made > airtight without making reference to keeping the other candidates in the > same order. > > My definitions before today, for Monotonicity & a fixed way of marking the > other > candidates, I call that my "unavoidable change" definitions. Today's > definition I call > my "if possible" definitions. > > Maybe the unavoidable-change definitions can be fixed. I'd previously felt > that > when we vote Smith higher in a ranking, the change in the other candidates' > marks that > keeps them in the same order is unavoidable, but any change from that is > avoidable. > Now that doesn't seem so justified to say, which is why I'm posting today's > definitions. > > What is the date or the message number for Forest's & Richard's latest > definition > of Monotonicity? > > Mike Ossipoff > > _________________________________________________________________ > MSN Photos is the easiest way to share and print your photos: > http://photos.msn.com/support/worldwide.aspx
