Bart wrote:
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. I reply: But you can't specify that the other candidates remain in the same relative order, since that would be violated in Plurality. Say your vote is on Jones, but then you vote Smith higher by moving your vote from Jones to Smith. Before you did that, you were voting Jones over the other candidates, and now you no longer are. You've changed the relative order of how you're voting the other candidates. Well I guess you could say that the meaning of a change in order doesn't include voting A equal to B when you'd previously voted A over B. But I don't know how well that would work when the criterion is applied. It seems better to allow for Plurality's change in order. Bart continued: 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 reply: Again, there's the problem that there's a good case for saying that, when you move your vote to Smith in Plurality, you're unavoidably changing the order in which you vote the other candidates, when you take that vote from Jones, whom you'd previously voted over everyone. Also, why allow any avoidable fudging of how we vote the other candidates? In Approval and CR it's possible to vote Smith higher without in any way changing how we mark the other candidates, and when we apply Monotonicity to those methods, isn't that the kind of ballot-change that we're talking about? Maybe it isn't necessary to say that if possible we don't change the other candidates' marks, but even if we can get by with that, in the sense that the criterion works ok that way, when we speak of voting Smith higher in Approval or CR, we don't mean also changing other people's ratings. Bart continued: 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. I reply: Yes it's a can of worms, but it's already open, and the question is how to put the worms back in the can. It's something that can't be ignored in Monotonicity, since with Plurality you can't vote Smith higher without changing how some other candidate is voted in relation to other candidates. (The candidate from whom you took your vote when you moved it to Smith). Mike _________________________________________________________________ MSN Photos is the easiest way to share and print your photos: http://photos.msn.com/support/worldwide.aspx
