MIKE OSSIPOFF wrote: > > Here's a short definition of Monotonicity, followed by supporting > definitions, followed by the long definition, with one fix added: > > > Short Monotonicity definition: > > If, by a certain set of ballots, Smith wins, then, if we modify > some of the ballots so that they vote Smith higher, that shouldn't > cause Smith to not win. > > [end of short Monotonicity definition] > > A ballot votes Smith over Jones if it marks them in such a way > that it's possible to contrive a configuration of other people's > votes such that, if we delete everyone but Smith & Jones from the > ballots, Smith is the unique winner if & only if we count that > ballot. > > A voter votes Smith higher if he changes how he marks Smith on > his ballot in such a way that it's possible to contrive > an initial way for that voter to mark the other candidates such > that his change in how he marks Smith causes Smith to be voted over > someone over whom Smith wasn't voted before the change.
I had a little trouble interpreting this at first, but I think I got it now. We initially mark Smith a certain way and the other candidates a certain way. Now we change how we mark Smith, and this may or may not force a change in how we mark the others (depending on the type of ballot). So there is at least one candidate, call him Jones, such that Jones is marked over Smith before the change, and Smith is marked over Jones after the change. Is that correct? This is a different approach to defining "voting Smith higher" than Forest and I were taking (we were using the term "favoring" rather than "voting higher"). There might be merit in this approach, although at the moment I'm not able to see any easy way to "formalize" the definition. Forest and I never referred to the marks on the ballots. But putting aside the issue of translating this to a mathematical definition, let's see how it holds up. What about CR ballots? If I initially mark my ballot Jones 100 Smith 60 Hitler 0 and if my change is to mark it Jones 100 Smith 90 Hitler 0 does this change not "vote Smith higher" (at least, given the usual interpretation of CR ballots)? According to the above definition, it doesn't. Yes, I realize that you can find other modifications of CR ballots that do "vote Smith higher" by the definition, and use only those modifications in testing for monotonicity. But aren't we then assuming that the method we are testing is well-behaved enough to cooperate with our tests, while the property we are testing for (nonmonotonicity) is itself a form of bad behavior? -- Richard
