# [EM] Short definition. Long definition fixed.

Richard Moore rmoore4 at home.com
Sat Jan 5 20:38:14 PST 2002

```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