# [EM] IRV's nonmonotonicity

Sat Mar 30 21:09:40 PST 2002

```Rob wrote:

>To violate monotonicity, an example must cause a winner to lose by having
>some voters uprank him or cause a loser to win by having some voters
>downrank him.  Alex's first Bill/George/Ross example and Adam's
>Al/George/Ralph example have the same problem.

My example again was

44% George > Al > Ralph
15% Al > George > Ralph
15% Al > Ralph > George
26% Ralph > Al > George

So if 5% of the George > Al > Ralph voters mirror the ballot entirely, to
Ralph>Al>George, they are also downranking George.  Of course this fails
the test of keeping the order of all other candidates the same.

It comes down to how you define monotonicity.  In these examples, dropping
win.  Intuitively, that's a non-monotonic result.  Would defining this
result as a failure of non-monotonicity cause some other result to be
misinterpreted as a failure of monotonicity?

I would simply define monotonicity this way:

"Lowering the ranked position of a losing candidate on some ballots cannot
cause that candidate to win, and raising the ranked position of a winning
candidate on some ballots cannot cause that candidate to lose."

At any rate, my example can be made to comply exactly with your (and
others') definition of non-monotonicity if the 5% of the voters that switch
to Ralph > George > Al were originally George > Ralph > Al.  That is,

5% George > Ralph > Al
39% George > Al > Ralph
15% Al > George > Ralph
15% Al > Ralph > George
26% Ralph > Al > George

And if the first faction lowers George down their ballots, this hands him
the election.  This complies with the exact definition, but it breaks us
out of the intuitive linear political spectrum, which makes the example a
hair less elegant in my book.  But if someone can show me a compelling
reason why my simple definition of monotonicity does not work, then rest
assured I will not bring up this example again.