# IRV's nonmonotonicity

Sun Mar 31 11:20:42 PST 2002

```The rest of my message (not sure why it didn't make the list; it's in my
out box) appears below.  I apologize for the junk message.

>Rob wrote:
>
>>  9:A>B>C
>>  8:B>C>A
>>  6:C>A>B
>>
>>A wins by all four methods.  Now four of the B>C>A voters switch to
>>C>A>B, which is "upranking" A by Adam's definition:
>>
>>  9:A>B>C
>>  4:B>C>A
>>10:C>A>B
>
>Now C wins.

Fair enough; my alternate definition does not work.  But the example still
shows a kind of failure of IRV, even if it is not technically a
non-monotonic result. And the slightly modified example (where the 5% that
change to Ralph first were originally George>Ralph>Al voters) is in fact a
non-monotonic result.

I still feel that intuitively, this is a failure of monotonicity, and the
monotonicity criterion could be defined to incorporate this example.  The
problem with my above definition was that we could change some ballots to
change the winner without moving the winner, and then pushing the winner up
in an irrelevant manner.  I also suppose that "raising the ranked position"
of a candidate becomes sort of ambiguous when lots of candidates move around.

"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, provided any
other changes to the ballot would not change the outcome of the election
were the candidate in question not moved from his or her original ranked
position."

I fully realize this is now a more complex definition than the original
monotonicity criterion.  I am trying to make the criterion definition
coincide with my intuitive understanding of what monotonicity means.  Since
that definition was a mouthful, here's a procedural definition:

1)  Take a set of identical ballots.
2)  Fix the position of the candidate in question, and all candidates
ranked equal to that candidate.
3) The relative rankings of candidates ranked above and below this
candidate can be moved, as long as this does not change the outcome of the
election.
4) If dropping the candidate in question lower can turn the candidate from
a winner to a loser, or raising the candidate higher can turn a loser into
a winner, then monotonicity is violated.

Since any election method that fails the above definition would (with the
right example) fail the original definition of monotonicity, the two must
be equivalent.  But this whole argument is therefore relatively
unimportant.  The only difference is whether we can say a particular
example shows failure of monotonicity.  For almost all purposes the more
complex definition is not worth it.

We can still trash IRV for it's failure to avoid "lesser of two evils" in
the case of relatively even candidates, which is really the most important,
fundamental flaw. This suggests why IRV does not really help third parties
grow. For example, in the case,

42% George > Al > Ralph
14% Al > George > Ralph
14% Al > Ralph > George
30% Ralph > Al > George

...Ralph voters clearly need to bury Ralph behind Al to avoid George.  It's
just Favorite Betrayal, but it comes up a lot more in IRV than in Condorcet
(or Approval, of course).

I think that the best thing to do, when arguing with IRV advocates, is to
introduce Condorcet at this point, and show them how smoothly it sorts
things out. It is extremely easy to come up with examples where IRV fails
and Condorcet succeeds. The inverse is not true; examples with cyclic
ambiguities that can trouble Condorcet only produce satisfying IRV results
by a twist of luck.  Arguing about monotonicity is often beside the point.