# IRV's nonmonotonicity

Rob LeGrand honky1998 at yahoo.com
Sat Mar 30 21:44:16 PST 2002

```Adam wrote:
> It comes down to how you define monotonicity.  In these examples, dropping
> your first place candidate down on your ballot causes him to
> 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."

I see this definition as too strict.  In particular, Dodgson, Schulze, Minmax
and Ranked Pairs would have to be considered nonmonotonic by this definition,
as the following example shows:

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.  I don't see this as a problem for the four methods; while A is
"upranked", C is too, which reduces B's pairwise victory over C enough for C to
win the election.  I think considering Dodgson, Schulze, Minmax and Ranked
Pairs as nonmonotonic makes the concept of monotonicity less useful.

For what it's worth, here's Hannu Nurmi's informal definition:

"Monotonicity requires that, if an alternative x which wins under a given
procedure gets more support and nothing else changes in the individual
preferences, then that alternative remains the winner after the change as
well."

--
Rob LeGrand
honky98 at aggies.org
http://www.onr.com/user/honky98/rbvote/calc.html

"Fifty-one percent of a nation can establish a totalitarian regime, suppress
minorities and still remain democratic." - Erik von Kuehnelt-Leddihn

". . . [T]he Bill of Rights does not come from the people and is not subject to
change by majorities.  It comes from the nature of things.  It declares the
inalienable rights of man not only against all government but also against the
people collectively." - Walter Lippmann

"The majority, oppressing an individual, is guilty of a crime, abuses its
strength, and by acting on the law of the strongest breaks up the foundations
of society." - Thomas Jefferson

"Individual rights are not subject to a public vote; a majority has no right to
vote away the rights of a minority; the political function of rights is
precisely to protect minorities from oppression by majorities (and the smallest
minority on earth is the individual)." - Ayn Rand

__________________________________________________
Do You Yahoo!?
Yahoo! Greetings - send holiday greetings for Easter, Passover
http://greetings.yahoo.com/

```