IRV's nonmonotonicity

Rob LeGrand honky1998 at
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:


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:


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

Rob LeGrand
honky98 at

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