# Rank methods, Participation, Consistency

Richard Moore rmoore4 at home.com
Sun Jan 13 21:29:06 PST 2002

```MIKE OSSIPOFF wrote:

> I can't name a method that fails Participation & Consistency and
> has been proven Monotonic. It seems to me that Ranked Pairs is widely
> assumed to be Monotonic, but I don't know if that's proved. The fact
> that Tideman liked RP suggests that he knew it was Monotonic, but
> I don't know. Has Markus shown that BeatpathWinner (Schulze's method)
> is Monotonic?

I've read the claim for RP being Monotonic, but never seen proof.

The general problem for Condorcet methods is this: If X is the CW, then
it is easy to show that any ballot change favoring X won't make X the
loser. If X is the winner, but not a CW, then it is possible to make
a ballot change favoring X that breaks up the Smith set. Let's say the
Smith set consists of W, X, Y, and Z, and that

W beats X
X beats Z
Y beats W and X
Z beats W and Y

Suppose the completion method picks X. If we find some ballots containing
"W>X" and swap those two candidates on those ballots, and this changes
things so that X beats W, then W is no longer in the Smith set, and we
have:

X beats Z
Y beats X
Z beats Y

Unless the method picks X again in this new configuration, it is not
monotonic. Let's suppose it does, so that Monotonicity is not violated
(yet).

Now suppose you started with this:

W beats Y
X beats W and Z
Y beats X
Z beats W and Y

Suppose this configuration has Y as the winner. Now find some ballots
with "W>Y", and reverse those two. Then, if this causes W to lose to Y,
you end up with

X beats Z
Y beats X
Z beats Y

which looks very familiar. If the {margins|winning votes} in this
final Smith set are identical to the ones in the final Smith set of
the original example, then X is picked again (assuming those numbers
are the only data used by the completion method). That violates
Monotonicity.

So, for a Condorcet method to be monotonic, it would be necessary that
the method is unable to produce identical {margins|winning vote} in
the final configuration of the two examples given. I suspect that is
impossible.

(Of course, we could have a Condorcet completion method that's based on

something other than just the margins or winning votes of the Smith
set candidates. It isn't even necessary for a Condorcet method to always
pick from the Smith set. It might be possible to find methods of that
sort that would be immune to this problem.)

> Of course, as I think you suggested, not much can be said about
> this till Monotonicity has a definite definition.

We do have a working definition for ranked voting methods. I see no reason
that that definition can't continue to be used for those methods. Also,
it shouldn't be too hard to write a definition for methods using other
types of ballots (plurality ballots, CR ballots, approval ballots). When
we find an all-purpose definition, the problem is then to make sure it
is backwards-compatible with the other definitions.

-- Richard

```