Chris Benham chrisbenham at bigpond.com
Fri Jan 2 11:03:02 PST 2004

```  On Sunday Dec.28, 2003,  I  wrote:

"Mono-add-top" is a Woodall criterion which says that adding ballots that all give
first-preference to X must not harm X. It is met by IRV and Margins, but not by WV.

On the same date Mike Ossipoff replied:
"I'm not necessarily denying that, but can you demonstrate that those
statements are correct?"

I assume that he was mainly referring to the comparison between Margins and WV.
(As IRV successively excludes the candidate which is highest ranked on the fewest ballots,
it is obvious logic that it cannot fail Mono-add-top. Mono-add-top is implied by Later-no-harm,
which IRV meets.)
In Woodall's very interesting paper,"Monotonicity and Single-Seat Election rules"
http://groups.yahoo.com/group/election-methods-list/files/wood1996.pdf
he defines the "Plurality" criterion:

"If some candidate X has strictly fewer votes in total than some other candidate Y has
first-preference votes, then X should not have a greater probability than Y of being elected."
(This is in the context of truncation being allowed,and a "vote" is a ranking above last place.)

It is met by WV and IRV, but not Margins.

One of Woodall's "Impossibility Theorems" says that you can't have all three of "Plurality",
Condorcet,  and Mono-add-top. He gives this example:

11:A>B
07:B
12:C

Margins elects A, and so violates Plurality (and also in this case Later-no-help, but that
criterion is incompatible with Condorcet, so bad luck).
WV meets Plurality and elects B. Adding 2 B>A ballots gives:

11:A>B
07:B
02:B>A
12:C

To meet Mono-add-top, WV must still elect B,  but it can't because A is now the Condorcet
Winner.
I surmise that Margins meets Mono-add-top, because it is not burdened with meeting Plurality,
and also according to Woodall Mono-add-top and Condorcet are not incompatible.

Chris Benham

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