# [Election-Methods] Which monotonicity?

Jonathan Lundell jlundell at pobox.com
Wed Jan 16 13:57:47 PST 2008

```On Jan 15, 2008, at 11:28 AM, Stéphane Rouillon wrote:

> I am fed up a bit with that discussion about non-monotonicity
> because it depends how
> monotonicity is defined.  IRV is monotonic when you consider adding
> or retrieving
> ballots with you preffered candidate as first choice. IRV is non-
> monotonic when
> you consider highering or lowering the positions of your preferred
> candidate on several ballots...
>
> Maybe some concision and precision in the definitions would help.
> Could we use 2 different names for monotonicities please?
> Or maybe they exist and I don't know these definitions...

You might check out Woodall's taxonomy:

Monotonicity. A candidate x should not be harmed if:

• (mono-raise) x is raised on some ballots without changing the
orders of the other candidates;
• (mono-raise-delete) x is raised on some ballots and all candidates
now below x on those ballots are deleted from them;
• (mono-raise-random) x is raised on some ballots and the positions
now below x on those ballots are filled (or left vacant) in any way
that results in a valid ballot;
• (mono-append) x is added at the end of some ballots that did not
previously contain x;
• (mono-sub-plump) some ballots that do not have x top are replaced
by ballots that have x top with no second choice;
• (mono-sub-top) some ballots that do not have x top are replaced by
ballots that have x top (and are otherwise arbitrary);
• (mono-add-plump) further ballots are added that have x top with no
second choice;
• (mono-add-top) further ballots are added that have x top (and are
otherwise arbitrary);
• (mono-remove-bottom) some ballots are removed, all of which have x
bottom, below all other candidates.
There is also the following property, which is not strictly a form of
monotonicity but is very close to it. It is an extension to multi-seat
elections of a property proposed by Moulin[3] for single-seat elections.
• Participation. The addition of a further ballot should not, for any
positive whole number k, reduce the probability that at least one
candidate is elected out of the first k candidates listed on that
ballot.

http://www.votingmatters.org.uk/ISSUE3/P5.HTM
```