[EM] Election-Methods Digest, Vol 76, Issue 7

[fsimmons at pcc.edu]

Thanks for the clarification, Kristopher.

Also, if I remember correctly, we have another kind of monotonicity that is
really a weak form of Participation, namely "mono add plump,"  which means that
if a new ballot is added that "bullet votes" for candidate A, then A's
probability of winning should not decrease.

It seems to me that this milder form of participation should be attainable.

Forest

> Date: Tue, 19 Oct 2010 08:39:53 +0200
> From: Kristofer Munsterhjelm
> To: kathy.dopp at gmail.com
> Cc: election-methods at lists.electorama.com
> Subject: Re: [EM] MCA on electowiki
> Kathy Dopp wrote:
>
> > The mathematical definition of increasing monotonicity says
> when I
> > increase the independent variable, the dependent variable likewise
> > increases (for voting, when I increase votes for a candidate, that
> > candidate's chance of winning increases.) Or the mathematical
> > definition of nondecreasing monotonicity says, when I increase the
> > independent variable, the dependent variable never decreases (for
> > voting when I increase votes for a candidate, the candidate's
> chances> of winning never decreases.)
> >
> > I would say by any standard normal mathematical definition of
> > monotonicity, if a voting method fails the Participation
> Criterion you
> > linked to, it also fails to be monotonic.
> >
> > Adding votes or increasing ranking for a candidate, should not cause
> > that candidate to lose whereas he otherwise might have won.
> To me,
> > that is just another way of stating nonmonotonicity.
>
> Using Woodall's terms, the full name of what we usually call
> "monotonicity" on this mailing list is "mono-raise". That is:
> monotonicity regarding raising (ranking higher) a candidate.
> There are
> many other forms of monotonicity: for instance, mono-add-top
> (adding a
> vote that ranks a candidate first shouldn't make the candidate
> lose),
> mono-append (adding a candidate to a truncated ballot should not
> make
> that candidate lose), and so on. See
> http://www.votingmatters.org.uk/ISSUE3/P5.HTM for the full list.
>
> Any of these might be called monotonicity criteria, since they
> involve
> situations where ballots are added or altered in a way that is
> seemingly
> favorable for the new candidate, and the method fails the
> criterion if
> the candidate loses.
>
> As for Participation, Woodall says: "There is also the following
> property, which is not strictly a form of monotonicity but is
> very close
> to it. (...) 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. ".
> It is, unfortunately, a very strict criterion. Only voting
> methods that
> consist of point systems with point system tiebreakers (not
> necessarily
> the same tiebreakers) can fulfill it. A point system is one
> where you
> give the first candidate on a ballot x points, the second y
> points, the
> third z points, etc. DAC/DSC is in this sense a series of point
> systems,
> each breaking ties of the last.
>
>
> In summing up: what we call "monotonicity" is just one form of
> monotonicity, that is true; and it is unfortunate but also true
> that
> most complex systems fail Participation.
>