[EM] MCA on electowiki

[Kristofer Munsterhjelm]

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.