Indeed, all forms of MCA satisfy mono-add-plump (unless a non-compliant method is used to choose the finalists for the runoff in MCA-IR or MCA-VR).
In fact, they satisfy an slightly stronger criterion, let's call it mono-add-antiplump. You cannot cause Y to win by adding a ballot which doesn't approve Y (that is, votes them at the lowest rating possible). Also, I agree: thanks, Kristopher, your explanation of monotonicity is great. JQ 2010/10/19 <[email protected]> > 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: [email protected] > > Cc: [email protected] > > 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. > > > ---- > Election-Methods mailing list - see http://electorama.com/em for list info >
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