[EM] (P1) and monotonicity for single-winner election systems and Condorcet.

[David Catchpole]

Very true. Well, at least Craig's 2-member FPTP rule and majority rules
are similar- and you get my drift (I hope!). More further down...

On Fri, 22 Oct 1999, Markus Schulze wrote:

> Dear David,
> 
> you wrote (21 Oct 1999):
> > Now, take for granted majority rules (what Craig has defined as 2-member
> > FPTP rule). That is, we already have a space of voting schema for which we
> > know the answer- those in which one candidate recieves more than half of
> > first preference votes (the candidate is most highly ranked by more than
> > half of voters who express some non-indifference between candidates).
> 
> Borda meets Craig Carey's 2-member FPTP rule. But nevertheless, if one
> candidate recieves more than half of the first preferences, then the Borda
> method still doesn't necessarily choose this candidate.
> 
> > It has now occured to me that it may be possible to demonstrate that
> > Condorcet is a necessary condition of monotonicity without assuming
> > majority rules ("2-candidate FPTP") but instead assuming the system is
> > neutral to candidates (a switch in candidates brings on a corresponding
> > change in results). However, this is going to take some effort, because it
> > involves 3!=6 "points" and 2^12 possible permutations of results.
> I believe that you have to presume explicitely that the used election
> method meets the majority criterion (i.e. that a candidate who is strictly
> prefered to every other candidate by more than half of the voters must be
> elected).
> 
> Markus Schulze

The challenge though is to work out whether the majority criterion may be
implied by other criteria such as monotonicity (obviously, where Condorcet
winners are elected, it is assured that the majority criterion is
satisfied). It would be an interesting result if this were the case.