[EM] Inferring a method from an MMC axiom

[Markus Schulze]

Dear Craig,

you wrote (10 March 2003):
> To Mr Schulze: did you find out why Mr Dummett had such a
> weak rule ?. That rule had a Floor(x) function on it, reducing
> the number of candidates that it said had to win. It seemed
> something that did not look desirable, and it seemed to be
> too distant from imposing a constraint.

Let's say that V > 0 is the number of voters, S > 0 is the number
of seats, and C > S is the number of candidates.

A "solid coalition" is a set of candidates with a set of voters
such that every voter in this set of voters strictly prefers every
candidate of this set of candidates to every candidate outside this
set of candidates.

Let's say that a given solid coalition consists of C0 candidates
and V0 > 0 voters. Then "Dummett's proportionality criterion for
solid coalitions" says that at least min{ceil((S+1)*V0/V)-1; C0}
candidates of this solid coalition must be elected.

The reason why Dummett's criterion is so weak is that it is a
conditio sine qua non: When a given multi-winner election method
doesn't meet Dummett's proportionality criterion for solid
coalitions then it isn't even a proportial election method due
to Dummett's theory.

Markus Schulze