[EM] Required Number Votes in a Matrix

[Kristofer Munsterhjelm]

On 05/20/2014 05:31 AM, DNOW1 at aol.com wrote:
> Many of the examples by folks have INCOMPLETE votes in a N x N matrix.

Er, they don't have incomplete votes in the matrix. They usually don't 
mention the matrix at all.

> That is improper.
> A mini Approval Boundary Example --
>
> 50 AB
> 1 B
> 49 Z
> 100
>
> IF Approval is being used, then B wins -- 51 B, 50 A, 49 Z.
> Choice A is obviously very hurt by the 2nd choice votes for B.
>
> IF all voters are required to use Number Votes for all choices, then
> there would be --
> Single office
> 50 ABZ
> 1 B??
> 49 Z??
> 100
>
> Who wins then -- with what election method ???

Some election methods can provide an outcome even with incomplete 
rankings. For instance, for Condorcet, consider:

50: A>B
1: B
49: Z

Then the raw matrix is:

    (against)
    A  B  Z
A  ## 50 50
B   1 ## 51
Z  49 49 ##

and the (derived) wv matrix is

    A  B  Z
A  ## 50 50
B   0 ## 51
Z   0  0 ##

so the ranking is: A gets first place (beats B and Z). B gets second 
(beats Z), and Z gets third (is beaten by both).

> Thus - If there are N choices, then should the first N-1 choices be
> deemed *YES* votes ???

The usual implicit Approval interpretation is that listed candidates are 
considered yes votes and those not listed are considered no.

For instance:

50: A>B

here the Approval information is "yes to A and B, no to Z".

With explicit approvals, this notation is sometimes used on the list:

2: A>>B>Z

meaning "yes to A, no to B and Z".

> How many of the *complex* election methods have ANY chance whatever of
> becoming such an Amendment ???
>
> See the chaos about the mere statutory math for apportioning USA Reps
> among the States.

The surprising thing there is that the apportionment math is *more* 
complex than it needs to be; probably for a combination of mathematical 
elegance (that didn't turn out to be so elegant after all) and political 
convenience.

Before the present Huntington-Hill system, the US apportionment used 
Webster's method, which can be pretty simply put: each state gets as 
many representatives as its state population divided by x, rounded off. 
The variable x is set so that the total comes out to the total number of 
seats in the House.