[EM] Multiwinner Method Yardstick (Gregory Nisbet)

[Raph Frank]

On Sat, Oct 11, 2008 at 6:44 AM, Greg Nisbet <gregory.nisbet at gmail.com> wrote:
<snip>

Basically, each voter gives a utility of

k=N
sum(1/k)
k=1

Where N is the number of candidates that the voter approved that won.

Pick the winning group that gives the maximum total sum.

> Multiwinner Method Yardstick
> Let's pretend Alice votes X = 99, Y = 12, Z = 35
<snip>

Proposal is sort ratings given to winners and find sum with the PAV weights

99 + 35/2 + 12/3 = 120.5

I suggested in a previous thread to use the following formula

f( sum up the ranting of approved candidates / max score allowed)

so

f( (99 + 35 + 12)/99 ) = f( 146/99 ) = f( 1.475 )

The f function is selected so that

f( 0 ) = 0
f( 1 ) = 1
f( 2 ) = 1 + 1/2 = 1.5
f( 3 ) = 1 + 1/2 + 1/3 = 1.83

i.e. it is the same the PAV sum.

It can be calculated approx using

f( x ) = ln(2*x+a)

a is (almost) a constant.  It could be calculated exactly for all the
integers and linearly interpolated.

This would give the exact answer for all integers.

This means that you get the same answer as PAV if everyone rates the
candidates as 0 or 99.

It has the advantage that if A,B and C are elected, then the following
voters will be considered equally happy

A: 0
B: 99
C: 99
Your proposal: 99 + 99/2 = 1.5
My suggestion: f( 2 ) = 1.5

A: 50
B: 50
C: 98

Your proposal: 98 + 50/2 + 50/3 = 139.7
My suggestion: f( 2 ) = 1.41


One problem with my suggestion (but not yours) is that it doesn't
handle non-normalised utilities very well.

For example, if you multiplied all the voters' ratings by a constant,
the ordering would not be changed under your system.

Likewise, if you added a constant to all the ratings, the ordering
would not be changed.