[EM] Ways to Evaluate Multiwinner Contests (Greg Nisbet)

[Greg Nisbet]

So far some nice ideas have been proposed for measuring how effective a
multiwinner method is.
All of the ones proposed are based on n, let n be a list of utility scores
for the candidates.
1. ln(2*sum(n)+a) we don't know what a is. It's probably pretty close to a
constant, let's just call it 1.
2. sum(n)*ln(2*len(n)+a) same comment

Note:
values for the ln(2*sum(n)+a) can be computed exactly if you have time on
your hands
f(1) = 1
f(2) = 1 + 1/2 => 1.5
f(3) = 1 + 1/2 + 1/3 => 1.83

3. this iterative procedure:
def sortpav(n):
    n.sort()
    n.reverse()
    ret = 0
    for y,x in enumerate(n):
        ret += x/float(y+1)
    return ret
Of course these three procedures can be used for Range Voting, but it is not
necessary to accept the validity and perfection of Range Voting to use these
as metrics.

Range Voting resembles utility summation with two fundamental differences 1)
people vote strategically and 2) a ceiling

So, now we move onto the multiwinner bayesian versions.
Unlike single winner, the methods for measuring regret so far proposed have
a floor, specifically 0. See, the ln of negative numbers includes pi*i. I do
not know how to handle imaginary numbers with regard to voting methods, so I
am going to outlaw negative numbers as the lazy solution to the problem.

If anyone has a multiwinner method to suggest, a criticism to the metric or
anything else before I run the simulation please let me know. I know better
than to call a vote on which metric to use, so if you have an opinion on one
of them please tell me.
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