Greg Nisbet wrote:
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
In a prior post, I determined a "constant" with the same purpose as a.
http://listas.apesol.org/pipermail/election-methods-electorama.com/2008-September/022453.html
That variable is about 1.773, increasing very slowly (1.7778 for x=14,
1.781 for x=1000), and the function is then
f(x) = ln(e + p * (x-1)),
where p is the "constant". In the post, I used the term "log" for "ln",
because that's how it is in C. To clear up potential confusion, I'll
state that I'm referring to the natural logarithm.
The Sainte-Laguë variant seems to be
f(x) = ln(sqrt(e^2 + q * (x-1)))
where q is about 7.1 (7 for x = 2, 7.095 for x = 9, 7.124 for x = 1000).
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.
Negative numbers shouldn't be used at all. If you look at it from the
PAV point of view, it's only sensible to not permit negative numbers.
Having negative numbers would mean that the voter in question got less
than no candidates of those that he approved of.
If you disagree and want to have negative numbers in a consistent way,
you have to find out what it means for something to have negative
utility, and how sensitive the metric should be to that.
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.
I'd say that there's a degree to which this metric is arbitrary, but I
already gave that argument in more detail in another reply. To show that
this really is multiwinner Bayesian regret, you'd have to show that
people's utility in having two they approved of is 1 + 1/2. As for
suggesting a multiwinner method, I'd suggest QPQ, which does well in my
own simulations, at least with honest voters. It's STV style, which
means that it uses ranked ballots and may be nonmonotonic - I don't know
what criteria it passes and what it fails.
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