At 05:33 PM 5/8/2008, Juho wrote:
(If there are e.g. two parties, one small and one large, the
probability of getting two small party supporters (that would elect
one of them to the next higher level) in a group of three is so small
that in the next higher level the number of small party supporters is
probably lower than at this level.)
Okay, let's do the math. Suppose the ratio of voters who are of some
group is p, where 0 < p < 1. If x is not-p, then the permutations and
probabilities for the four possibilities of 0 members, 1 member, 2
members, and three members, are:
xxx, (1-p)^3 = P(0)
xxp, xpx, pxx, 3 * (1 - p)^2 * p = P(1)
xpp, pxp, ppx, 3 * (1 - p)* p^2 = P(2)
ppp, p^3 = P(3)
expanding those,
P(0) = 1 - 3p +3p^2 -p^3
P(1) = 3p -6p^2 +3p^3
P(2) = 3p^2 - 3p^3
P(3) = p^3.
To check, the sum simplifies to 1. These four are the only possibilities.
If the group selects based on majority p, then we have a p choice
with P(2) and P(3). That occurs with probability
3p^2 -2p^3.
If p = 0.1, then the probability of a group choosing a p
representative is .03 - .002 equals .028.
p is 10% of the population, but is represented in the next layer with
only 2.8% of the elected representatives. And then the same
phenomenon occurs in the next layer, etc., with the proportion of p
declining more rapidly with each layer. I get 0.23% for the next
layer. With many layers, as is necessary for this system to represent
a large population the proportion of p rapidly approaches zero, and
it becomes extraordinarily unlikely for the minority to be
represented at all, even with an Assembly of, say, 100 members or
more. And that is already a fairly large assembly, in my opinion.
Assemblies that large tend to function mostly in committee.
Now, perhaps my math is wrong, I'm rusty and all that, and I make
mistakes even when I understand clearly what to do. Mr. Gohlke, do
you care to look at this?
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