Raph Frank wrote:
On Sat, May 22, 2010 at 11:39 PM, <[email protected]> wrote:
Satisfaction Approval Voting - A Better Proportional Representation Electoral
Method
One way to generalize Proportional Approval voting to range ballots is by
finding the most natural smooth extension of the function f that takes each
natural number n to the sum
f(n) = 1 + 1/2 + ... + 1/n.
It turns out that we can extend f(n) to all positive real values of n via the
integral
Integral from zero to one of (1-t^n)/(1-t) with respect to t
Interesting. The previous suggestion was to use logs and a correction.
That is probably because the integral approaches the natural logarithm
as n goes to infinity. For integer values, f(n) is the nth harmonic
number H_n, and
lim (n -> inf) H_n = ln(n) + gamma
where gamma is Euler's constant (0.5772...)
For instance, ln(10) + gamma = 2.879801 whereas H_10 is 2.828968.
That, in turn, is related to that the integral of 1/n is ln(n).
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