But luck in [0,1] would even make less sense as luck^n (as it goes to 0), while as luck in [0,1000] could scale a successful startup by the factor of up to 1000.
What I meant by 'n' was a weighting. I wasn't clear.
Let's say you have 4 factors a, b, c, d.
If we are summing them then we can just sum them straight:
score = a + b + c + d
If we want to place more importance on some of them we weight them, with (constant) weights w, x, y, z by doing this:
score = w * a + x * b + y * c + z * d.
If it's a multiplicative relationship however then we multiply straight:
score = a * b * c * d
If we tried add weighting in the same way:
score = w * a * x * b * y * c * z * d
Then all we've done is scale the previous score by (w * x * y * z) which is constant, so hasn't actually affected any ranking.
To weight a multiplicative relationship requires the weights to be exponents:
score = a^w * b^x * c^y * d^z
The scores can be in [0..1] or [1..N], the increased weights still increase relative power of the input parameter.
If you take logs it's clear why.
log(score) = w * log(a) + ... + z * log(d) which is our weighting when summing again, and log is monotonic (proof of which is left to the reader), log(S1) < log(S2) implies S1 < S2.
