I have recently encountered an interesting phenomenon on SO reputation system:
Let $f(n)$ denote the current score of the $n$th best user.
A sample that I collected at a given moment:
- $f(k)=k \implies k\approx7920$
- $f(k)=2k \implies k\approx5470$
- $f(k)=4k \implies k\approx3740$
- $f(k)=8k \implies k\approx2545$
- $f(k)=16k \implies k\approx1695$
- $f(k)=32k \implies k\approx1115$
The ratio appears approximately $1.5$ in all cases.
Now, obviously, the SO reputation system wasn't design to yield a relative scoring system amongst users, but an absolute one, based on some arbitrary table of "score per event".
Nevertheless, in the overall perspective given above, this scoring system appears to be giving a very tight relation between the scores of different users.
Of course, I should probably conduct the measurements above over a period of time, but it feels as if there's a "guiding hand" behind this.
I doubt it is made intentionally by "SO managers", so my question is, could it be the reflection of the law of large numbers within a wisdom of the crowd based system?
This scaling is usually referred to as a power law. To see why, note that one assumes that for every nonnegative integer $i$, $k=a/r^i$ solves the equation $f(k)=2^ik$, for some fixed $a\gt0$ and $r\gt1$, that is, $f(a/r^i)=a2^i/r^i$ for every $i$.
If $x=a/r^i$ then $i=\log(a/x)/\log r$ and $a2^i/r^i=x2^i=x\mathrm e^{i\log2}=x(a/x)^{\log2/\log r}$, hence the functional equation translates as $$f(x)=bx^c,$$ with $b=a^{\log2/\log r}$ and $c=1-(\log2/\log r)$. Since $r\gt1$, $c$ can take every value $c\lt1$. If $r=1.5$ then $c=-0.709511$.