The US House of Representatives uses the
method to figure out how many representatives each state should
have. First you give each state one representative, because even the
smallest state is guaranteed one, and then you assign the remaining
spots, one at a time, to whichever state has the highest "priority".
Priority being ratio of its population to the geometric mean of the
number of seats it currently holds and the number it would hold if it
received this extra seat:
Ps = state population
Rs = state reps
Where does this come from? I had a shot at deriving it, and it
actually makes a lot of sense. First, we restate the problem has one
of error minimization. At every stage, we want to assign the next seat
wherever it would most minimize representational inaccuracy.
Current error is, summed over all states:
Pt = total population
Rt = total (target) reps
| Pt Ps |
| -- - -- | * Ps
| Rt Rs |
For each state we might give a seat to, the effect that would have on
total error is:
| Pt Ps | | Pt Ps |
| -- - ---- | * Ps - | -- - -- | * Ps
| Rt Rs+1 | | Rt Rs |
We would like to identify the state that minimizes this quantity.
Since we are adding representatives one by one, Pt/Rt
will always be greater than Ps/Rs  and we can remove
the absolute value and distribute the Ps.
PtPs PsPs PtPs PsPs
---- - ---- - ---- + ----
Rt Rs+1 Rt Rs
Cancel the PtPs/Rt and we have:
---- - ----
Combine the two fractions and cancel again:
Since we're trying to identify the state that minimizes the quantity,
we can instead identify the state that minimizes its square root:
Which is in the prioritization of Huntington-Hill.
I initially tried to derive this from squared error, which did not
work and ended up with an enormous amount of scribbles on paper.
 This is not quite true, as we get to assigning the very last
representatives, but I think it still works?
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I belive this is an apportionment algorithm. They're interesting (mathematically) and like lots of other similar things, seem to inevitably involve tradeoffs among various reasonable criteria we'd all like to be met.
They're a neat way to allocate money amounts in a way that doesn't annoy accountsnts tho!