## Why Huntington-Hill? |
October 24th, 2020 |

math, politics |

Ps = state population Rs = state reps Ps --------------- sqrt(Rs*(Rs+1))

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`

[1] 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:

PsPs PsPs ---- - ---- Rs Rs+1

Combine the two fractions and cancel again:

PsPs --------- Rs*(Rs+1)

Since we're trying to identify the state that minimizes the quantity, we can instead identify the state that minimizes its square root:

Ps --------------- sqrt(Rs*(Rs+1))

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.

[1] This is not quite true, as we get to assigning the very last
representatives, but I think it still works?

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