Editing Apportionment paradox (section)

===Alabama paradox===
The Alabama paradox was the first of the apportionment paradoxes to be discovered. The US House of Representatives is [[United States Constitution|constitutionally]] required to allocate seats based on population counts, which are required every 10 years. The [[United States congressional apportionment|size of the House]] is set by statute.

After the [[United States Census, 1880|1880 census]], C. W. Seaton, chief clerk of the [[United States Census Bureau]], computed [[United States Congressional Apportionment|apportionments]] for all House sizes between 275 and 350, and discovered  that Alabama would get eight seats with a House size of 299 but only seven with a House size of 300.<ref name=Stein2008/>{{rp|228–231}} In general the term ''Alabama paradox'' refers to any apportionment scenario where increasing the total number of items would decrease one of the shares. A similar exercise by the Census Bureau after the [[United States Census, 1900|1900 census]] computed apportionments for all House sizes between 350 and 400: Colorado would have received three seats in all cases, except with a House size of 357 in which case it would have received two.<ref>{{cite web|first = Alex |last = Bogomolny |date = January 2002 |url = http://www.cut-the-knot.org/ctk/Democracy.shtml |title = The Constitution and Paradoxes |work = Cut The Knot! }}</ref>

The following is a simplified example (following the [[largest remainder method]]) with three states and 10 seats and 11 seats.

{| class="wikitable" width="500px"
! colspan="2" | !! colspan="2" | With 10 seats !! colspan="2" | With 11 seats
|-
! State !! Population !! Fair share !! Seats !! Fair share !! Seats
|-
| A || align="right" | 6 || align="right" | 4.286 || align="right" | 4 || align="right" | 4.714 || align="right" | 5
|-
| B || align="right" | 6 || align="right" | 4.286 || align="right" | 4 || align="right" | 4.714 || align="right" | 5
|-
| C || align="right" | 2 || align="right" | 1.429 || align="right" | 2 || align="right" | 1.571 || align="right" | 1
|}

Observe that state C's share decreases from 2 to 1 with the added seat.

In this example of a 10% increase in the number of seats, each state's share increases by 10%. However, increasing the number of seats by a fixed % increases the fair share more for larger numbers (i.e., large states more than small states). In particular, large A and B had their fair share increase faster than small C. Therefore, the fractional parts for A and B increased faster than those for C. In fact, they overtook C's fraction, causing C to lose its seat, since the Hamilton method allocates according to which states have the largest fractional remainder.

The Alabama paradox gave rise to the axiom known as [[house monotonicity]], which says that, when the house size increases, the allocations of all states should weakly increase.