Alabama paradox
The Alabama paradox was the first of the apportionment paradoxes to be discovered. After the 1880 census, C. W. Seaton, chief clerk of the U. S. Census Office, computed apportionments for all House sizes between 275 and 350, and discovered that Alabama would get 8 seats with a House size of 299 but only 7 with a House size of 300. 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 toy example with 4 states and 323 seats (assuming that the Hamilton method is used) is as follows:
| State | Size | Fair share | Seats |
| A | 5670 | 183.141 | 183 |
| B | 3850 | 124.355 | 124 |
| C | 420 | 13.566 | 14 |
| D | 60 | 1.938 | 2 |
With 324 seats:
| State | Size | Fair share | Seats |
| A | 5670 | 183.708 | 184 |
| B | 3850 | 124.740 | 125 |
| C | 420 | 13.608 | 13 |
| D | 60 | 1.944 | 2 |
Observe that state C's share decreases from 14 to 13. The reason this occurs starts with the fact that increasing the number of seats increases the fair share faster for the large states than for the small states. Hence, large A and B had their fair share increase faster than small C. Therefore, the decimal parts for A and B increased faster than those for C. In fact, they overtook C's decimal part, causing C to lose its seat, since the Hamilton method examines which states have the largest decimal part.
See also Apportionment paradoxes for more information on apportionment Paradox
Reference Methods of apportionment (pdf file)
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