## 17.4 Outline

- Introduction
- Alabama paradox
- Population paradox
- New states paradox
- Balinski and Young’s impossibility theorem

## 17.4 Essential Ideas

**Alabama Paradox**

A reapportionment in which an increase in the total number of seats results in a loss of seats for some state is called the **Alabama paradox.**

**Population Paradox
**

When there is a fixed number of seats, a reapportionment that causes a state to lose a seat to another state even though the percent increase in the population of the state of the state that loses the seat is larger than the percent increase of the state that wins the seat is called the **population paradox.**

**New States Paradox
**

A reapportionment in which an increase in the total number of seats causes a shift in the apportionment of the existing states is called the **new states paradox.**

**Balinski and Young Impossibility Theorem
**

Any apportionment plan that does not violate the quota rule must produce paradoxes. Recall, the **quota rule** says that the number assigned to each represented unit must be either the standard quota rounded down to the nearest integer, or the standard quota rounded up to the nearest integer. Also, any apportionment plan that does not produce paradoxes must violate the quota rule.