Apportionment

US Constitution Article I, Section 1, Clause 3:

"Representatives … shall be apportioned among the several States … within this Union, according to their respective Numbers, which shall be determined by adding to the whole Number of free Persons, including those bound to Service for a Term of Years, and excluding Indians not taxed, three fifths of all other Persons. … The Number of Representatives shall not exceed one for every thirty Thousand, but each State shall have at Least one Representative;"

The apportionment problem is a type of fair division method. It arises because it is highly unlikely that any state’s fair share of the House of Representatives will be a whole number. The apportionment problem arises when we are required to round fractions so that their sum is maintained at some constant value.

In the 1790 census, the US population was 3, 615, 920. The most populous state was the commonwealth of Virginia, with 630, 560 people, while Delaware had the least number of people, with 55, 540. At the time the House had 105 seats to apportion. Thus Virginia’s fair share would be 18.310 and that of Delaware 1.613:

Apportionment Everywhere

- Resource Allocation
- Political Seat Allocations
- States (proportional to their populations)
- Parties (proportional to votes they get in an election)

Professor |
Current Salary |
Increased by 5% |
Lower Quota |
Apportioned salary |

A |
43,100 |
45,255 |
45,000 |
45,000 |

B |
42,150 |
44,257 |
44,000 |
44,000 |

C |
10,000 |
10,500 |
10,000 |
11,000 |

Totals |
95,250 |
100,012 |
99,000 |
100,000 |

Basic Terms

Let **P** be the
total population, **M** the total
number of seats to be apportioned, and **p** the
population of a state.

**Standard
Divisor** = P/**M**

**Standard
quota ** = **p**/(P/**M**)= (**pM**)/P

**Lower quota**is the standard quota rounded down**Upper quota**is the standard quota rounded up**Quota Rule**to allocate to each state either its lower or upper quota.

Hamilton's Method (1852 - 1901)

- Calculate each state's standard quota
- Give to each state its lower quota
- If there are leftover seats distribute them one at a time to the states with the largest fractional parts.

Problems with Hamilton's Method:

- Alabama Paradox:
- Population Paradox
- New Sate Paradox

An increase in house size causes a state’s apportionment to decrease

A State’s population increases, while other states maintained their populations, and yet the state apportionment decreases.

A new State is admitted into the union with its fair share of seats and yet one other states loses seats for another.

Jefferson's Method (1790-1842)

- Find a divisor,
**D**, such that when each states modified quota (**p**/**D**) is rounded downward, the total is the exact number of seats to be apportioned. - Apportion to each state its modified lower quota.

Problems with Jefferson's Method

- Violation of the upper quota.
- Shows favoritism towards larger (population wise) states.

Adam's Method (never implemented)

- Find a divisor,
**D**, such that when each states modified quota (**p**/**D**) is rounded upward, the total is the exact number of seats to be apportioned. - Apportion to each state its modified upper quota.

Problems with Adams' Method

- Violation of the lower quota.
- Shows favoritism towards smaller (population wise) states.

Webster's Method (1842-1852, 1911-1942)

- Find a divisor,
**D**, such that when each state’s modified quota (**p**/**D**) is rounded the conventional way (to the nearest integer), the total is the exact number of seats to be apportioned. - Apportion to each state its modified quota rounded to the nearest integer.

Problems with Webster's Method

- Violation, though very rarely, of the quota rule.

Huntington-Hill rule

**Cutoff for the **Huntington-Hill method

**H** =

In other words round up if the modified quota is greater than
**H** and round down
if the modified quota is less than **H**.

Huntington-Hill Method of Apportionment (1941- to date)

- Find a divisor,
**D**, such that when each states modified quota (**p**/**D**) is rounded according to the Huntington-Hill rule, the total is the exact number of seats to be apportioned. - Apportion to each state its modified quota rounded using the Huntington-Hill rules.

Balinski and Young's Impossibility Theorem

It is mathematically impossible for an apportionment method to be perfect. There is no apportionment method that satisfies the quota rule and avoids paradoxes.