Chapter 4
4.1
 The Apportionment Problem: A special kind of fair division.
4.2
 Standard Division: Population (P), Number of seats to be apportioned (M), P/M is
the standard divisor.
 Standard Quota: fraction of the total number of seats that each state would be
entitled to if fractional seats were possible. State X’s population / SD.
 Lower Quota: Standard quota rounded down.
 Upper Quota: Standard quota rounded up.
4.3
 Hamilton’s Method and the Quota Rule

Step1: Calculate each state’s standard quota.

Step 2: Give each state its lower quota.

Step 3: Give the surplus to the states with the largest fractional parts.

Fails Neutral criteria: Every state has the same opportunity for favorable
apportionment. (Favors larger states)
 Quota Rule:

Apportionment should be either its upper quota or its lower quota.

An apportionment method that guarantees that every state will be
apportioned either to its lower quota or upper quota satisfies the rule.

Violations: lower-quota violation and upper-quota violation.
4.4
 Alabama Paradox: An increase in the total number of seats being apportioned, in
and of itself, forces a state to lose one of its seats.
4.5
 Population Paradox: Occurs when a state X loses a seat to state Y even though
X’s population grew at a higher rate than Y’s.
 New State Paradox: The addition of a new state with its fair share of seats affects
the apportionment of other states.
4.6
 Jefferson’s Method: We need to use a modified divisor (trial and error) that will
give us new modified quotas that, when rounded down, will total the exact
number of seats to be apportioned.

Step1: Find the modified divisor (D, smaller than the standard divisor)
such that when each state’s modified quota is rounded down, the total is
the exact number of seats to be apportioned.

Step2: Apportion to each state its modified lower quota.
 Jefferson’s method was the very first apportionment method used by the U.S.
House of Representatives. (Violates the quota rule)
4.7
 Adam’s Method

Step 1: Find the modified divisor D such that when each state’s modified
quota is rounded upward, the total is the exact number of seats to be
apportioned.

Step 2: Apportion to each state its modified upper quota.
 Adam’s violations are all lower quota violations.
4.8
 Webster’s Method: Round quotas to the nearest integer.

Step 1: Find the modified divisor D such that when each state’s modified
quota is rounded the conventional way the total is the exact number of
seats to be apportioned.

Step 2: Apportion to each state its modified quota rounded the
conventional way.

May violate the quota rule but rare.

Comes closest to satisfying all main requirements for fairness.
 Balinski and Young’s impossibility theorem: There cannot be a perfect
apportionment method. Any method that does not violate the quota rule must
produce paradoxes and vice versa.
Appendix 1
 The Huntington-Hill Method:

Comparable to the Webster Method: Find modified quotas and round
some down and some upward. The difference is the cutoff point for
rounding. L: lower quota
L+1: upper quota

Cutoff for Webster’s: L + (L+1) / 2

Cutoff for Huntington-Hill: Square root of (L x (L+1)) “geometric mean”
“arithmetic mean”