This document discusses different methods for apportioning congressional seats among US states based on population: Hamilton, Jefferson, Webster, and Hill-Huntington. Hamilton's method allocates remaining seats to states with the largest fractional quotas. Jefferson's method favors larger states by allocating extra seats to those with the largest critical divisors. Webster's method rounds quotas neutrally. Hill-Huntington has been used since 1940, allocating extra seats to states with the largest critical divisors based on geometric means. The best method minimizes differences in representative shares or district populations among states.

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ApportionmentApportionment
• Apportionment Problem: is to round a set of fractions
so that their sum is maintained at its original value.
• Apportionment Method: the rounding procedure used in
the apportionment problem.
• Hamilton
• Jefferson
• Webster
• Huntington-Hill

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Apportionment - DefinitionsApportionment - Definitions
• Alabama Paradox: occurs when a state loses a
seat due to an increase in house size.
• Population Paradox: occurs when one state’s
population increases and it’s apportionment
decreases while simultaneously another state’s
population increases proportionately less, or
decreases and it’s apportionment increases.
• Quota Condition: occurs if in every situation
each state’s apportionment is equal to either its
lower quota or its upper quota.

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Hamilton Method of Largest FractionsHamilton Method of Largest Fractions
• Each state receives either its lower quota or
its upper quota depending on the fractional
parts.
• Calculate the individual quotas qi .
• Determine individual tentative apportionments
ni, and find their sum.
• If the sum is less than h, allot the remaining
seats to be apportioned , one each, to the states
whose quotas have the largest fractional parts. It
is possible that a tie will be encountered, but in
practice this rarely occurs
 iq
 iq

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Hamilton Method of Largest FractionsHamilton Method of Largest Fractions

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Divisor MethodsDivisor Methods
• Determine each state’s apportionment by
dividing it’s population by a common divisor,
d, and rounding the resulting quotient. Divisor
methods differ in how rounding is done.
• Jefferson Method – favors larger states
• Webster Method - neutral
• Hill-Huntington Method

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Jefferson MethodJefferson Method
• Calculate the individual quotas, qi.
• Determine the individual tentative apportionments, ni ,
using the formula in the table below and find their sum.
• If the sum is bigger (less than) h, determine which state
is to receive (lose) the extra seat as follows:
– Calculate the critical divisor, di , for each state.
– Select the state with the largest (smallest)critical divisor to
gain (lose) seats.
– Compute the adjusted quota for each state.
– Calculate the new tentative apportionment.
– Begin process again if needed.

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Jefferson Method - ExampleJefferson Method - Example

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Webster MethodWebster Method
• Rounds the quota to the nearest whole number. rounds
up when fractional part is greater than or equal to ½ and
round down when the fractional part is less than ½.
• Calculate the individual quotas, qi .
• Determine the individual tentative apportionments, ni ,
using the formula in the table below and find their sum.
• If the sum is bigger (less) than h, determine which state is
to receive (lose) the extra seat as follows:
– Calculate the critical divisor, di , for each state.
– Select the state with the largest (smallest) critical divisor to gain
(lose) seats.
– Compute the adjusted quotas for each state.
– Calculate the new tentative apportionment.
– Begin process again if needed.

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Webster Method - ExampleWebster Method - Example

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Hill-Huntington MethodHill-Huntington Method
• Used to apportion the House since 1940.
• Uses the geometric mean to determine rounding:
• Calculate the individual quotas, qi .
• Determine
• Determine the individual tentative apportionments, ni, using
the formula in the table below and find their sum.
• If the sum is less than or bigger than h, determine which state
is to receive the extra seat as follows:
– Calculate the critical divisor, di , for each state.
– Select the state with the largest (smallest) critical divisor to gain (lose)
seats.
– Compute the adjusted quotas for each state.
– Calculate the new tentative apportionment.
– Begin process again if needed.
 
 



<
≥
== *
*
,
,
qqq
qqq
qn
ii
ii
ii
   ii qqq =*

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Hill-Huntington Method - ExampleHill-Huntington Method - Example

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Summary of Apportionment MethodsSummary of Apportionment Methods
Table showing which methods satisfy the Alabama condition (AC),
Population condition (PC), quota condition (QC), and which favor
the larger states (LS).

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Which Divisor Method is Best?Which Divisor Method is Best?
• Representative share: represents the share of a
congressional seat given to each citizen of the state. For
each state, compute the number of microseats (106
ai )/pi
where ai is the final apportionment given to state i. Take
the difference between the largest and smallest quotient to
get the absolute difference. Do this for each method. The
method with the smallest absolute difference is considered
the best method.

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Which Divisor Method is Best?Which Divisor Method is Best?
• District Population: represents the average population of a
congressional district in the state. For each state, compute
pi/ai where is the final apportionment given to state i. Take
the difference between the largest and smallest quotient to
get the absolute difference. Do this for each method. The
method with the smallest absolute difference is considered
the best method.

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Which Divisor Method is Best?Which Divisor Method is Best?
• Relative Difference: Developed by Huntington is a
compromise between a. and c. above. For each method,
given two numbers A and B (in microseats), with A>B, find
the absolute difference A-B. Then find the relative
difference which defines the relative inequity:
• The method yielding the smallest relative difference is
considered the best method.
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B
BA −