VIII: Apportionment Quick Summary
Apportionment is a fair division process
used to divide identical, indivisible objects among units which may be entitled to unequal
shares.
For convenience, we will call the
"objects", SEATS and the "units" among which the objects are divided,
STATES.
TERMINOLOGY
The Standard Divisor (SD) is
the average number of people per seat over the entire population
The Standard Quota (SQ) for each state is
the number of seats a state would be entitled to if states could receive a fractional part
of a seat.
- SQ = State Population / SD (each state has
a SQ)
A Modified Divisor (MD) is a divisor
(relatively near the Standard Divisor) chosen in an attempt to make the "Rounded
Quotas" exactly equal to the number of seats to be apportioned.
The Modified Quota (MQ) for each state is
the same as the Standard Quota except that the calculation is done using the Modified
Divisor instead of the Standard Divisor.
- MQ = State Population / MD (Each state has
a MQ.)
Quota is used to refer to either a
Standard Quota or a Modified Quota.
A Rounding Rule provides instructions on
how Quotas are to be rounded.
A Rounded Quota is a Quota that has been
rounded either to the nearest whole number below it (Lower Quota) or above it (Upper
Quota).
A Lower Quota is a Quota rounded down to
the nearest whole number.
An Upper Quota is a Quota rounded up to
the nearest whole number.
An apportionment method that always gives each
state a number of seats equal to the Upper or Lower Quota (of its Standard Quota) is
called a Quota Rule.
If it is possible for an apportionment method to
give a state a number of seats that is not equal to the Upper or Lower Quota (of its
Standard Quota), then that apportionment method is said to violate quota.
SOME PROBLEMS WITH APPORTIONMENT METHODS
- Violating Quota
- Paradoxes
The Alabama Paradox: An increase in the total
number of seats to be apportioned causes a state to lose a seat.
The Population Paradox: An increase in a
state’s population causes it to lose a seat.
The New States Paradox: Adding a new state with
its fair share of seats affects the number of seats apportioned to other states.
SOME APPORTIONMENT METHODS
Hamilton’s Method (see procedure below): The only method we study that
doesn’t use Modified Divisors/Quotas. It does not violate quota but is susceptible to
the Alabama, Population, and New States paradox.
Jefferson’s
Method (see procedure below):
One of what are often called "divisor" methods (methods that use Modified
Divisors/Quotas). It, like all divisor methods, violates quota but is not susceptible to
any of the paradoxes above.
Adams’
Method (see procedure below):
Another "divisor" method (method that uses Modified Divisors/Quotas). It, like
all divisor methods, violates quota but is not susceptible to any of the paradoxes above.
Webster’s
Method (see procedure below):
Another "divisor" method (method that uses Modified Divisors/Quotas). It, like
all divisor methods, violates quota but is not susceptible to any of the paradoxes above.
Huntington-Hill
Method (see procedure below): A
final "divisor" method (method that uses Modified Divisors/Quotas). It, like all
divisor methods, violates quota but is not susceptible to any of the paradoxes above.
ROUNDING RULES USED BY THE VARIOUS
APPORTIONMENT METHODS
Round Quotas DOWN to the nearest whole number
(Lower Quota): used by Hamilton’s Method and Jefferson’s Method
Round Quotas UP to the nearest whole number
(Upper Quota): used by Adams' Method
Round Quotas based on the arithmetic mean
(UQ+LQ)/2: used by Webster’s Method
Round Quotas based on the geometric mean : used by the Huntington-Hill
Method
CARRYING
OUT AN APPORTIONMENT PROCEDURE
STEP 1: STANDARD DIVISOR
SD = Total Pop / # of seats
STEP 2: STANDARD QUOTAS (for each
state)
SQ = State Pop / SD
STEP 3: ROUNDED QUOTAS (for each
state)
Use the appropriate Rounding Rule to get the
Rounded Quotas from the Standard Quotas:
- For HM, the Rounding Rule is to always
round DOWN to Lower Quotas.
- For JM, the Rounding Rule is to always
round DOWN to Lower Quotas.
- For AM, the Rounding Rule is to always
round UP to Upper Quotas.
- For WM, the Rounding Rule is to round UP if
the computed Quota is greater than or equal to the arithmetic mean, and round DOWN if not.
- For HHM, the rounding rule is to round UP
if the computed Quota is greater than or equal to the geometric mean, and round DOWN if not.
STEP 4: FINAL APPORTIONMENT (for each state)
- For HM: If Lower Quotas do not
apportion enough seats, add the proper number of seats by giving an additional seat to
those states whose Divisors have the largest fractional parts.
- All other methods: If Rounded Quotas do not
apportion enough seats, modify the divisor and use this Modified Divisor to compute
Modified Quotas and then go back to STEP 3.
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