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

  • SD = Total Population / # of Seats

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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