Apportionment paradox
Encyclopedia
An apportionment paradox exists when the rules for apportionment
in a political system produce results which are unexpected or seem to violate common sense
.
To apportion is to divide into parts according to some rule, the rule typically being one of proportion
. Certain quantities, like milk, can be divided in any proportion whatsoever; others, such as horses, cannot—only whole numbers will do. In the latter case, there is an inherent tension between our desire to obey the rule of proportion as closely as possible and the constraint restricting the size of each portion to discrete values. This results, at times, in unintuitive observations, or paradox
es.
Several paradoxes related to apportionment, also called fair division, have been identified. In some cases, simple adjustments to an apportionment methodology can resolve observed paradoxes. Others, such as those relating to the United States House of Representatives
, call into question notions that mathematics
alone can provide a single, fair resolution.
paradox was discovered in 1880, when it was found that increasing the total number of seats would decrease Alabama's share from 8 to 7. There was more to come: when Oklahoma
became a state in 1907, a recomputation of apportionment showed that the number of seats due to other states would be affected even though Oklahoma would be given a fair share of seats and the total number of seats increased by that number.
The method for apportionment used during this period, originally put forth by Alexander Hamilton
but not adopted until 1852, was as follows (after meeting the requirements of the United States Constitution, wherein each state must be allocated at least one seat in the House of Representatives, regardless of population):
, proved that any method of apportionment will result in paradoxes whenever there are three or more parties (or states, regions, etc.). More precisely, their theorem states that there is no apportionment system that has the following properties (as the example we take the division of seats between parties in a system of proportional representation
):
required to allocate seats based on population counts, which are required every 10 years. The size of the House
is set by statute.
After the 1880 census
, C. W. Seaton, chief clerk of the United States Census Bureau
, computed apportionments
for all House sizes between 275 and 350, and discovered that Alabama would get 8 seats with a House size of 299 but only 7 with a House size of 300. In general the term Alabama paradox refers to any apportionment scenario where increasing the total number of items would decrease one of the shares. A similar exercise by the Census Bureau after the 1900 census
computed apportionments for all House sizes between 350 and 400: Colorado would have received three seats in all cases, except with a House size of 357 in which case it would have received two.
The following is a simplified example (following the largest remainder method
) with three states and 10 seats and 11 seats.
Observe that state C's share decreases from 2 to 1 with the added seat.
This occurs because increasing the number of seats increases the fair share faster for the large states than for the small states. In particular, large A and B had their fair share increase faster than small C. Therefore, the fractional parts for A and B increased faster than those for C. In fact, they overtook C's fraction, causing C to lose its seat, since the Hamilton method examines which states have the largest fraction.
When two states have populations increasing at different rates, a small state with rapid growth can lose a legislative seat to a big state with slower growth.
The paradox arises because of rounding in the procedure for dividing the seats. See the apportionment rules for the United States Congress for an example.
Apportionment (politics)
Apportionment is the process of allocating political power among a set of principles . In most representative governments, political power has most recently been apportioned among constituencies based on population, but there is a long history of different approaches.The United States Constitution,...
in a political system produce results which are unexpected or seem to violate common sense
Common sense
Common sense is defined by Merriam-Webster as, "sound and prudent judgment based on a simple perception of the situation or facts." Thus, "common sense" equates to the knowledge and experience which most people already have, or which the person using the term believes that they do or should have...
.
To apportion is to divide into parts according to some rule, the rule typically being one of proportion
Proportionality (mathematics)
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...
. Certain quantities, like milk, can be divided in any proportion whatsoever; others, such as horses, cannot—only whole numbers will do. In the latter case, there is an inherent tension between our desire to obey the rule of proportion as closely as possible and the constraint restricting the size of each portion to discrete values. This results, at times, in unintuitive observations, or paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
es.
Several paradoxes related to apportionment, also called fair division, have been identified. In some cases, simple adjustments to an apportionment methodology can resolve observed paradoxes. Others, such as those relating to the United States House of Representatives
United States House of Representatives
The United States House of Representatives is one of the two Houses of the United States Congress, the bicameral legislature which also includes the Senate.The composition and powers of the House are established in Article One of the Constitution...
, call into question notions that mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
alone can provide a single, fair resolution.
History
The AlabamaAlabama
Alabama is a state located in the southeastern region of the United States. It is bordered by Tennessee to the north, Georgia to the east, Florida and the Gulf of Mexico to the south, and Mississippi to the west. Alabama ranks 30th in total land area and ranks second in the size of its inland...
paradox was discovered in 1880, when it was found that increasing the total number of seats would decrease Alabama's share from 8 to 7. There was more to come: when Oklahoma
Oklahoma
Oklahoma is a state located in the South Central region of the United States of America. With an estimated 3,751,351 residents as of the 2010 census and a land area of 68,667 square miles , Oklahoma is the 28th most populous and 20th-largest state...
became a state in 1907, a recomputation of apportionment showed that the number of seats due to other states would be affected even though Oklahoma would be given a fair share of seats and the total number of seats increased by that number.
The method for apportionment used during this period, originally put forth by Alexander Hamilton
Alexander Hamilton
Alexander Hamilton was a Founding Father, soldier, economist, political philosopher, one of America's first constitutional lawyers and the first United States Secretary of the Treasury...
but not adopted until 1852, was as follows (after meeting the requirements of the United States Constitution, wherein each state must be allocated at least one seat in the House of Representatives, regardless of population):
- First, the fair share of each state, i.e. the proportional share of seats that each state would get if fractional values were allowed, is computed.
- Next, the fair shares are rounded down to whole numbers, resulting in unallocated "leftover" seats. These seats are allocated, one each, to the states whose fair share exceeds the rounded-down number by the highest amount.
Impossibility result
In 1982 two mathematicians, Michel Balinski and Peyton YoungPeyton Young
Hobart Peyton Young is an American game theorist known for his contributions to evolutionary game theory and its application to the study of institutional and technological change, as well as the theory of learning in games...
, proved that any method of apportionment will result in paradoxes whenever there are three or more parties (or states, regions, etc.). More precisely, their theorem states that there is no apportionment system that has the following properties (as the example we take the division of seats between parties in a system of proportional representation
Proportional representation
Proportional representation is a concept in voting systems used to elect an assembly or council. PR means that the number of seats won by a party or group of candidates is proportionate to the number of votes received. For example, under a PR voting system if 30% of voters support a particular...
):
- It follows the quota rule: Each of the parties gets one of the two numbers closest to its fair share of seats (if the party's fair share is 7.34 seats, it gets either 7 or 8).
- It does not have the Alabama paradox: If the total number of seats is increased, no party's number of seats decreases.
- It does not have the population paradox: If party A gets more votes and party B gets fewer votes, no seat will be transferred from A to B.
Alabama paradox
The Alabama paradox was the first of the apportionment paradoxes to be discovered. The US House of Representatives is constitutionallyUnited States Constitution
The Constitution of the United States is the supreme law of the United States of America. It is the framework for the organization of the United States government and for the relationship of the federal government with the states, citizens, and all people within the United States.The first three...
required to allocate seats based on population counts, which are required every 10 years. The size of the House
United States congressional apportionment
United States congressional apportionment is the process by which seats in the United States House of Representatives are redistributed amongst the 50 states following each constitutionally mandated decennial census. Each state is apportioned a number of seats which approximately corresponds to its...
is set by statute.
After the 1880 census
United States Census, 1880
The United States Census of 1880 was the tenth United States Census conducted by the Census Bureau during June 1880. It was the first time that women were permitted to be enumerators...
, C. W. Seaton, chief clerk of the United States Census Bureau
United States Census Bureau
The United States Census Bureau is the government agency that is responsible for the United States Census. It also gathers other national demographic and economic data...
, computed apportionments
United States congressional apportionment
United States congressional apportionment is the process by which seats in the United States House of Representatives are redistributed amongst the 50 states following each constitutionally mandated decennial census. Each state is apportioned a number of seats which approximately corresponds to its...
for all House sizes between 275 and 350, and discovered that Alabama would get 8 seats with a House size of 299 but only 7 with a House size of 300. In general the term Alabama paradox refers to any apportionment scenario where increasing the total number of items would decrease one of the shares. A similar exercise by the Census Bureau after the 1900 census
United States Census, 1900
The Twelfth United States Census, conducted by the Census Office on June 1, 1900, determined the resident population of the United States to be 76,212,168, an increase of 21.0 percent over the 62,979,766 persons enumerated during the 1890 Census....
computed apportionments for all House sizes between 350 and 400: Colorado would have received three seats in all cases, except with a House size of 357 in which case it would have received two.
The following is a simplified example (following the largest remainder method
Largest remainder method
The largest remainder method is one way of allocating seats proportionally for representative assemblies with party list voting systems...
) with three states and 10 seats and 11 seats.
| | With 10 seats | | With 11 seats | ||||
|---|---|---|---|---|---|
| State | Population | Fair share | Seats | Fair share | Seats |
| A | 6 | 4.286 | 4 | 4.714 | 5 |
| B | 6 | 4.286 | 4 | 4.714 | 5 |
| C | 2 | 1.429 | 2 | 1.571 | 1 |
Observe that state C's share decreases from 2 to 1 with the added seat.
This occurs because increasing the number of seats increases the fair share faster for the large states than for the small states. In particular, large A and B had their fair share increase faster than small C. Therefore, the fractional parts for A and B increased faster than those for C. In fact, they overtook C's fraction, causing C to lose its seat, since the Hamilton method examines which states have the largest fraction.
New states paradox
Given a fixed number of total representatives (as determined by the United States House of Representatives), adding a new state would in theory reduce the number of representatives for existing states, as under the United States Constitution each state is entitled to at least one representative regardless of its population. However, because of how the particular apportionment rules deal with rounding methods, it is possible for an existing state to get more representatives than if the new state were not added.Population paradox
The population paradox is a counterintuitive result of some procedures for apportionment.When two states have populations increasing at different rates, a small state with rapid growth can lose a legislative seat to a big state with slower growth.
The paradox arises because of rounding in the procedure for dividing the seats. See the apportionment rules for the United States Congress for an example.