Editing Apportionment paradox

{{Short description|Pathological behavior by an apportionment rule}}
An '''apportionment paradox''' is a situation where an [[Apportionment (politics)|apportionment]]—a rule for dividing discrete objects according to some [[proportionality (mathematics)|proportional relationship]]—produces results that violate notions of [[common sense]] or [[justice|fairness]].

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 the desire to obey the rule of proportion as closely as possible and the constraint restricting the size of each portion to discrete values.

Several paradoxes related to apportionment and [[fair division]] have been identified. In some cases, simple adjustments to an apportionment methodology can resolve observed paradoxes. However, as shown by the Balinski–Young theorem, it is not always possible to provide a perfectly fair resolution that satisfies all competing fairness criteria.<ref name=Stein2008>{{cite book |last = Stein |first = James D. |title = How Math Explains the World: A Guide to the Power of Numbers, from Car Repair to Modern Physics |location = New York |publisher = Smithsonian Books |year = 2008 |isbn = 9780061241765 }}</ref>{{rp|227–235}}

==History==
An example of the apportionment paradox known as "the [[Alabama]] paradox" was discovered in the context of [[United States congressional apportionment]] in 1880,<ref name=Stein2008/>{{rp|228–231}} when census calculations found that if the total number of seats in the [[United States House of Representatives|House of Representatives]] were hypothetically increased, this would decrease Alabama's seats from 8 to 7. An actual impact was observed in 1900, when Virginia lost a seat to Maine, even though Virginia's population was growing more rapidly: this is an example of the population paradox.<ref name=Stein2008/>{{rp|231–232}} In 1907, when [[Oklahoma]] became a state, New York lost a seat to Maine, thus the name "the new state paradox".<ref name=Stein2008/>{{rp|232–233}}<ref name=Caulfield>{{cite journal |last = Caulfield |first = Michael J. |date = November 2010  |url = https://www.maa.org/publications/periodicals/convergence/apportioning-representatives-in-the-united-states-congress-paradoxes-of-apportionment |title = Apportioning Representatives in the United States Congress - Paradoxes of Apportionment |journal = Convergence |publisher = Mathematical Association of America |doi = 10.4169/loci003163 |doi-broken-date = 1 November 2024 }}</ref>

The [[Hamilton method|method for apportionment]] used during this period, originally put forth by [[Alexander Hamilton]], but vetoed by [[George Washington]] and not adopted until 1852,<ref name=Stein2008/>{{rp|228}} was as follows:

* First, the fair share of each state is computed, i.e. the proportional share of seats that each state would get if fractional values were allowed. 
* Second, each state receives as many seats as the whole number portion of its fair share.
* Third, any state whose fair share is less than one receives one seat, regardless of population, as required by the [[United States Constitution]].
* Fourth, any remaining seats are distributed, one each, to the states whose fair shares have the highest fractional parts.

The Hamilton method replaced a rounding method proposed by [[Thomas Jefferson]],<ref name=Stein2008/>{{rp|228}} and was itself replaced by the [[Huntington–Hill method]] in 1941.<ref name=Stein2008/>

==Examples of paradoxes==
{{expand section|with=[[:sk:Lexmann paradox|Lexmann paradox]]|date=September 2023}}
===Alabama paradox===
The Alabama paradox was the first of the apportionment paradoxes to be discovered. The US House of Representatives is [[United States Constitution|constitutionally]] required to allocate seats based on population counts, which are required every 10 years. The [[United States congressional apportionment|size of the House]] is set by statute.

After the [[United States Census, 1880|1880 census]], C. W. Seaton, chief clerk of the [[United States Census Bureau]], computed [[United States Congressional Apportionment|apportionments]] for all House sizes between 275 and 350, and discovered  that Alabama would get eight seats with a House size of 299 but only seven with a House size of 300.<ref name=Stein2008/>{{rp|228–231}} 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 [[United States Census, 1900|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.<ref>{{cite web|first = Alex |last = Bogomolny |date = January 2002 |url = http://www.cut-the-knot.org/ctk/Democracy.shtml |title = The Constitution and Paradoxes |work = Cut The Knot! }}</ref>

The following is a simplified example (following the [[largest remainder method]]) with three states and 10 seats and 11 seats.

{| class="wikitable" width="500px"
! colspan="2" | !! colspan="2" | With 10 seats !! colspan="2" | With 11 seats
|-
! State !! Population !! Fair share !! Seats !! Fair share !! Seats
|-
| A || align="right" | 6 || align="right" | 4.286 || align="right" | 4 || align="right" | 4.714 || align="right" | 5
|-
| B || align="right" | 6 || align="right" | 4.286 || align="right" | 4 || align="right" | 4.714 || align="right" | 5
|-
| C || align="right" | 2 || align="right" | 1.429 || align="right" | 2 || align="right" | 1.571 || align="right" | 1
|}

Observe that state C's share decreases from 2 to 1 with the added seat.

In this example of a 10% increase in the number of seats, each state's share increases by 10%. However, increasing the number of seats by a fixed % increases the fair share more for larger numbers (i.e., large states more than 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 allocates according to which states have the largest fractional remainder.

The Alabama paradox gave rise to the axiom known as [[house monotonicity]], which says that, when the house size increases, the allocations of all states should weakly increase.

===Population paradox===
{{See also|State-population monotonicity}}
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.

Some of the earlier Congressional apportionment methods, such as Hamilton, could exhibit the population paradox. In 1900, Virginia lost a seat to Maine, even though Virginia's population was growing more rapidly.<ref name=Stein2008/>{{rp|231–232}} However, divisor methods such as the current method do not.{{cn|date=March 2024}}

===New states paradox===
{{See also|Coherence (fairness)}}
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. Also, even if the number of members in the House of Representatives is increased by the number of Representatives in the new state, a pre-existing state could lose a seat because of how the particular apportionment rules deal with rounding methods. In 1907, when [[Oklahoma]] became a state, it was given a fair share of seats and the total number of seats increased by that number. The House increased from 386 to 391 members. A recomputation of apportionment affected the number of seats because of other states: New York lost a seat while Maine gained one.<ref name=Stein2008/>{{rp|232–233}}<ref name=Caulfield/>

The New States paradox gave rise to the axiom known as [[Coherence (fairness)|coherence]], which says that, whenever an apportionment rule is activated on a subset of the states, with the subset of seats allocated to them, the outcome should be the same as in the grand solution.

==Balinski–Young theorem==
In 1983, two mathematicians, [[Michel Balinski]] and [[Peyton Young]], proved that any method of apportionment that does not violate the [[quota rule]] will result in paradoxes whenever there are four or more parties (or states, regions, etc.).<ref name=":0">{{cite book |title=Fair Representation: Meeting the Ideal of One Man, One Vote |last1=Balinski |first1 = Michel L. |last2 = Young |first2 = H. Peyton |year=1982 |location = New Haven |publisher=Yale University Press |isbn=0-300-02724-9 |url-access=registration |url=https://archive.org/details/fairrepresentati00bali }}</ref><ref name=":1">{{cite book |title=Fair Representation: Meeting the Ideal of One Man, One Vote  |edition=2nd |last1=Balinski |first1=Michel L.  |last2=Young |first2 = H. Peyton |year=2001 |location = Washington, DC |publisher=Brookings Institution Press |isbn=0-8157-0111-X }}
</ref> More precisely, their theorem states that there is no apportionment system that has the following properties for more than three states<ref name=Stein2008/>{{rp|233–234}}  (as the example we take the division of seats between parties in a system of [[proportional representation]]):
* It avoids violations of the quota rule: Each of the parties gets one of the two numbers closest to its fair share of seats. For example, if a party's fair share is 7.34 seats, it must get either 7 or 8 seats to avoid a violation; any other number will violate the rule.
* 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.

It is of note that any method of apportionment free of the Population Paradox will always be free of Alabama Paradox. The converse is not true, however. Webster's method can be free of incoherence and maintain quota when there are three states. All sensible methods satisfy both criteria in the trivial two-state case.<ref name=":0" /><ref name=":1" />

They show a [[proof of impossibility]]: apportionment methods may have a subset of these properties, but cannot have all of them:
* A method may be free of both the Alabama paradox and the population paradox. These methods are [[Highest averages method|divisor methods]], and [[Huntington–Hill method|Huntington–Hill]], the method currently used to apportion House of Representatives seats, is one of them. However, these methods will necessarily fail to always follow quota in other circumstances.
* No method may always follow quota and be free of the population paradox.<ref>{{cite web |url = http://pure.iiasa.ac.at/id/eprint/1338/1/WP-80-131.pdf |first1 = Michel L. |last1 = Balinski |first2 = H. Peyton |last2 = Young |title = The Theory of Apportionment |date = September 1980 |work = Working Papers |publisher = International Institute for Applied Systems Analysis |id = WP-80-131 }}</ref> 

The division of seats in an election is a prominent cultural concern. In 1876, the United States [[1876 United States presidential election|presidential election]] turned on the method by which the remaining fraction was calculated. [[Rutherford Hayes]] received 185 electoral college votes, and [[Samuel Tilden]] received 184. Tilden won the popular vote. With a different rounding method the final electoral college tally would have reversed.<ref name=Stein2008/>{{rp|228}} However, many mathematically analogous situations arise in which quantities are to be divided into discrete equal chunks.<ref name=Stein2008/>{{rp|233}} The Balinski–Young theorem applies in these situations: it indicates that although very reasonable approximations can be made, there is no mathematically rigorous way to reconcile the small remaining fraction while complying with all the competing fairness elements.<ref name=Stein2008/>{{rp|233}} In general, the response from mathematicians has been to abandon the [[quota rule]] as the less-important property, accepting that apportionment errors may sometimes slightly exceed one seat.

A method may follow quota and be free of the Alabama paradox. Balinski and Young constructed a method that does so, although it is not in common political use.<ref>{{cite journal |last1=Balinski |first1=Michel L.  |last2= Young |first2 = H. Peyton |date=November 1974 |title=A New Method for Congressional Apportionment |journal=Proceedings of the National Academy of Sciences |volume=71 |issue=11  |pages=4602–4606 |doi=10.1073/pnas.71.11.4602|pmc=433936 |pmid=16592200|bibcode=1974PNAS...71.4602B |doi-access=free }}</ref>

==References==
{{reflist}}

==External links==
*[https://www.cut-the-knot.org/ctk/Democracy.shtml The Constitution and Paradoxes]
*[https://www.ams.org/featurecolumn/archive/apportionII3.html Apportionment: Balinski and Young's Contribution]

{{Decision theory paradoxes}}

{{DEFAULTSORT:Apportionment paradox}}
[[Category:Apportionment (politics)]]
[[Category:Decision-making paradoxes]]