Editing Apportionment paradox (section)

==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>