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In 1983, two mathematicians, [[Michel Balinski]] and Peyton Young, proved that any method of [[Party-list proportional representation | apportionment method]] will result in paradoxes whenever there arefor three or more parties (or states, regions, etc.).<ref>
{{cite book |title=Fair Representation: Meeting the Ideal of One Man, One Vote |last=Balinski |first=M |author2=Young HP |year=1982 |publisher=Yale Univ Pr |isbn=0-300-02724-9 |url-access=registration |url=https://archive.org/details/fairrepresentati00bali }}</ref><ref>
{{cite book |title=Fair Representation: Meeting the Ideal of One Man, One Vote |edition=2nd |last=Balinski |first=M |author2=Young HP|year=2001 |publisher=Brookings Institution Press |isbn=0-8157-0111-X }}
</ref> The theorem shows that any possible method used to allocate the remaining fraction will necessarily fail to always follow quota. More precisely, their theorem states that there is no apportionment system that haswith the following 3 properties for all house sizes:<ref name="Stein2008">Stein JD. How Math Explains the World: A Guide to the Power of Numbers, from Car Repair to Modern Physics. Smithsonian. Apr 22, 2008. {{ISBN|9780061241765}}</ref> (as the example we take the division of seats between parties in a system of [[proportional representation]]):
* It avoids violations of the [[Quota rule | Quota Rule]]: Each of the partiesparty 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 has [[HousePopulation monotonicity criterion |Pairwise Housepopulation monotonicity]]: If theparty totalA numbergets ofmore seats is increasedvotes, with no change in party B's numbervotes, A will not lose a ofseat seatsto decreasesB.
* It has [[Population monotonicity]]: If party A gets more votes and party B gets fewer votes, no seat will be transferred from A to B.
 
== Specific Systemssystems==
 
Methods may have a subset of these properties, but can't have all of them:
* AThe methodonly maymethods bethat are free of both the Alabamapopulation paradox andare the population paradox. These methods are [[Highest averages method|divisor methods]],.<ref name="Smith">{{cite web |url=http://rangevoting.org/Apportion.html |title=Apportionment and rounding schemes |last=Smith |first=WD}}</ref> and [[W: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 tothe alwaysquota follow quotarule in othersome circumstanceselections.▼
* 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 |last=Balinski |first=M |author2=Young HP |year=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}}</ref>▼
** Such failures can be extremely rare under a good apportionment system--so far, no apportionment of the US congress under Huntington-Hill or Webster would have failed to maintain quota.
▲* A method may be free of both the Alabama paradox and the population paradox. These methods are [[Highest averages method|divisor methods]],<ref name=Smith>{{cite web |url=http://rangevoting.org/Apportion.html |title=Apportionment and rounding schemes |last=Smith |first=WD}}</ref> and [[W: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.
* [[Largest remainder method |Largest remainder methods]] like [[Hamilton method|Hamilton]] obey the quota rule but fail pairwise population monotonicity.
* No method may always follow quota and be free of the population paradox.<ref name="Smith" />
* [[Largest remainder method | Largest remainder methods]] obey Quota Rules but have neither of the other two criteria. The [[Hamilton method]] of apportionment is actually a largest-remainder method which is specifically defined as using the Hare Quota
 
Many mathematically analogous situations arise in which quantities are to be divided into discrete equal chunks. The Balinski–Young theorem applies in these situations: it indicates that although very reasonable approximations can be made, there is no mathematically rigorous way in which to reconcile the small remaining fraction while complying with all the competing fairness elements.
 
== House monotonicity ==
▲*The ABalinski-Young methodtheorem mayis followsometimes quotaquoted andwith bean freeunnecessary ofcondition thecalled Alabama[[House monotonicity]], which says that adding more seats should not cause a party to lose seats. However, this condition turns out to be paradoxirrelevant. Balinski and Young have constructed a method that doessatisfies both quota and house somonotonicity, although it is not in common political use.<ref>{{cite journal |last=Balinski |first=M |author2=Young HP |year=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}}</ref> It is also possible to satisfy both population and house monotonicity (with divisor methods); thus, house monotonicity is compatible with either of the other two criteria.
 
However, the quota rule is incompatible with population monotonicity.
 
==History==
 
The division of seats in an election is a prominent cultural concern. In 1876, the United States [[W: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. However, many mathematically analogous situations arise in which quantities are to be divided into discrete equal chunks. The Balinski–Young theorem applies in these situations: it indicates that although very reasonable approximations can be made, there is no mathematically rigorous way in which to reconcile the small remaining fraction while complying with all the competing fairness elements.
 
== Notes ==
 
The quota rule is incompatible with resistingimmunity to [[vote management]]. See [[Divisor method#Notes]] for an example.
 
For [[proportional representation]], the paradox can theoretically be sidestepped by giving party representatives slightly different voting power to correct for any error. For instance, if 13.57% of the voters voted for party A, the total voting power of party A's representatives could be set to 13.57% of the total. However, it makes counting votes much more opaque. This solution is not employed by any current political assembly.
 
==Related==
 
* [[HousePopulation monotonicity criterion]]
* [[House monotonicity criterion |House monotonicity]]
* [[Alabama paradox]]
* [[Quota rule]]
 
==Further Reading==
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