Balinski–Young theorem: Difference between revisions

From electowiki
Content added Content deleted
VisualWikitext
No edit summary
No edit summary
 
(7 intermediate revisions by 3 users not shown)
Line 1: Line 1:


In 1983, two mathematicians, Michel Balinski and Peyton Young, proved that any method of [[Party-list proportional representation | apportionment]] will result in paradoxes whenever there are three or more parties (or states, regions, etc.).<ref>
In 1983, two mathematicians, [[Michel Balinski]] and Peyton Young, proved that any [[Party-list proportional representation |apportionment method]] will result in paradoxes for three or more parties (or states).<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 |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 }}
{{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 has the following 3 properties <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]]):
</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 with the following 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>
* It avoids violations of the [[W:quota rule | 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.
* [[Quota rule]]: Each party 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; any other number will violate the rule.
* It has [[House monotonicity criterion | House monotonicity]]: If the total number of seats is increased, no party's number of seats decreases.
* [[Population monotonicity|Pairwise population monotonicity]]: If party A gets more votes, with no change in party B's votes, A will not lose a seat to B.
* It has [[W: Apportionment_paradox#Population_paradox | Population monotonicity]]: If party A gets more votes and party B gets fewer votes, no seat will be transferred from A to B.


== Specific Systems==
== Specific systems==


Methods may have a subset of these properties, but can't have all of them:
Methods may have a subset of these properties, but can't have all of them:
⚫
* The only methods that are free of the population paradox are the [[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> [[W:Huntington-Hill method|Huntington-Hill]], the method currently used to apportion House of Representatives seats, is one of them. However, these methods will fail the quota rule in some elections.
⚫
* 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/>
* 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 Balinski-Young theorem is sometimes quoted with an unnecessary condition called [[House monotonicity]], which says that adding more seats should not cause a party to lose seats. However, this condition turns out to be irrelevant. Balinski and Young have constructed a method that satisfies both quota and house monotonicity, although it is not in common 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==
==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.
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.


== Notes ==
== Notes ==


The quota rule is incompatible with resisting [[vote management]]. See [[Divisor method#Notes]] for an example.
The quota rule is incompatible with immunity 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==
==Related==


* [[House monotonicity criterion]]
* [[Population monotonicity]]
* [[House monotonicity criterion |House monotonicity]]
* [[Alabama paradox]]
* [[Quota rule]]


==Further Reading==
==Further Reading==
Line 37: Line 46:
[[Category:Voting theory]]
[[Category:Voting theory]]
[[Category:Voting system criteria]]
[[Category:Voting system criteria]]
[[Category:Proportionality-related concepts]]

Latest revision as of 04:21, 20 February 2024

In 1983, two mathematicians, Michel Balinski and Peyton Young, proved that any apportionment method will result in paradoxes for three or more parties (or states).[1][2] 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 with the following properties for all house sizes:[3]

  • Quota rule: Each party 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; any other number will violate the rule.
  • Pairwise population monotonicity: If party A gets more votes, with no change in party B's votes, A will not lose a seat to B.

Specific systems

Methods may have a subset of these properties, but can't have all of them:

  • The only methods that are free of the population paradox are the divisor methods.[4] Huntington-Hill, the method currently used to apportion House of Representatives seats, is one of them. However, these methods will fail the quota rule in some elections.
    • 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.
  • Largest remainder methods like Hamilton obey the quota rule but fail pairwise population monotonicity.
  • No method may always follow quota and be free of the population paradox.[4]

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 Balinski-Young theorem is sometimes quoted with an unnecessary condition called House monotonicity, which says that adding more seats should not cause a party to lose seats. However, this condition turns out to be irrelevant. Balinski and Young have constructed a method that satisfies both quota and house monotonicity, although it is not in common use.[5] 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 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.

Notes

The quota rule is incompatible with immunity 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

Further Reading

References

  1. ↑ Balinski, M; Young HP (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. Yale Univ Pr. ISBN 0-300-02724-9.
  2. ↑ Balinski, M; Young HP (2001). Fair Representation: Meeting the Ideal of One Man, One Vote (2nd ed.). Brookings Institution Press. ISBN 0-8157-0111-X.
  3. ↑ 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
  4. ↑ a b Smith, WD. "Apportionment and rounding schemes".
  5. ↑ Balinski, M; Young HP (1974). "A new method for congressional apportionment". Proceedings of the National Academy of Sciences. 71 (11): 4602–4606. doi:10.1073/pnas.71.11.4602. PMC 433936. PMID 16592200.
Retrieved from "https://electowiki.org/wiki/Balinski–Young_theorem?oldid=18115"