Editing Proportional representation (section)

==== Biproportional apportionment ====
{{Main|Biproportional apportionment}}

Biproportional [[apportionment (politics)|apportionment]] aims to achieve proportionality in two dimensions, for example: proportionality by region and proportionality by party. There are several mathematical methods to attain biproportionality.

One method is called [[iterative proportional fitting]] (IPF). It was proposed for elections by the mathematician [[Michel Balinski]] in 1989, and first used by the city of [[Zürich]] for its council elections in February 2006, in a modified form called "new Zürich apportionment" (''Neue Zürcher Zuteilungsverfahren''). Zürich had had to modify its party list PR system after the Swiss Federal Court ruled that its smallest [[Ward (electoral subdivision)|wards]], as a result of population changes over many years, unconstitutionally disadvantaged smaller political parties. With biproportional apportionment, the use of open party lists has not changed, but the way winning candidates are determined has. The proportion of seats due to each party is calculated according to their overall citywide vote, and then the district winners are adjusted to conform to these proportions. This means that some candidates, who would otherwise have been successful, can be denied seats in favor of initially unsuccessful candidates, in order to improve the relative proportions of their respective parties overall. This peculiarity is accepted by the Zürich electorate because the resulting city council is proportional and all votes, regardless of district magnitude, now have equal weight. The system has since been adopted by other Swiss cities and [[Cantons of Switzerland|cantons]].<ref name="pukZna">{{Cite journal |last=Pukelsheim |first=Friedrich |date=September 2009 |title=Zurich's New Apportionment |url=http://www.uni-augsburg.de/pukelsheim/2008e-en.pdf |journal=German Research |volume=31 |issue=2 |pages=10–12 |doi=10.1002/germ.200990024 |access-date=10 August 2014 |ref=ZurichNA}}</ref><ref name="balinskiFMV">{{Cite journal |last=Balinski |first=Michel |author-link=Michel Balinski |date=February 2008 |title=Fair Majority Voting (or How to Eliminate Gerrymandering) |url=http://www.maa.org/programs/maa-awards/writing-awards/fair-majority-voting-or-how-to-eliminate-gerrymandering |journal=The American Mathematical Monthly |volume=115 |issue=2 |pages=97–113 |doi=10.1080/00029890.2008.11920503 |access-date=10 August 2014 |ref=FMV |s2cid=1139441}}</ref>

Balinski has proposed another variant called [[fair majority voting]] (FMV) to replace single-winner plurality/majoritarian electoral systems, in particular the system used for the [[US House of Representatives]]. FMV introduces proportionality without changing the method of voting, the number of seats, or the{{snd}}possibly gerrymandered{{snd}}district boundaries. Seats would be apportioned to parties in a proportional manner at the [[U.S. state|state]] level.<ref name="balinskiFMV" /> In a related proposal for the [[UK parliament]], whose elections are contested by many more parties, the authors note that parameters can be tuned to adopt any degree of proportionality deemed acceptable to the electorate. In order to elect smaller parties, a number of constituencies would be awarded to candidates placed fourth or even fifth in the constituency{{snd}} unlikely to be acceptable to the electorate, the authors concede{{snd}} but this effect could be substantially reduced by incorporating a third, regional, apportionment tier, or by specifying minimum thresholds.<ref>{{Cite journal |last1=Akartunalı |first1=Kerem |last2=Knight |first2=Philip A. |date=June 2017 |title=Network models and biproportional rounding for fair seat allocations in the UK elections |journal=Annals of Operations Research |publisher=[[University of Strathclyde]] |volume=253 |issue=1 |pages=1–19 |doi=10.1007/s10479-016-2323-0 |s2cid=30623821 |issn=0254-5330 |doi-access=free }}</ref>