Editing Proportional representation (section)

==Measuring disproportionality==
Exact proportionality has a single unambiguous definition: the seat shares must exactly equal the vote shares, measured as [[seats-to-votes ratio]]. When this condition is violated, the allocation is disproportional, and it may be interesting to examine the degree of disproportionality{{snd}}the degree to which the number of seats won by each party differs from that of a perfectly proportional outcome. This degree does not have a single unambiguous definition. Some common disproportionality indexes are:<ref name="karpov2008">{{cite journal |last1=Karpov |first1=Alexander |date=2008 |title=Measurement of disproportionality in proportional representation systems |journal=Mathematical and Computer Modelling |volume=48 |issue=9–10 |pages=1421–1438 |doi=10.1016/j.mcm.2008.05.027 |doi-access=free}}</ref>

* The [[Gallagher Index]]{{snd}}involves squaring the difference between each party's vote share and seat share, and finding the square root of half of the sum of these amounts.
* [[Wasted vote]], which counts votes cast for parties that did not obtain any seats (or votes not used to elect anyone at the district level, in systems where district-level electing is all that is used)
* The [[Sainte-Laguë Index]]{{snd}}where the squared discrepancy from ideal [[seats-to-votes ratio]] is weighted equally for each voter.

Disproportionality changes from one election to another depending on voter behavior and size of electoral threshold or natural threshold. This is seen in the number of wasted votes in New Zealand.<ref name="results">{{cite web |url=https://electionresults.govt.nz/electionresults_2020/statistics/index.html |title=2020 General Election{{snd}}Official Results and Statistics |publisher= [[Electoral Commission (New Zealand)|Electoral Commission]] |date= 30 November 2020}}</ref> In [[2005 New Zealand general election]] every party receiving more than 1 percent of the votes acquired seats due to every party at that level getting at least one seat in first-past-the-post voting. This election thus saw far fewer wasted votes compared to other elections when only the most popular parties took district seats. {{Graph:Chart|width=400|height=160|type=rect|xAxisTitle=Year of New Zealand general election |yAxisTitle=wasted vote in %|yAxisMin=0|yAxisMax=10
|x=1996, 1999, 2002, 2005, 2008, 2011, 2014, 2017, 2020
|y1Title=wasted vote|y1=7.54,6.03,4.89, 1.5, 6.46, 3.31, 6.1, 4.62, 7.71
|colors=#99ccff}}

Different indexes measure different concepts of disproportionality. Some disproportionality concepts have been mapped to [[social welfare function]]s.<ref name="wada2016">
{{cite journal |last1=Wada |first1=Junichiro |title=Apportionment behind the veil of uncertainty |journal=The Japanese Economic Review |date=2016 |volume=67 |issue=3 |pages=348–360 |doi=10.1111/jere.12093 |s2cid=156608434}}</ref>

Disproportionality indexes are sometimes used to evaluate existing and proposed electoral systems. For example, the [[Parliament of Canada|Canadian Parliament]]'s [[Canadian House of Commons Special Committee on Electoral Reform|2016 Special Committee on Electoral Reform]] recommended that if the existing election system was replaced, the new system should be designed to achieve "a Gallagher score of 5 or less". This low level of disproportionality is consistently achieved in European PR<ref>{{cite journal |url=https://www.researchgate.net/publication/349468714_New_indexes_for_measuring_electoral_disproportionality |last1=Arredondo |first1=Verónica |last2=Martínez-Panero
|first2=Miguel |last3=Palomares |first3=A. |last4=Peña |first4=Teresa |last5=Ramirez |first5=V. |date=2020 |title=New indexes for measuring electoral disproportionality |journal=Revista Electrónica de Comunicaciones y Trabajos de ASEPUMA |volume=21 |number=2 |doi=10.24309/recta.2020.21.2.05 |p=372}}</ref> but is much lower than was produced in the [[2015 Canadian federal election|2015 Canadian election]] under [[first-past-the-post voting]], where the Gallagher index was 12.<ref name="CanadaErreFinalReport">{{cite web |author=Canadian House of Commons Special Committee on Electoral Reform |title=Strengthening Democracy in Canada: Principles, Process and Public Engagement for Electoral Reform |url=http://www.ourcommons.ca/DocumentViewer/en/42-1/ERRE/report-3 |date=December 2016 |author-link=Canadian House of Commons Special Committee on Electoral Reform}}</ref>