Social:Loosemore–Hanby index

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Short description: A measurement of difference between votes received and seats won

The Loosemore–Hanby index measures disproportionality of electoral systems, how much the principle of one person, one vote is violated.[1] It computes the absolute difference between votes cast and seats obtained using the formula:

Script error: No such module "in5".[math]\displaystyle{ LH=\frac{1}{2}\sum_{i=1}^n|v_i-s_i| }[/math],
where [math]\displaystyle{ v_i }[/math] is the vote share and [math]\displaystyle{ s_i }[/math] the seat share of party [math]\displaystyle{ i }[/math] such that [math]\displaystyle{ \Sigma_i v_i = \Sigma_i s_i = 1 }[/math], and [math]\displaystyle{ n }[/math] is the overall number of parties.[2]

This index is minimized by the largest remainder (LR) method with the Hare quota. Any apportionment method that minimizes it will always apportion identically to LR-Hare. Other methods, including the widely used divisor methods such as the Webster/Sainte-Laguë method or the D'Hondt method minimize the Sainte-Laguë index instead.

The index is named after John Loosemore and Victor J. Hanby, who first published the formula in 1971 in a paper entitled "The Theoretical Limits of Maximum Distortion: Some Analytic Expressions for Electoral Systems". Along with Douglas W. Rae's, the formula is one of the two most cited disproportionality indices.[3] Whereas the Rae index measures the average deviation, the Loosemore–Hanby index measures the total deviation. Michael Gallagher used least squares to develop the Gallagher index, which takes a middle ground between the Rae and Loosemore–Hanby indices.[4]

The LH index is related to the Schutz index of inequality, which is defined as[math]\displaystyle{ S = \frac{1}{2} \sum_{i=1}^{n} |e_i - a_i|, }[/math]where [math]\displaystyle{ e_i }[/math] is the expected share of individual [math]\displaystyle{ i }[/math] and [math]\displaystyle{ a_i }[/math] her allocated share. Under the LH index, parties take the place of individuals, vote shares replace expectation shares, and seat shares allocation shares. The LH index is also related to the dissimilarity index of segregation. All three indexes are special cases of the more general [math]\displaystyle{ \Delta }[/math] index of dissimilarity.[5] The LH index is related to the amount of wasted vote, which only measures the difference between votes cast and seats obtained for parties which did not obtain any seats.

The complement of the LH index is called Party Total Representativity,[6] also called Rose index R. The Rose index is typically expressed in % and can be calculated by subtracting the LH index from 1:[7]

Script error: No such module "in5".[math]\displaystyle{ R=1-LH=1-\frac{1}{2}\sum_{i=1}^n|v_i-s_i| }[/math].

Example of calculating distortion

Netherlands

This table uses the 2021 Dutch general election result.[8] The Netherlands uses a nationwide party list system, with seats allocated by the D'Hondt method. The low figure achieved through this calculation suggests the election was very proportional.

Party Vote Share (%) Seat Share (%) Absolute Difference (%)
VVD 21.87 22.67 0.80
D66 15.02 16.00 0.98
PVV 10.79 11.33 0.54
CDA 9.50 10.00 0.50
SP 5.98 6.00 0.02
PvdA 5.73 6.00 0.27
GL 5.16 5.33 0.17
FvD 5.02 5.33 0.31
PvdD 3.84 4.00 0.16
CU 3.37 3.33 0.04
Volt 2.42 2.00 0.42
JA21 2.37 2.00 0.37
SGP 2.07 2.00 0.07
DENK 2.03 2.00 0.03
50+ 1.02 0.67 0.35
BBB 1.00 0.67 0.33
BIJ1 0.84 0.67 0.17
Others 1.97 0.00 1.97
Total of absolute differences 4.58 %
Total / 2 2.29 %

Application

Europe

The following table displays a calculation of the Rose Index by Nohlen of the most, or second most, recent legislative election in each European country prior to 2009. This calculation ranges from 0-100, with 100 being the most proportional score possible, and 0 the least. Parties which received less than 0.5% of the vote were not included.[9]

Country Rose Index
Albania 46.6[lower-alpha 1]
Andorra 89.1[lower-alpha 2]
Austria 95.6
Belarus N/A
Belgium 90.5
Bosnia and Herzegovina 82.0
Bulgaria 93.4
Croatia 87.2
Cyprus 96.1
Czechia 91.1
Denmark 99.0
Estonia 94.0
Finland 94.6
France 77.4
Germany 94.2
Great Britain 80.0
Greece 90.9
Hungary N/A
Iceland 96.7
Ireland 90.3
Italy 95.1
Latvia 89.7
Liechtenstein 98.3
Lithuania 91.5
Luxembourg 95.5
Macedonia N/A
Malta 98.0
Moldova 83.6
Monaco 68.8
Montenegro N/A
Netherlands 96.7
Norway 96.1
Poland 53.3
Portugal 91.5
Romania N/A
Russia 70.0
San Marino 98.5
Serbia 90.8
Slovakia 89.1
Slovenia 90.1
Spain 93.2
Sweden 95.7
Switzerland 96.2
Ukraine 81.6

Notes

  1. Calculated using aggregated plurality results
  2. Calculated only using the proportional part of the system

Software implementation

See also

  • Efficiency gap

References

  1. Loosemore, John; Hanby, Victor J. (October 1971). "The Theoretical Limits of Maximum Distortion: Some Analytic Expressions for Electoral Systems". British Journal of Political Science (Cambridge University Press) 1 (4): 467–477. doi:10.1017/S000712340000925X. 
  2. Cortona, Pietro Grilli di; Manzi, Cecilia; Pennisi, Aline; Ricca, Federica; Simeone, Bruno (1999). Evaluation and Optimization of Electoral Systems. SIAM. ISBN 978-0-89871-422-7. https://books.google.com/books?id=A1tNCQg2QKEC. 
  3. Grofman, Bernard (1999). Elections in Japan, Korea, and Taiwan Under the Single Non-transferable Vote: The Comparative Study of an Embedded Institution. University of Michigan Press. ISBN 0-472-10909-X. https://books.google.com/books?id=ry26lbfP16sC. 
  4. Lijphart, Arend; Grofman, Bernard (2007). The Evolution of Electoral and Party Systems in the Nordic Countries. Algora Publishing. ISBN 978-0-87586-168-5. https://books.google.com/books?id=oigZTGYdQb4C. 
  5. Agesti, Alan (2002). Categorical Data Analysis. Wiley. ISBN 0-471-36093-7. https://books.google.com/books?id=hpEzw4T0sPUC. 
  6. "Voting matters, Issue 10: pp 7-10". http://www.votingmatters.org.uk/ISSUE10/P6.HTM. 
  7. Fry, Vanessa; McLean, Iain (1991). "A note on rose's proportionality index". Electoral Studies 10: 52–59. doi:10.1016/0261-3794(91)90005-D. https://doi.org/10.1016/0261-3794(91)90005-D. 
  8. "Tweede Kamer 17 maart 2021 (House of Representatives 17 March 2021)". 17 March 2021. https://www.verkiezingsuitslagen.nl/verkiezingen/detail/TK20210317. 
  9. Stöver, Philip; Nohlen, Dieter (2010) (in English). Elections in Europe: A Data Handbook. Nomos. ISBN 9783832956097.