Loosemore–Hanby index

The Loosemore–Hanby index[1] (also called Rose index) measures disproportionality of electoral systems. It computes the absolute difference between votes cast and seats obtained using the formula:[2]

LH={\frac {1}{2}}\sum _{i=1}^{n}|v_{i}-s_{i}|

,

where $v_{i}$ is the vote share and $s_{i}$ the seat share of party $i$ such that $\Sigma _{i}v_{i}=\Sigma _{i}s_{i}=1$ , and $n$ is the overall number of parties.

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 other disproportionality indexes 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]^: 292 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]^: 85

The LH index is related to the Schutz index of inequality, which is defined as

S={\frac {1}{2}}\sum _{i=1}^{n}|e_{i}-a_{i}|,

where $e_{i}$ is the expected share of individual $i$ and $a_{i}$ 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 $\Delta$ index of dissimilarity. [5]^: 229

The complement of the LH index is called Party Total Representativity.[6]

Software implementation

Rose index in PolRep, an R package.

References

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. JSTOR 193346.
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.
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.
Lijphart, Arend; Grofman, Bernard (2007). The Evolution of Electoral and Party Systems in the Nordic Countries. Algora Publishing. ISBN 978-0-87586-168-5.
Agesti, Alan (2002). Categorical Data Analysis. Wiley. ISBN 0-471-36093-7.
"Voting matters, Issue 10: pp 7-10". www.votingmatters.org.uk. Retrieved 2021-04-18.

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[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. JSTOR 193346.

[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.

[grofman-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.

[lg-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.

[5] Agesti, Alan (2002). Categorical Data Analysis. Wiley. ISBN 0-471-36093-7.

[6] "Voting matters, Issue 10: pp 7-10". www.votingmatters.org.uk. Retrieved 2021-04-18.