List of indices
In the case of a block system, disproportionality is not measured within individual blocks, but rather based on the total number of votes and seats received.
- \(p\): Number of parties
- \(v_i\): voteshare of party \(i\)
- \(s_i\): seatshare of party \(i\)
| Index | ID in {PRcalc} | Rule |
|---|---|---|
| D’Hondt (Gallagher 1991) | "dhondt" |
\(\mbox{max}_i \frac{s_i}{v_i}\) |
| Monroe (1994) | "monroe" |
\(\sqrt{\frac{\sum_i (s_i - v_i)^2}{1 + \sum_i v_i^2}}\) |
| Maximum absolute deviation | "maxdev" |
\(\mbox{max}_i\{|s_i - v_i|\}\) |
| Max-Min ratio | "mm_ratio" |
\(\frac{\max{\frac{v_i}{s_i}}}{\min{\frac{v_i}{s_i}}}\) |
| Rae (1967) | "rae" |
\(\frac{1}{p}\sum_i |s_i - v_i|\) |
| Loosemore & Hanby (1971) | "lh" |
\(\frac{1}{2}\sum_i |s_i - v_i|\) |
| Grofman | "grofman" |
\(\frac{1}{e}\sum_i |s_i - v_i|; e = \frac{1}{\sum_i v_i^2}\) |
| Lijphart | "lijphart" |
\(\frac{|s_a - v_a| + |s_b - v_b|}{2}; v_a > v_b > ...\) |
| Gallagher (1991) | "gallagher" |
\(\sqrt{\frac{1}{2}\sum_i (s_i - v_i)^2}\) |
| Generalized Gallagher | "g_gallagher" |
\(\sqrt[k]{\frac{1}{k}\sum_i (s_i - v_i)^k}\) |
| Gatev | "gatev" |
\(\sqrt{\frac{\sum_i(s_i - v_i)^2}{\sum_i(s_i^2 + v_i^2)}}\) |
| Ryabtsev | "ryabtsev" |
\(\sqrt{\frac{\sum_i(s_i - v_i)^2}{\sum_i(s_i + v_i)^2}}\) |
| Szalai (Stewart 2006) | "szalai" |
\(\sqrt{\frac{1}{p}\sum_i \bigl(\frac{s_i - v_i}{s_i + v_i}\bigr)^2}\) |
| Weighted Szalai (Stewart 2006) | "w_szalai" |
\(\sqrt{\frac{1}{2}\sum_i \frac{(s_i - v_i)^2}{s_i + v_i}}\) |
| Aleskerov & Platonov | "ap" |
\(\frac{\sum_i k_i \frac{s_i}{v_i}}{\sum_i k_i}; k_i = \mathbb{I}\bigl(\frac{s_i}{v_i} > 1\bigr)\) |
| Gini coefficient | "gini" |
\(1\) |
| Atkinson | "atkinson" |
\(1 - \Bigl[\sum_i v_i \bigl(\frac{s_i}{v_i}\bigr)^{(1 - \eta)} \Bigr]^{\frac{1}{1-\eta}}\) |
| Generalized Entropy | "entropy" |
\(\frac{1}{\alpha^2 - \alpha}\Bigl[ \sum_i v_i \bigl( \frac{s_i}{v_i} \bigr)^\alpha - 1 \Bigr]\) |
| Sainte-Laguë (1910) | "sl" |
\(\sum_i \frac{(s_i - v_i)^2}{v_i}\) |
| Cox & Shugart | "cs" |
\(\frac{\sum_i (s_i - \bar{s})(v_i - \bar{v})}{\sum_i(v_i - \bar{v})^2}\) |
| Farina (Kestelman 2005) | "farina" |
\(\mbox{arccos}\Bigl[ \frac{\sum_i s_i v_i}{\sqrt{\sum_i s_i^2 \sum_i v_i^2}} \Bigr]\frac{10}{9}\) |
| Ortona | "ortona" |
\(\frac{\sum_i |s_i - v_i|}{\sum_i |u_i - v_i|}\) |
| Cosine Dissimilarity | "cd" |
\(1 - \frac{\sum_i s_i v_i}{\sqrt{\sum_i s_i^2}\sqrt{\sum_i v_i^2}}\) |
| Lebeda’s RR (Mixture D’Hondt) | "rr" |
\(1 - \frac{1}{\mbox{max}_i \frac{s_i}{v_i}}\) |
| Lebeda’s ARR | "arr" |
\(\frac{1}{p}\Bigl( 1 - \frac{1}{\mbox{max}_i \frac{s_i}{v_i}} \Bigr)\) |
| Lebeda’s SRR | "srr" |
\(\sqrt{\sum_i\Bigl(v_i - \frac{s_i}{\mbox{max}_i \frac{s_i}{v_i}}\Bigr)^2}\) |
| Lebeda’s WDRR | "wdrr" |
\(\frac{1}{3}\Bigl(\Bigl(\sum_i |v_i - s_i|\Bigr) + \Bigl(\sum_i 1 - \frac{1}{\mbox{max}_i \frac{s_i}{v_i}} \Bigr)\Bigr)\) |
| Kullback-Leibler Surprise | "kl" |
\(\sum_{s_i > 0}s_i \mbox{ln} \frac{s_i}{v_i}\) |
| Likelihood Ratio Statistic | "lr" |
\(2\sum_i v_i \mbox{ln} \frac{v_i}{s_i}\) |
| \(\chi^2\) | "chisq" |
\(\sum_{s_i > 0}\frac{(v_i - s_i)^2}{s_i}\) |
| Hellinger Distance | "hellinger" |
\(\frac{1}{\sqrt{2}}\sqrt{\sum_i(\sqrt{s_i} - \sqrt{v_i})^2}\) |
| \(\alpha\)-divergence | "ad" |
See Note 2 |
If as_disprop = FALSE (default), the calculation is based on each party’s total votes and total seats. This is fine when calculating disproportionality in the allocation of seats for proportional representation system. However, in a proportional representation system, there can be many parties that receive a positive number of votes but no seats. In this case, Max-Min ratio is not a useful indicator.
However, when measuring malapportionment, Max-Min ratio is one of the useful indicators. In this case, as_disprop = FALSE must be specified or it will not be calculated correctly.
\[ D(\alpha) = \begin{cases} \sum_i \bigl(s_i \mbox{ln} \bigl(\frac{s_i}{v_i}\bigr)\bigr) & \mbox{if}\quad\alpha = 1,\\ \sum_i \bigl(v_i \mbox{ln} \bigl(\frac{v_i}{s_i}\bigr)\bigr) & \mbox{if}\quad\alpha = 0,\\ \sum_i \bigl(v_i \frac{1}{\alpha (\alpha - 1)}\bigr) \Bigl[\bigl(\frac{s_i}{v_i}\bigr)^\alpha - 1\Bigr] & \mbox{otherwise.}\\ \end{cases} \]
- \(0\log(0) = 0\)