Editing D'Hondt method (section)

==Approximate proportionality under D'Hondt==
The D'Hondt method approximates proportionality by minimizing the largest [[seats-to-votes ratio]] among all parties.<ref name="Sainte1910">{{cite journal |author=André Sainte-Laguë |title=La représentation Proportionnelle et la méthode des moindres carrés |url=http://www.numdam.org/article/ASENS_1910_3_27__529_0.pdf |journal=Annales Scientifiques de l'École Normale Supérieure |publisher=l'École Normale Supérieure |volume=27 |year=1910}}</ref>
This ratio is also known as the advantage ratio. In contrast, the average seats-to-votes ratio is optimized by the [[Webster/Sainte-Laguë method]].
For party <math>p \in \{1,\dots,P\}</math>, where <math>P</math> is the overall number of parties, the advantage ratio is

<math display="block">a_p=\frac{s_p}{v_p},</math>

where
*<math>s_p</math> – the seat share of party <math>p</math>, <math>s_p \in [0,1],\;\sum_p s_p = 1</math>,
*<math>v_p</math> – the vote share of party <math>p</math>, <math>v_p \in [0,1],\;\sum_p v_p = 1</math>.

The largest advantage ratio,

<math display="block">\delta = \max_p a_p,</math>

captures how over-represented is the most over-represented party.

The D'Hondt method assigns seats so that this ratio attains its smallest possible value,

<math display="block">\delta^* = \min_{\mathbf{s} \in \mathcal{S}} \max_p a_p,</math>

where <math>\mathbf{s}=\{s_1,\dots,s_P\}</math> is a seat allocation 
from the set of all allowed seat allocations <math>\mathcal{S}</math>.
Thanks to this, as shown by Juraj Medzihorsky,<ref name="Medzihorsky2019" /> the D'Hondt method splits the votes into exactly proportionally represented ones and residual ones. The overall fraction of residual votes is

<math display="block">\pi^* = 1 - \frac{1}{\delta^*}.</math>

The residuals of party {{mvar|p}} are

<math display="block">r_p = v_p - (1-\pi^*) s_p,\; r_p \in [0, v_p], \sum_p\,r_p=\pi^*.</math>

For illustration, continue with the above example of four parties. The advantage ratios of the four parties are 1.2 for A, 1.1 for B, 1 for C, and 0 for D. The reciprocal of the largest advantage ratio is {{math|1=1/1.15 = 0.87 = 1 &minus; π{{sup|*}}}}. The residuals as shares of the total vote are 0% for A, 2.2% for B, 2.2% for C, and 8.7% for party D. Their sum is 13%, i.e., {{math|1=1 &minus; 0.87 = 0.13}}. The decomposition of the votes into represented and residual ones is shown in the table below.

{|class="wikitable"
|+ Allocation of eight seats under the D'Hondt method
! Party 
! Vote<br />share 
! Seat<br />share 
! Advantage<br />ratio 
! Residual<br />votes 
! Represented<br />votes
|- style="text-align: left;"
|| A || align="right" | 43.5% || align="right" | 50.0% || style="text-align: center" | 1.15 || style="text-align: right" | 0.0% || style="text-align: right" |  43.5%
|-
|| B || style="text-align: right" | 34.8% || style="text-align: right" | 37.5% || style="text-align: center" | 1.08 || style="text-align: right" | 2.2% || style="text-align: right" |  32.6%
|-
|| C || style="text-align: right" | 13.0% || style="text-align: right" | 12.5% || style="text-align: center" | 0.96 || style="text-align: right" | 2.2% || style="text-align: right" |  10.9%
|-
|| D || style="text-align: right" |  8.7% || style="text-align: right" |  0.0% || style="text-align: center" | 0.00 || style="text-align: right" | 8.7% || style="text-align: right" |   0.0%
|-
! Total || 100% || 100% || — || style="text-align: right" | 13% || style="text-align: right" | 87%
|}