Social:Seats-to-votes ratio
The seats-to-votes ratio,[1] also known as the advantage ratio,[2] is a measure of equal representation of voters. The equation for seats-to-votes ratio for a political party i is:
- [math]\displaystyle{ \mathrm{a_i} = s_i/v_i }[/math],
where [math]\displaystyle{ v_i }[/math] is fraction of votes and [math]\displaystyle{ s_i }[/math] is fraction of seats.
In the case both seats and votes are represented as fractions or percentages, then every voter has equal representation if the seats-to-votes ratio is 1. The principle of equal representation is expressed in slogan one man, one vote and relates to proportional representation.
Related is the votes-per-seat-won,[3] which is inverse to the seats-to-votes ratio.
Relation to disproportionality indices
The Sainte-Laguë Index is a disproportionality index derived by applying the Pearson's chi-squared test to the seats-to-votes ratio,[4] the Gallagher index has a similar formula.
Seats-to-votes ratio for seat allocation
Different apportionment methods such as Sainte-Laguë method and D'Hondt method differ in the seats-to-votes ratio for individual parties.
Seats-to-votes ratio for Sainte-Laguë method
The Sainte-Laguë method optimizes the seats-to-votes ratio among all parties [math]\displaystyle{ i }[/math] with the least squares approach. The difference of the seats-to-votes ratio and the ideal seats-to-votes ratio for each party is squared, weighted according to the vote share of each party and summed up:
[math]\displaystyle{ error = \sum_i {v_i*\left(\frac{s_i}{v_i}-1\right)^2} }[/math]
It was shown[2] that this error is minimized by the Sainte-Laguë method.
Seats-to-votes ratio for D'Hondt method
The D'Hondt method approximates proportionality by minimizing the largest seats-to-votes ratio among all parties.[2] The largest seats-to-votes ratio, which measures how over-represented the most over-represented party among all parties is:
[math]\displaystyle{ \delta = \max_i a_i, }[/math]
The D'Hondt method minimizes the largest seats-to-votes ratio by assigning the seats,[5]
[math]\displaystyle{ \delta^* = \min_{\mathbf{s} \in \mathcal{S}} \max_i a_i, }[/math]
where [math]\displaystyle{ \mathbf{s} }[/math] is a seat allocation from the set of all allowed seat allocations [math]\displaystyle{ \mathcal{S} }[/math].
Notes
- ↑ Niemi, Richard G. "Relationship between Votes and Seats: The Ultimate Question in Political Gerrymandering." UCLA L. Rev. 33 (1985): 185.
- ↑ 2.0 2.1 2.2 Sainte-Laguë, André. "La représentation proportionnelle et la méthode des moindres carrés." Annales scientifiques de l'école Normale Supérieure. Vol. 27. 1910.
- ↑ General Election 2019: Turning votes into seats, Published Friday, 10 January, 2020, Roderick McInnes, UK Parliament, House of Commons Library
- ↑ Goldenberg, Josh, and Stephen D. Fisher. "The Sainte-Laguë index of disproportionality and Dalton’s principle of transfers." Party Politics 25.2 (2019): 203-207.
- ↑ Juraj Medzihorsky (2019). "Rethinking the D'Hondt method". Political Research Exchange 1 (1): 1625712. doi:10.1080/2474736X.2019.1625712.
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