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:

\mathrm {a_{i}} =S_{i}/V_{i}

,

where $V_{i}$ is fraction of votes and $S_{i}$ 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]

Relation to seat allocation methods

Different seat allocation methods such as D'Hondt method and Sainte-Laguë method differ in the seats-to-votes ratio for individual parties. The proportionality of seat allocation methods can be proven by calculating the seats-to-votes ratio.[2]

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:

\delta =\max _{i}a_{i},

The D'Hondt method minimizes the largest seats-to-votes ratio by assigning the seats,

\delta ^{*}=\min _{\mathbf {s} \in {\mathcal {S}}}\max _{i}a_{i},

where $\mathbf {s}$ is a seat allocation from the set of all allowed seat allocations ${\mathcal {S}}$ . The D'Hondt method splits the votes into exactly proportionally represented ones and residual ones,[5] minimizing the overall amount of the residuals in the process. The overall fraction of residual votes is

\pi ^{*}=1-{\frac {1}{\delta ^{*}}}.

Notes

Niemi, Richard G. "Relationship between Votes and Seats: The Ultimate Question in Political Gerrymandering." UCLA L. Rev. 33 (1985): 185.
André Sainte-Laguë (1910). "La représentation Proportionnelle et la méthode des moindres carrés" (PDF). Annales Scientifiques de l'École Normale Supérieure. l'École Normale Supérieure. 27.
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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[Niemi-1] Niemi, Richard G. "Relationship between Votes and Seats: The Ultimate Question in Political Gerrymandering." UCLA L. Rev. 33 (1985): 185.

[Sainte1910-2] André Sainte-Laguë (1910). "La représentation Proportionnelle et la méthode des moindres carrés" (PDF). Annales Scientifiques de l'École Normale Supérieure. l'École Normale Supérieure. 27.

[3] General Election 2019: Turning votes into seats, Published Friday, 10 January, 2020, Roderick McInnes, UK Parliament, House of Commons Library

[Goldenberg-4] 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.

[Medzihorsky2019-5] Juraj Medzihorsky (2019). "Rethinking the D'Hondt method". Political Research Exchange. 1 (1): 1625712. doi:10.1080/2474736X.2019.1625712.