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 $\mathrm {v_{i}}$ 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] 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 $i$ 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:

$error=\sum _{i}{v_{i}*\left({\frac {s_{i}}{v_{i}}}-1\right)^{2}}$

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:

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

The D'Hondt method minimizes the largest seats-to-votes ratio by assigning the seats,^[5]

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

where $\mathbf {s}$ is a seat allocation from the set of all allowed seat allocations ${\mathcal {S}}$ .

Notes

^ Niemi, Richard G. "Relationship between Votes and Seats: The Ultimate Question in Political Gerrymandering." UCLA L. Rev. 33 (1985): 185.
^ ^a ^b ^c 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; Fisher, Stephen D. (2019). "The Sainte-Laguë index of disproportionality and Dalton's principle of transfers". Party Politics. 25 (2): 203–207. doi:10.1177/1354068817703020.
^ Juraj Medzihorsky (2019). "Rethinking the D'Hondt method". Political Research Exchange. 1 (1): 1625712. doi:10.1080/2474736X.2019.1625712.

[Niemi-1] Niemi, Richard G. "Relationship between Votes and Seats: The Ultimate Question in Political Gerrymandering." UCLA L. Rev. 33 (1985): 185.

[sainte-2] Sainte-Laguë, André. "La représentation proportionnelle et la méthode des moindres carrés." Annales scientifiques de l'école Normale Supérieure. Vol. 27. 1910.

[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; Fisher, Stephen D. (2019). "The Sainte-Laguë index of disproportionality and Dalton's principle of transfers". Party Politics. 25 (2): 203–207. doi:10.1177/1354068817703020.

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

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