Seats-to-votes ratio explained

The seats-to-votes ratio,^[1] also known as the advantage ratio, is a measure of equal representation of voters. The equation for seats-to-votes ratio for a political party i is:

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,^[2] 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,^[3] 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

See main article: articles. 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:

$$error\; =\; \backslash sum\_i$$

It was shown^[4] that this error is minimized by the Sainte-Laguë method.

Seats-to-votes ratio for D'Hondt method

See main article: articles. The D'Hondt method approximates proportionality by minimizing the largest seats-to-votes ratio among all parties.^[4] The largest seats-to-votes ratio, which measures how over-represented the most over-represented party among all parties is:

$$\backslash delta\; =\; \backslash max\_i\; a\_i,$$

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

$$\backslash delta^*\; =\; \backslash min\_\; \backslash max\_i\; a\_i,$$

where

s is a seat allocation from the set of all allowed seat allocations l{S} .

Notes and References

1. https://heinonline.org/HOL/LandingPage?handle=hein.journals/uclalr33&div=13 Niemi, Richard G. "Relationship between Votes and Seats: The Ultimate Question in Political Gerrymandering." UCLA L. Rev. 33 (1985): 185.
2. https://commonslibrary.parliament.uk/general-election-2019-turning-votes-into-seats/ General Election 2019: Turning votes into seats, Published Friday, 10 January, 2020, Roderick McInnes, UK Parliament, House of Commons Library
3. The Sainte-Laguë index of disproportionality and Dalton's principle of transfers . 10.1177/1354068817703020 . 2019 . Goldenberg . Josh . Fisher . Stephen D. . Party Politics . 25 . 2 . 203–207 .
4. http://www.numdam.org/item/10.24033/asens.627.pdf Sainte-Laguë, André. "La représentation proportionnelle et la méthode des moindres carrés." Annales scientifiques de l'école Normale Supérieure. Vol. 27. 1910.
5. Juraj Medzihorsky . Rethinking the D'Hondt method . Political Research Exchange . 1 . 1 . 1625712 . 2019 . 10.1080/2474736X.2019.1625712 . free.