The seats-to-votes ratio, 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 {\displaystyle \mathrm {a_{i}} =s_{i}/v_{i}} ,where v i {\displaystyle \mathrm {v_{i}} } is fraction of votes cast for that party and s i {\displaystyle s_{i}} is fraction of seats won by that party.
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. The seats-to-votes ratio is used as the basis for the Gallagher index method of analyzing proportionality or disproportionality.
Related is the votes-per-seat-won, which is inverse to the seats-to-votes ratio.
Also related are the principles of one man one vote and representation by population.
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 to parties
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
Main article: Sainte-Laguë method § Proportionality under Sainte-Laguë method
The Sainte-Laguë method optimizes the seats-to-votes ratio among all parties i {\displaystyle i} with the least squares approach.
Disproportionality, the difference of the parties' 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:
e r r o r = ∑ i v i ∗ ( s i v i − 1 ) 2 {\displaystyle error=\sum _{i}{v_{i}*\left({\frac {s_{i}}{v_{i}}}-1\right)^{2}}}
It was shown5 that this error is minimized by the Sainte-Laguë method.
Seats-to-votes ratio for D'Hondt method
Main article: D'Hondt method § Approximate proportionality under D'Hondt
The D'Hondt method approximates proportionality by minimizing the largest seats-to-votes ratio among all parties.6 The largest seats-to-votes ratio, which measures how over-represented the most over-represented party among all parties is:
δ = max i a i , {\displaystyle \delta =\max _{i}a_{i},}
The D'Hondt method minimizes the largest seats-to-votes ratio by assigning the seats,7
δ ∗ = min s ∈ S max i a i , {\displaystyle \delta ^{*}=\min _{\mathbf {s} \in {\mathcal {S}}}\max _{i}a_{i},}
where s {\displaystyle \mathbf {s} } is a seat allocation from the set of all allowed seat allocations S {\displaystyle {\mathcal {S}}} .
Notes
References
Niemi, Richard G. "Relationship between Votes and Seats: The Ultimate Question in Political Gerrymandering." UCLA L. Rev. 33 (1985): 185. https://heinonline.org/HOL/LandingPage?handle=hein.journals/uclalr33&div=13 ↩
Sainte-Laguë, André. "La représentation proportionnelle et la méthode des moindres carrés." Annales scientifiques de l'école Normale Supérieure. Vol. 27. 1910. http://www.numdam.org/item/10.24033/asens.627.pdf ↩
General Election 2019: Turning votes into seats, Published Friday, 10 January, 2020, Roderick McInnes, UK Parliament, House of Commons Library https://commonslibrary.parliament.uk/general-election-2019-turning-votes-into-seats/ ↩
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. https://doi.org/10.1177%2F1354068817703020 ↩
Sainte-Laguë, André. "La représentation proportionnelle et la méthode des moindres carrés." Annales scientifiques de l'école Normale Supérieure. Vol. 27. 1910. 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. http://www.numdam.org/item/10.24033/asens.627.pdf ↩
Juraj Medzihorsky (2019). "Rethinking the D'Hondt method". Political Research Exchange. 1 (1): 1625712. doi:10.1080/2474736X.2019.1625712. https://doi.org/10.1080%2F2474736X.2019.1625712 ↩