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:

${\displaystyle \mathrm {a_{i}} =s_{i}/v_{i}}$,

where ${\displaystyle \mathrm {v_{i}} }$ is fraction of votes and ${\displaystyle s_{i}}$ 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

Main article: Sainte-Laguë method § Proportionality under Sainte-Laguë method

The Sainte-Laguë method optimizes the seats-to-votes ratio among all parties ${\displaystyle i}$ 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:

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

It was shown ^[2] that this error is minimized by the Sainte-Laguë method.

Seats-to-votes ratio for D'Hondt method

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. ^[2] The largest seats-to-votes ratio, which measures how over-represented the most over-represented party among all parties is:

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

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

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

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

Notes

1. Niemi, Richard G. "Relationship between Votes and Seats: The Ultimate Question in Political Gerrymandering." UCLA L. Rev. 33 (1985): 185.
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
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.
5. Juraj Medzihorsky (2019). "Rethinking the D'Hondt method". Political Research Exchange. 1 (1): 1625712. doi: 10.1080/2474736X.2019.1625712 .