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.

## 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 ${\displaystyle i}$ 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

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. ^ 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.
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.