Seats-to-votes ratio

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:

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

where $\mathrm {v_{i}}$ is fraction of votes cast for that party and $s_{i}$ 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, 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 $i$ 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:

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

It was shown 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. 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,

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

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