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 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, 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, 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 $$i$$ 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:

$$error = \sum_i {v_i*\left(\frac{s_i}{v_i}-1\right)^2}$$

It was shown that this error is minimized by the Sainte-Laguë method.

Seats-to-votes ratio for D'Hondt method
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 $$\mathcal{S}$$.