Seats-to-votes ratio

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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:


where is fraction of votes and 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[edit]

The Sainte-Laguë Index is a disproportionality index derived by applying the Pearson's chi-squared test to the seats-to-votes ratio.[4]

Relation to seat allocation methods[edit]

Different seat allocation methods such as D'Hondt method and Sainte-Laguë method differ in the seats-to-votes ratio for individual parties. The proportionality of seat allocation methods can be proven by calculating the seats-to-votes ratio.[2]

Seats-to-votes ratio for D'Hondt method[edit]

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:

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

where is a seat allocation from the set of all allowed seat allocations . The D'Hondt method splits the votes into exactly proportionally represented ones and residual ones,[5] minimizing the overall amount of the residuals in the process. The overall fraction of residual votes is


  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 André Sainte-Laguë (1910). "La représentation Proportionnelle et la méthode des moindres carrés" (PDF). Annales Scientifiques de l'École Normale Supérieure. l'École Normale Supérieure. 27.
  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.