Sainte-Laguë method

Sainte-Laguë method
Name	Sainte-Laguë method
Invented by	André Sainte-Laguë
Used in	Germany, New Zealand, Norway, Sweden, Bosnia and Herzegovina, Latvia, Kosovo
Purpose	Party-list proportional representation
Related methods	D'Hondt method, Webster method

Sainte-Laguë method. The Sainte-Laguë method is a highest averages method for allocating seats in party-list proportional representation systems. It is named after the French mathematician André Sainte-Laguë, who described it in 1910. The method is designed to produce more proportional results than the D'Hondt method, particularly favoring smaller political parties. It is used in the national elections of several countries, including Germany and the Scandinavian countries.

Description and formula

The Sainte-Laguë method allocates seats sequentially to parties based on a calculated quotient. For a given party, its total votes are divided by a series of odd divisors: 1, 3, 5, 7, and so on. The highest resulting quotients across all parties, after each seat is allocated, win the available seats in a legislature. This process continues until all seats in an electoral district are filled. The mathematical formula for the quotient is V/(2s+1), where V represents the party's total votes and s is the number of seats the party has already been allocated. This procedure is functionally equivalent to the Webster method used for apportionment in the United States House of Representatives during the 19th century. The method's core algorithm ensures a proportional distribution of mandates among competing political parties.

Example calculation

Consider an election in a constituency with 5 seats contested by three parties: Party A with 52,000 votes, Party B with 31,000 votes, and Party C with 17,000 votes. The first seat is awarded to Party A, as its quotient is 52,000 (52,000/1). For the second seat, Party A's new quotient is 52,000/3 ≈ 17,333, while Party B's is 31,000/1 = 31,000, so Party B wins the seat. The third seat sees quotients of ~17,333 for Party A, 31,000/3 ≈ 10,333 for Party B, and 17,000/1 for Party C; thus, Party C wins. The fourth seat calculation yields ~17,333 for Party A, ~10,333 for Party B, and 17,000/3 ≈ 5,667 for Party C, so Party A gains another seat. For the final seat, Party A's quotient becomes 52,000/5 = 10,400, competing against ~10,333 for Party B and ~5,667 for Party C, awarding the seat to Party A. The final allocation is 3 seats for Party A, 1 for Party B, and 1 for Party C.

Properties and comparison with other methods

The Sainte-Laguë method is considered more favorable to smaller parties than the D'Hondt method, which uses divisors 1, 2, 3, 4... This difference arises because the odd divisors increase the relative cost of each additional seat for larger parties. Consequently, the method minimizes the overall Gallagher index of disproportionality in many theoretical models. Compared to the Largest remainder method with the Hare quota, Sainte-Laguë tends to be less susceptible to the Alabama paradox and other apportionment paradoxes. It generally satisfies the quota rule in practice, meaning a party's seat total will usually be within one seat of its ideal Hare quota. However, like all highest averages methods, it can slightly favor medium-sized parties in certain distributions of votes.

Modifications and variants

A common modification is the **first divisor method**, where the first divisor is adjusted from 1 to a higher number, such as 1.4. This variant, used in Norway and Sweden, introduces an effective electoral threshold to prevent very small parties from gaining representation. Another adjustment involves using a modified sequence like 1.4, 3, 5, 7..., which is the standard in German Bundestag elections for distributing seats at the state level. Some electoral systems, like that of New Zealand Parliament, use the pure Sainte-Laguë method for converting party votes into seats after applying a nationwide threshold. The Danish Folketing employs a similar system with a compensatory tier to ensure national proportionality.

Usage in electoral systems

The Sainte-Laguë method is the principal seat allocation formula in numerous national parliaments. It is used for all seat distributions in the New Zealand House of Representatives under the Mixed-member proportional representation system. In Europe, it is employed in Germany (for state-list seats in the Bundestag), Norway (for the Storting), Sweden (for the Riksdag), and Bosnia and Herzegovina (for the House of Representatives). It is also found in the electoral laws of Latvia (Saeima), Kosovo (Assembly of the Republic of Kosovo), and Nepal (House of Representatives). Furthermore, many proportional representation systems in local and regional elections, such as those for the Scottish Parliament, have considered or adopted this method for its fairness.

Mathematical rationale

The Sainte-Laguë method can be derived from the goal of minimizing the squared differences between a party's share of votes and its share of seats, a principle known as the method of least squares. Mathematically, it seeks to minimize the sum over all parties of (V_i/V_total - S_i/S_total)², where V is votes and S is seats. This objective function leads directly to the divisor sequence of odd integers. The method is also optimal in the sense of being **unbiased**; it does not systematically favor larger or smaller parties across a wide range of electoral outcomes, unlike the D'Hondt method which exhibits a large-party bias. The relationship between the Sainte-Laguë method and the Webster method highlights its foundation in rounding each party's ideal seat share to the nearest integer, based on the arithmetic mean criterion.

Category:Electoral systems Category:Apportionment methods Category:Voting theory