Proportional Representation

Frontmatter

Chapter 1. Exposing Methods: The 2009 European Parliament Elections

Abstract

The multitude of apportionment methods that is available for the translation of vote counts into seat numbers is exemplified. The examples are taken from the 2009 European Parliament elections.

Friedrich Pukelsheim

Chapter 2. Imposing Constitutionality: The 2009 Bundestag Election

Abstract

Electoral systems veer between constitutional demands and political desires on the one hand side, and procedural rules and practical manageability on the other. The diverse requirements are exemplified by the 2009 election of the German Bundestag and its underlying Federal Election Law.

Friedrich Pukelsheim

Chapter 3. From Reals to Integers: Rounding Functions, Rounding Rules

Abstract

A rounding function is a function mapping positive quantities into integers. Prominent examples are the floor function, the ceiling function, the commercial rounding function, and the even-number rounding function. Every rounding function induces a sequence of jumppoints, called signposts, where it advances from one integer to the next.

Friedrich Pukelsheim

Chapter 4. Divisor Methods of Apportionment: Divide and Round

Abstract

Apportionment methods are procedures to allocate a preordained number of seats proportionately to vote counts, census figures, or similar quantities. Apportionment methods must be anonymous, balanced, concordant, decent, and exact. Beyond these organizing principles the central issue is proportionality.

Friedrich Pukelsheim

Chapter 5. Quota Methods of Apportionment: Divide and Rank

Abstract

Another important family of apportionment methods is quota methods. Relying on a fixed divisor of some intrinsic persuasiveness, called quota, they follow the motto “Divide and rank”. The most prominent member of the family, the Hare-quota method with residual fit by greatest remainders, is discussed in the first part of the chapter.

Friedrich Pukelsheim

Chapter 6. Targeting the House Size: Discrepancy Distribution

Abstract

Technical aspects are discussed that are common to divisor methods and to quota methods. The methods start with an initial seat apportionment possibly missing the target house size by some discrepancy. The range of variation of the discrepancy is analyzed.

Friedrich Pukelsheim

Chapter 7. Favoring Some at the Expense of Others: Seat Biases

Abstract

A party’s seat bias is a quantitative measure assessing the deviation of the number of seats apportioned to the party from the party’s ideal share of seats. Seat bias formulas for stationary divisor methods are calculated when parties are ordered by their vote strengths. The formulas are rather telling, and entail manifold consequences.

Friedrich Pukelsheim

Chapter 8. Preferring Stronger Parties to Weaker Parties: Majorization

Abstract

Majorization provides a partial order of apportionment methods. When passing from one method to another, every group of stronger parties gets more seats and the complementary group of weaker parties fewer seats, or they keep what they have. Specifically, one divisor method majorizes another if and only if their signpost ratios are strictly increasing.

Friedrich Pukelsheim

Chapter 9. Securing System Consistency: Coherence and Paradoxes

Abstract

Apportionment methods are assessed from the collective viewpoint whether all variables act together in a consistent manner. A decisive requirement is coherence, demanding that the solution for an apportionment problem as a whole agrees with the solutions of all embedded subproblems.

Friedrich Pukelsheim

Chapter 10. Appraising Electoral Equality: Goodness-of-Fit Criteria

Abstract

Perfect electoral equality is a conceptual ideal defying reality, a certain degree of inequality must be practically tolerated. Ways and means are discussed how to numerically evaluate the residue of disproportionality that apportionment methods inevitably carry along. Three approaches are explored.

Friedrich Pukelsheim

Chapter 11. Tracing Peculiarities: Vote Thresholds and Majority Clauses

Abstract

Various sorts of vote thresholds are studied in detail. The minimum vote share and the maximum vote share that are compatible with a given number of seats are determined. Particular cases are the threshold of representation, and the threshold of exclusion. Amendments are presented ensuring that a party with a straight majority of votes, however narrow, is guaranteed a straight majority of seats.

Friedrich Pukelsheim

Chapter 12. Truncating Seat Ranges: Minimum-Maximum Restrictions

Abstract

Some electoral systems restrict the eventual seat numbers to respect a guaranteed minimum or, occasionally, to obey a preordained maximum. Minimum restrictions or maximum restrictions form an obstacle for quota methods, but are accommodated easily by divisor methods.

Friedrich Pukelsheim

Chapter 13. Proportionality and Personalization: BWG 2013

Abstract

The 2013 amendment of the German Federal Election Law (Bundeswahlgesetz 2013, BWG 2013) is described by example. The Bundestag’s seat apportionment obeys strict proportionality by political parties. All second votes in the country acquire practically equal success values. Thereafter a party’s country-wide seats are allocated among its various state-lists of nominees in a way granting precedence to the direct seats won via first votes in single-seat constituencies.

Friedrich Pukelsheim

Chapter 14. Representing Districts and Parties: Double Proportionality

Abstract

Double-proportional methods achieve fairness in two directions, the geographical division of the country and the political division of the electorate. Initially all seats are apportioned among districts proportionately to population figures and, independently, among parties proportionately to vote counts.

Friedrich Pukelsheim

Backmatter