Mathematical aspects of degressive proportionality

We analyze properties of apportionment functions in context of the problem of allocating seats in the European Parliament. Necessary and sufficient conditions for apportionment functions are investigated. Some exemplary families of apportionment functions are specified and the corresponding partitions of the seats in the European Parliament among the Member States of the European Union are presented. Although the choice of the allocation functions is theoretically unlimited, we show that the constraints are so strong that the acceptable functions lead to rather similar solutions.Comment: several minor corrections, revised version 10 pages in two column style, one figure and two tables include

Proposal for Measure of Degressive Proportionality

AbstractDegressive proportionality is an intermediary solution between equality and proportionality. Taking this fact into account, the article proposes a measure of degression of the degressively proportional division. The defined measure was used, among other instances, in allocation based on classical proposals of seat distribution in European Parliament by Pukelsheim and Ramirez as well as the distribution of seats during the Parliamentary term of 2014- 2019. The outcome is confronted with other measures functioning in the literature

The Proposal of Allocation of Seats in the European Parliament – The Shifted Root

AbstractThe issue of allocation of seats in the European Parliament among the Member States of the EU has been the subject of several studies and proposals of algorithms of allocation. The rejection by the Parliament of the Cambridge Compromise means that the issue is still pertinent. This article presents a new method of forming the composition of the EP based on the well-known algorithms of Pukelsheim and Ramirez, called the shifted root. The composition of the Parliament obtained with the use of this method is also given, followed by a discussion of the results

Degressive representation of Member States in the European Parliament 2019-2024

Primary law of the European Union demands that the allocation of the seats of the European Parliament between the Member States must obey the principle of degressive proportionality. The principle embodies the political aim that the more populous states agree to be underrepresented in order to allow the less populous states to be better represented. This paper reviews four allocation methods achieving this goal: the Cambridge Compromise, the Power Compromise, the Modified Cambridge Compromise, and the 0.5-DPL Method. After a year of committee deliberations, Parliament decreed on 7 February 2018 an allocation of seats for the 2019 elections that realizes degressive proportionality, but otherwise lacks methodological grounding. The allocation emerged from haggling and bargaining behind closed doors

Multi-Winner Voting with Approval Preferences

Approval-based committee (ABC) rules are voting rules that output a fixed-size subset of candidates, a so-called committee. ABC rules select committees based on dichotomous preferences, i.e., a voter either approves or disapproves a candidate. This simple type of preferences makes ABC rules widely suitable for practical use. In this book, we summarize the current understanding of ABC rules from the viewpoint of computational social choice. The main focus is on axiomatic analysis, algorithmic results, and relevant applications.Comment: This is a draft of the upcoming book "Multi-Winner Voting with Approval Preferences

Multiwinner Voting with Fairness Constraints

Multiwinner voting rules are used to select a small representative subset of candidates or items from a larger set given the preferences of voters. However, if candidates have sensitive attributes such as gender or ethnicity (when selecting a committee), or specified types such as political leaning (when selecting a subset of news items), an algorithm that chooses a subset by optimizing a multiwinner voting rule may be unbalanced in its selection -- it may under or over represent a particular gender or political orientation in the examples above. We introduce an algorithmic framework for multiwinner voting problems when there is an additional requirement that the selected subset should be "fair" with respect to a given set of attributes. Our framework provides the flexibility to (1) specify fairness with respect to multiple, non-disjoint attributes (e.g., ethnicity and gender) and (2) specify a score function. We study the computational complexity of this constrained multiwinner voting problem for monotone and submodular score functions and present several approximation algorithms and matching hardness of approximation results for various attribute group structure and types of score functions. We also present simulations that suggest that adding fairness constraints may not affect the scores significantly when compared to the unconstrained case.Comment: The conference version of this paper appears in IJCAI-ECAI 201

A power-weighted variant of the EU27 Cambridge Compromise

The Cambridge Compromise composition of the European Parliament allocates five base seats to each Member State's citizenry, and apportions the remaining seats proportionately to population figures using the divisor method with rounding upwards and observing a 96 seat capping. The power-weighted variant avoids the capping step, proceeding instead by a progressive non-linear downweighting of the population figures until the largest State is allocated exactly 96 seats. The pertinent calculations of the variant are described, and its relative constitutional merits are discussed

Multi-Winner Voting with Approval Preferences

From fundamental concepts and results to recent advances in computational social choice, this open access book provides a thorough and in-depth look at multi-winner voting based on approval preferences. The main focus is on axiomatic analysis, algorithmic results and several applications that are relevant in artificial intelligence, computer science and elections of any kind. What is the best way to select a set of candidates for a shortlist, for an executive committee, or for product recommendations? Multi-winner voting is the process of selecting a fixed-size set of candidates based on the preferences expressed by the voters. A wide variety of decision processes in settings ranging from politics (parliamentary elections) to the design of modern computer applications (collaborative filtering, dynamic Q&A platforms, diversity in search results, etc.) share the problem of identifying a representative subset of alternatives. The study of multi-winner voting provides the principled analysis of this task. Approval-based committee voting rules (in short: ABC rules) are multi-winner voting rules particularly suitable for practical use. Their usability is founded on the straightforward form in which the voters can express preferences: voters simply have to differentiate between approved and disapproved candidates. Proposals for ABC rules are numerous, some dating back to the late 19th century while others have been introduced only very recently. This book explains and discusses these rules, highlighting their individual strengths and weaknesses. With the help of this book, the reader will be able to choose a suitable ABC voting rule in a principled fashion, participate in, and be up to date with the ongoing research on this topic