There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The rapidly growing field of computational social choice, at the intersection of computer science and economics, deals with the computational aspects of collective decision making. This handbook, written by thirty-six prominent members of the computational social choice community, covers the field comprehensively. Chapters devoted to each of the field's major themes offer detailed introductions. Topics include voting theory (such as the computational complexity of winner determination and manipulation in elections), fair allocation (such as algorithms for dividing divisible and indivisible goods), coalition formation (such as matching and hedonic games), and many more. Graduate students, researchers, and professionals in computer science, economics, mathematics, political science, and philosophy will benefit from this accessible and self-contained book.

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Handbook of Computational Social Choice

We investigate two systems of fully proportional representation suggested by Chamberlin & Courant and Monroe. Both systems assign a representative to each voter so that the "sum of misrepresentations" is minimized. The winner determination problem for both systems is known to be NP-hard, hence this work aims at investigating whether there are variants of the proposed rules and/or specific electorates for which these problems can be solved efficiently. As a variation of these rules, instead of minimizing the sum of misrepresentations, we considered minimizing the maximal misrepresentation introducing effectively two new rules. In the general case these "minimax" versions of classical rules appeared to be still NP-hard.

We investigated the parameterized complexity of winner determination of the two classical and two new rules with respect to several parameters. Here we have a mixture of positive and negative results: e.g., we proved fixed-parameter tractability for the parameter the number of candidates but fixed-parameter intractability for the number of winners.

For single-peaked electorates our results are overwhelmingly positive: we provide polynomial-time algorithms for most of the considered problems. The only rule that remains NP-hard for single-peaked electorates is the classical Monroe rule.

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On the computation of fully proportional representation

Multiwinner Voting: A New Challenge for Social Choice Theory

An increasing number of decisions regarding the daily lives of human beings are being controlled by artificial intelligence (AI) algorithms in spheres ranging from healthcare, transportation, and education to college admissions, recruitment, provision of loans and many more realms. Since they now touch on many aspects of our lives, it is crucial to develop AI algorithms that are not only accurate but also objective and fair. Recent studies have shown that algorithmic decision-making may be inherently prone to unfairness, even when there is no intention for it. This paper presents an overview of the main concepts of identifying, measuring and improving algorithmic fairness when using AI algorithms. The paper begins by discussing the causes of algorithmic bias and unfairness and the common definitions and measures for fairness. Fairness-enhancing mechanisms are then reviewed and divided into pre-process, in-process and post-process mechanisms. A comprehensive comparison of the mechanisms is then conducted, towards a better understanding of which mechanisms should be used in different scenarios. The paper then describes the most commonly used fairness-related datasets in this field. Finally, the paper ends by reviewing several emerging research sub-fields of algorithmic fairness.

Algorithmic Fairness

The goal of this paper is to propose and study properties of multiwinner voting rules which can be consider as generalisations of single-winner scoring voting rules. We consider SNTV, Bloc, k-Borda, STV, and several variants of Chamberlin--Courant's and Monroe's rules and their approximations. We identify two broad natural classes of multiwinner score-based rules, and show that many of the existing rules can be captured by one or both of these approaches. We then formulate a number of desirable properties of multiwinner rules, and evaluate the rules we consider with respect to these properties.

Properties of Multiwinner Voting Rules

We consider the following problem: There is a set of items (e.g., movies) and a group of agents (e.g., passengers on a plane); each agent has some intrinsic utility for each of the items. Our goal is to pick a set of K items that maximize the total derived utility of all the agents (i.e., in our example we are to pick K movies that we put on the plane's entertainment system). However, the actual utility that an agent derives from a given item is only a fraction of its intrinsic one, and this fraction depends on how the agent ranks the item among the chosen, available, ones. We provide a formal specification of the model and provide concrete examples and settings where it is applicable. We show that the problem is hard in general, but we show a number of tractability results for its natural special cases.

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Finding a collective set of items: from proportional multirepresentation to group recommendation

A committee selection rule (or, multiwinner voting rule) is a mapping that takes a collection of strict preference rankings and a positive integer k as input, and outputs one or more subsets of candidates of size k. In this paper we consider committee selection rules that can be viewed as generalizations of single-winner scoring rules, including SNTV, Bloc, k-Borda, STV, as well as several variants of the Chamberlin–Courant rule and the Monroe rule and their approximations. We identify two natural broad classes of committee selection rules, and show that many of the existing rules belong to one or both of these classes. We then formulate a number of desirable properties of committee selection rules, and evaluate the rules we consider with respect to these properties.

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Properties of multiwinner voting rules.

Achieving fully proportional representation

The goal of multi-winner elections is to choose a fixed-size committee based on voters’ preferences. An important concern in this setting is representation: large groups of voters with cohesive preferences should be adequately represented by the election winners. Recently, Aziz et al. proposed two axioms that aim to capture this idea: justified representation (JR) and its strengthening extended justified representation (EJR). In this paper, we extend the work of Aziz et al. in several directions. First, we answer an open question of Aziz et al., by showing that Reweighted Approval Voting satisfies JR for k = 3; 4; 5, but fails it for k >= 6. Second, we observe that EJR is incompatible with the Perfect Representation criterion, which is important for many applications of multi-winner voting, and propose a relaxation of EJR, which we call Proportional Justified Representation (PJR). PJR is more demanding than JR, but, unlike EJR, it is compatible with perfect representation, and a committee that provides PJR can be computed in polynomial time if the committee size divides the number of voters. Moreover, just like EJR, PJR can be used to characterize the classic PAV rule in the class of weighted PAV rules. On the other hand, we show that EJR provides stronger guarantees with respect to average voter satisfaction than PJR does.

Piotr Skowron

Papers

Properties of Multiwinner Voting Rules

Finding a collective set of items: from proportional multirepresentation to group recommendation

Properties of multiwinner voting rules.

Achieving fully proportional representation

Proportional Justified Representation