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

We study two influential voting rules proposed in the 1890s by Phragmen and Thiele, which elect a committee of k candidates which proportionally represents the voters. Voters provide their preferences by approving an arbitrary number of candidates. Previous work has proposed proportionality axioms satisfied by Thiele's rule (now known as Proportional Approval Voting, PAV) but not by Phragmen's rule. By proposing two new proportionality axioms (laminar proportionality and priceability) satisfied by Phragmen but not Thiele, we show that the two rules achieve two distinct forms of proportional representation. Phragmen's rule ensures that all voters have a similar amount of influence on the committee, and Thiele's rule ensures a fair utility distribution. Thiele's rule is a welfarist voting rule (one that maximizes a function of voter utilities). We show that no welfarist rule can satisfy our new axioms, and we prove that no such rule can satisfy the core. Conversely, some welfarist fairness properties cannot be guaranteed by Phragmen-type rules. This formalizes the difference between the two types of proportionality. We then introduce an attractive committee rule which satisfies a property intermediate between the core and extended justified representation (EJR). It satisfies laminar proportionality, priceability, and is computable in polynomial time. We show that our new rule provides a logarithmic approximation to the core. On the other hand, PAV provides a factor-2 approximation to the core, and this factor is optimal for rules that are fair in the sense of the Pigou--Dalton principle. The full version of the paper is available at http://arxiv.org/pdf/1911.11747.pdf.

Proportionality and the Limits of Welfarism

We study two influential voting rules proposed in the 1890s by Phragmen and Thiele, which elect a committee or parliament of k candidates which proportionally represents the voters. Voters provide their preferences by approving an arbitrary number of candidates. Previous work has proposed proportionality axioms satisfied by Thiele's rule (now known as Proportional Approval Voting, PAV) but not by Phragmen's rule. By proposing two new proportionality axioms (laminar proportionality and priceability) satisfied by Phragmen but not Thiele, we show that the two rules achieve two distinct forms of proportional representation. Phragmen's rule ensures that all voters have a similar amount of influence on the committee, and Thiele's rule ensures a fair utility distribution. 
Thiele's rule is a welfarist voting rule (one that maximizes a function of voter utilities). We show that no welfarist rule can satisfy our new axioms, and we prove that no such rule can satisfy the core. Conversely, some welfarist fairness properties cannot be guaranteed by Phragmen-type rules. This formalizes the difference between the two types of proportionality. We then introduce an attractive committee rule which satisfies a property intermediate between the core and extended justified representation (EJR). It satisfies laminar proportionality, priceability, and is computable in polynomial time. We show that our new rule provides a logarithmic approximation to the core. On the other hand, PAV provides a factor-2 approximation to the core, and this factor is optimal for rules that are fair in the sense of the Pigou--Dalton principle.

We consider voting under metric preferences: both voters and candidates are associated with points in a metric space, and each voter prefers candidates that are closer to her to ones that are further away. In this setting, it is often desirable to select a candidate that minimizes the sum of distances to the voters. However, common voting rules operate on voters’ preference rankings and therefore may be unable to identify the best candidate. A relevant measure of the quality of a voting rule is then its distortion, defined as the worst-case ratio between the performance of a candidate selected by the rule and that of an optimal candidate. Anshelevich, Bhardwaj and Postl (2015) show that some popular rules such as Borda and Plurality do badly in this regard: their distortion scales linearly with the number of candidates. On the positive side, Anshelevich et al. identify a few voting rules whose distortion is bounded by a constant; however, these rules are rarely used in practice. In this paper, we analyze the distortion of two widely used (classes of) voting rules, namely, scoring rules and Single Transferable Vote (STV). We show that all scoring rules have super-constant distortion, answering a question that was left open by Anshelevich et al.; however, we identify a scoring rule whose distortion is asymptotically better than that of Plurality and Borda. For STV, we obtain an upper bound of O(lnm), where m is the number of candidates, as well as a super-constant lower bound; thus, STV is a reasonable, though not a perfect rule from this perspective.

Social Choice Under Metric Preferences: Scoring Rules and STV.

We investigate how robust the results of committee elections are to small changes in the input preference orders, depending on the voting rules used. We find that for typical rules the effect of making a single swap of adjacent candidates in a single preference order is either that (1) at most one committee member might be replaced, or (2) it is possible that the whole committee will be replaced. We also show that the problem of computing the smallest number of swaps that lead to changing the election outcome is typically NP-hard, but there are natural FPT algorithms. Finally, for a number of rules we assess experimentally the average number of random swaps necessary to change the election result.

Piotr Skowron

Papers

Proportionality and the Limits of Welfarism

Proportionality and the Limits of Welfarism

Social Choice Under Metric Preferences: Scoring Rules and STV.

Robustness Among Multiwinner Voting Rules

Multi-attribute proportional representation