We consider approval-based committee voting, i.e. the setting where each voter approves a subset of candidates, and these votes are then used to select a fixedsize set of winners (committee). We propose a natural axiom for this setting, which we call justified representation (JR). This axiom requires that if a large enough group of voters exhibits agreement by supporting the same candidate, then at least one voter in this group has an approved candidate in the winning committee. We show that for every list of ballots it is possible to select a committee that provides JR. However, it turns out that several prominent approval-based voting rules may fail to output such a committee. In particular, while Proportional Approval Voting (PAV) always outputs a committee that provides JR, Reweighted Approval Voting (RAV), a tractable approximation to PAV, does not have this property. We then introduce a stronger version of the JR axiom, which we call extended justified representation (EJR), and show that PAV satisfies EJR, while other rules we consider do not; indeed, EJR can be used to characterize PAV within the class of weighted PAV rules. We also consider several other questions related to JR and EJR, including the relationship between JR/EJR and core stability, and the complexity of the associated algorithmic problems.

We generalize the classic problem of fairly allocating indivisible goods to the problem of fair public decision making, in which a decision must be made on several social issues simultaneously, and, unlike the classic setting, a decision can provide positive utility to multiple players. We extend the popular fairness notion of proportionality (which is not guaranteeable) to our more general setting, and introduce three novel relaxations -proportionality up to one issue, round robin share, and pessimistic proportional share -that are also interesting in the classic goods allocation setting. We show that the Maximum Nash Welfare solution, which is known to satisfy appealing fairness properties in the classic setting, satisfies or approximates all three relaxations in our framework. We also provide polynomial time algorithms and hardness results for finding allocations satisfying these axioms, with or without insisting on Pareto optimality.

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We study fair allocation of indivisible chores (i.e., items with non-positive value) among agents with additive valuations. An allocation is deemed fair if it is (approximately) equitable, which means that the disutilities of the agents are (approximately) equal. Our main theoretical contribution is to show that there always exists an allocation that is simultaneously equitable up to one chore (EQ1) and Pareto optimal (PO), and to provide a pseudopolynomial-time algorithm for computing such an allocation. In addition, we observe that the Leximin solution-which is known to satisfy a strong form of approximate equitability in the goods setting-fails to satisfy even EQ1 for chores. It does, however, satisfy a novel fairness notion that we call equitability up to any duplicated chore. Our experiments on synthetic as well as real-world data obtained from the Spliddit website reveal that the algorithms considered in our work satisfy approximate fairness and e ciency properties signi cantly more often than the algorithm currently deployed on Spliddit.

We consider the problem of fairly dividing a collection of indivisible goods among a set of players. Much of the existing literature on fair division focuses on notions of individual fairness. For instance, envy-freeness requires that no player prefer the set of goods allocated to another player to her own allocation. We observe that an algorithm satisfying such individual fairness notions can still treat groups of players unfairly, with one group desiring the goods allocated to another. Our main contribution is a notion of group fairness, which implies most existing notions of individual fairness. Group fairness (like individual fairness) cannot be satisfied exactly with indivisible goods. Thus, we introduce two “up to one good” style relaxations. We show that, somewhat surprisingly, certain local optima of the Nash welfare function satisfy both relaxations and can be computed in pseudo-polynomial time by local search. Our experiments reveal faster computation and stronger fairness guarantees in practice.

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In the late 19th century, Lars Edvard Phragmén proposed a load-balancing approach for selecting committees based on approval ballots. We consider three committee voting rules resulting from this approach: two optimization variants one minimizing the maximal load and one minimizing the variance of loads —and a sequential variant. We study Phragmén's methods from an axiomatic point of view, focussing on justified representation and related properties that have recently been introduced by Aziz et al. (2015a) and Sánchez-Fernández et al. (2017). We show that the sequential variant satisfies proportional justified representation, making it the first known polynomial-time computable method with this property. Moreover, we show that the optimization variants satisfy perfect representation. We also analyze the com- putational complexity of Phragmén's methods and provide mixed-integer programming based algorithms for computing them.

We study a dynamic social choice problem in which an alternative is chosen at each round according to the reported valuations of a set of agents. In the interests of obtaining a solution that is both efficient and fair, we aim to maximize the long-term Nash welfare, which is the product of all agents' utilities. We present and analyze two greedy algorithms for this problem, including the classic Proportional Fair (PF) algorithm. We analyze several versions of the algorithms and how they relate, and provide an axiomatization of PF. Finally, we evaluate the algorithms on data gathered from a computer systems application.