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We consider a voting scenario in which the resource to be voted upon may consist of both indivisible and divisible goods. This setting generalizes both the well-studied model of multiwinner voting and the recently introduced model of cake sharing. Under approval votes, we propose two variants of the extended justified representation (EJR) notion from multiwinner voting, a stronger one called EJR for mixed goods (EJR-M) and a weaker one called EJR up to 1 (EJR-1). We extend three multiwinner voting rules to our setting—GreedyEJR, the method of equal shares (MES), and proportional approval voting (PAV)—and show that while all three generalizations satisfy EJR-1, only the first one provides EJR-M. In addition, we derive tight bounds on the proportionality degree implied by EJR-M and EJR-1, and investigate the proportionality degree of our proposed rules. [ABSTRACT FROM AUTHOR]

The original article can be found online at https://doi.org/10.1007/s00355-024-01511-8.Correction: Social Choice and Welfarehttps://doi.org/10.1007/s00355-024-01511-8In this article, the University name in the affiliation 1 incorrectly published as University of New South Wales Sydney, but should have been University of New South Wales.The original article has been corrected.Publisher's NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.By Xinhang Lu; Jannik Peters; Haris Aziz; Xiaohui Bei and Warut SuksompongReported by Author; Author; Author; Author; Author [Extracted from the article]

Proportional representation (PR) is often discussed in voting settings as a major desideratum. For the past century or so, it is common both in practice and in the academic literature to look towards the single transferable vote (STV) rule as the solution for achieving PR. Some of the most prominent electoral reform movements around the globe are pushing for the adoption of STV. It has been termed a major open problem to design a voting rule that satisfies the same PR properties as STV and better monotonicity properties. In this paper, we first present a taxonomy of proportional representation axioms for general weak order preferences, some of which generalise and strengthen previously introduced concepts. We then present a rule called the expanding approvals rule (EAR) that satisfies properties stronger than the central PR axiom satisfied by STV, can handle indifferences in a convenient and computationally efficient manner, and also satisfies better candidate monotonicity properties. In view of this, our proposed rule seems to be a compelling solution for achieving proportional representation in voting settings. [ABSTRACT FROM AUTHOR]

We consider stability concepts for random matchings where agents have preferences over objects and objects have priorities for the agents. When matchings are deterministic, the standard stability concept also captures the fairness property of no (justified) envy. When matchings can be random, there are a number of natural stability and fairness concepts that coincide with stability and no envy whenever matchings are deterministic. We formalize known stability concepts for random matchings for a general setting that allows weak preferences and weak priorities, unacceptability, and an unequal number of agents and objects. We then present a clear taxonomy of the stability concepts and identify logical relations between them. Finally, we present a transformation from the most general setting to the most restricted setting, and show how almost all our stability concepts are preserved by that transformation. [ABSTRACT FROM AUTHOR]

We consider the problem of assigning objects probabilistically among a group of agents who may have multi-unit demands. Each agent has linear preferences over the (set of) objects. The most commonly used extension of preferences to compare probabilistic assignments is by means of stochastic dominance, which leads to corresponding notions of envy-freeness, efficiency, and strategy-proofness. We show that equal treatment of equals, efficiency, and strategy-proofness are incompatible. Moreover, anonymity, neutrality, efficiency, and weak strategy-proofness are incompatible. If we strengthen weak strategy-proofness to weak group strategy-proofness, then when agents have single-unit demands, anonymity, neutrality, efficiency, and weak group strategy-proofness become incompatible. [ABSTRACT FROM AUTHOR]

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 fixed-size set of winners (committee). We propose a natural axiom for this setting, which we call justified representation ( $$\mathrm {JR}$$ ). 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 $$\mathrm {JR}$$ . However, it turns out that several prominent approval-based voting rules may fail to output such a committee. In particular, while Proportional Approval Voting ( $$\mathrm {PAV}$$ ) always outputs a committee that provides $$\mathrm {JR}$$ , Sequential Proportional Approval Voting ( $$\mathrm {SeqPAV}$$ ), which is a tractable approximation to $$\mathrm {PAV}$$ , does not have this property. We then introduce a stronger version of the $$\mathrm {JR}$$ axiom, which we call extended justified representation ( $$\mathrm {EJR}$$ ), and show that $$\mathrm {PAV}$$ satisfies $$\mathrm {EJR}$$ , while other rules we consider do not; indeed, $$\mathrm {EJR}$$ can be used to characterize $$\mathrm {PAV}$$ within the class of weighted $$\mathrm {PAV}$$ rules. We also consider several other questions related to $$\mathrm {JR}$$ and $$\mathrm {EJR}$$ , including the relationship between $$\mathrm {JR}$$ / $$\mathrm {EJR}$$ and core stability, and the complexity of the associated computational problems. [ABSTRACT FROM AUTHOR]

The undercut procedure was presented by Brams et al. (Soc Choice Welf 39:615-631, ) as a procedure for identifying an envy-free allocation when agents have preferences over sets of objects. We point out some shortcomings of the undercut procedure. We then simplify the undercut procedure of Brams et al. and show that it works under more general conditions where agents may express indifference between objects and they may not necessarily have responsive preferences over sets of objects. Finally, we show that the procedure works even if agents have unequal claims. [ABSTRACT FROM AUTHOR]