We consider a model where a subset of candidates must be selected based on voter preferences, subject to general constraints that specify which subsets are feasible. This model generalizes committee elections with diversity constraints, participatory budgeting (including constraints specifying how funds must be allocated to projects from different pools), and public decision-making. Axioms of proportionality have recently been defined for this general model, but the proposed rules apply only to approval ballots, where each voter submits a subset of candidates she finds acceptable. We propose proportional rules for cardinal ballots, where each voter assigns a numerical value to each candidate corresponding to her utility if that candidate is selected. In developing these rules, we also introduce methods that produce proportional rankings, ensuring that every prefix of the ranking satisfies proportionality.
We consider fair allocation of indivisible items under additive utilities. When the utilities can be negative, the existence and complexity of an allocation that satisfies Pareto optimality and proportionality up to one item (PROP1) is an open problem. We show that there exists a strongly polynomial-time algorithm that always computes an allocation satisfying Pareto optimality and proportionality up to one item even if the utilities are mixed and the agents have asymmetric weights. We point out that the result does not hold if either of Pareto optimality or PROP1 is replaced with slightly stro…
当效用是可加的时，我们在之前的论文（Bogomolnaia et al.“Dividing Goods or Bads under Additive Utilities”）中发现了许多相似之处，但也发现了我们在划分（私人）商品或坏品时，熟悉的竞争规则（收入相等）的行为中令人惊讶的差异。在这两种情况下，该规则都会在效率边界上挑选出效用（或非效用）产品的临界点，但如果我们分享商品，则只有一个临界点，而在坏的情况下，可能会有许多个临界点。我们将这种分析扩展到混合项目的公平划分：每个项目可以被一些参与者视为好的，而被其他人视为坏的，具有相应的正或负的边际效用。我们发现，混合物品的划分可以归结为一个全好问题或全坏问题的变体，无论是规范上还是计算上：特别是划分非一次性物品的任务必须对每个人都是好消息，或者对每个人都是坏消息。如果至少有一个可行的效用配置文件是积极的，竞争规则挑选唯一的最大值的产品（积极的）效用。如果没有可行的效用曲线是正的，这个规则会在有效边界上选择所有非效用产品的临界点。
We consider fair division problems where indivisible items arrive one-by-one in an online fashion and are allocated immediately to agents who have additive utilities over these items. Many existing offline mechanisms do not work in this online setting. In addition, many existing axiomatic results often do not transfer from the offline to the online setting. For this reason, we propose here three new online mechanisms, as well as consider the axiomatic properties of three previously proposed online mechanisms. In this paper, we use these mechanisms and characterize classes of online mechanisms …
We consider a fair division model in which agents have positive, zero and negative utilities for items. For this model, we analyse one existing fairness property - EFX - and three new and related properties - EFX$_0$, EFX$^3$ and EF1$^3$ - in combination with Pareto-optimality. With general utilities, we give a modified version of an existing algorithm for computing an EF1$^3$ allocation. With $-α/0/α$ utilities, this algorithm returns an EFX$^3$ and PO allocation. With absolute identical utilities, we give a new algorithm for an EFX and PO allocation. With $-α/0/β$ utilities, this algorithm a…
We study voting rules for participatory budgeting, where a group of voters collectively decides which projects should be funded using a common budget. We allow the projects to have arbitrary costs, and the voters to have arbitrary additive valuations over the projects. We formulate an axiom (Extended Justified Representation, EJR) that guarantees proportional representation to groups of voters with common interests. We propose a simple and attractive voting rule called the Method of Equal Shares that satisfies this axiom for arbitrary costs and approval utilities, and that satisfies the axiom …
We study two influential voting rules proposed in the 1890s by Phragmén 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 Phragmén's rule. By proposing two new proportionality axioms (laminar proportionality and priceability) satisfied by Phragmén but not Thiele, we show that the two rules achieve two distinct forms of p…
Motivated by civic problems such as participatory budgeting and multiwinner elections, we consider the problem of public good allocation: Given a set of indivisible projects (or candidates) of different sizes, and voters with different monotone utility functions over subsets of these candidates, the goal is to choose a budget-constrained subset of these candidates (or a committee) that provides fair utility to the voters. The notion of fairness we adopt is that of core stability from cooperative game theory: No subset of voters should be able to choose another blocking committee of proportiona…
We study a resource allocation setting where $m$ discrete items are to be divided among $n$ agents with additive utilities, and the agents' utilities for individual items are drawn at random from a probability distribution. Since common fairness notions like envy-freeness and proportionality cannot always be satisfied in this setting, an important question is when allocations satisfying these notions exist. In this paper, we close several gaps in the line of work on asymptotic fair division. First, we prove that the classical round-robin algorithm is likely to produce an envy-free allocation p…
