13,460 research outputs found

    On Fair Division for Indivisible Items

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    We consider the task of assigning indivisible goods to a set of agents in a fair manner. Our notion of fairness is Nash social welfare, i.e., the goal is to maximize the geometric mean of the utilities of the agents. Each good comes in multiple items or copies, and the utility of an agent diminishes as it receives more items of the same good. The utility of a bundle of items for an agent is the sum of the utilities of the items in the bundle. Each agent has a utility cap beyond which he does not value additional items. We give a polynomial time approximation algorithm that maximizes Nash social welfare up to a factor of e^{1/{e}} ~~ 1.445. The computed allocation is Pareto-optimal and approximates envy-freeness up to one item up to a factor of 2 + epsilon

    Fair Division of Indivisible Items

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    This paper analyzes criteria of fair division of a set of indivisible items among people whose revealed preferences are limited to rankings of the items and for whom no side payments are allowed. The criteria include refinements of Pareto optimality and envy-freeness as well as dominance-freeness, evenness of shares, and two criteria based on equally-spaced surrogate utilities, referred to as maxsum and equimax. Maxsum maximizes a measure of aggregate utility or welfare, whereas equimax lexicographically maximizes persons' utilities from smallest to largest. The paper analyzes conflicts among the criteria along possibilities and pitfalls of achieving fair division in a variety of circumstances.FAIR DIVISION; ALLOCATION OF INDIVISIBLE ITEMS; PARETO OPTIMALITY; ENVY-FREENESS; LEXICOGRAPHIC MAXIMUM

    Paradoxes of Fair Division

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    Two or more players are required to divide up a set of indivisible items that they can rank from best to worst. They may, as well, be able to indicate preferences over subsets, or packages, of items. The main criteria used to assess the fairness of a division are efficiency (Pareto-optimality) and envy-freeness. Other criteria are also suggested, including a Rawlsian criterion that the worst-off player be made as well off as possible and a scoring procedure, based on the Borda count, that helps to render allocations as equal as possible. Eight paradoxes, all of which involve unexpected conflicts among the criteria, are described and classified into three categories, reflecting (1) incompatibilities between efficiency and envy-freeness, (2) the failure of a unique efficient and envy-free division to satisfy other criteria, and (3) the desirability, on occasion, of dividing up items unequally. While troublesome, the paradoxes also indicate opportunities for achieving fair division, which will depend on the fairness criteria one deems important and the trade-offs one considers acceptable.FAIR DIVISION; ALLOCATION OF INDIVISIBLE ITEMS; ENVY-FREENESS; PARETO- OPTIMALITY; RAWLSIAN JUSTICE; BORDA COUNT.

    Fair Division of a Graph

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    We consider fair allocation of indivisible items under an additional constraint: there is an undirected graph describing the relationship between the items, and each agent's share must form a connected subgraph of this graph. This framework captures, e.g., fair allocation of land plots, where the graph describes the accessibility relation among the plots. We focus on agents that have additive utilities for the items, and consider several common fair division solution concepts, such as proportionality, envy-freeness and maximin share guarantee. While finding good allocations according to these solution concepts is computationally hard in general, we design efficient algorithms for special cases where the underlying graph has simple structure, and/or the number of agents -or, less restrictively, the number of agent types- is small. In particular, despite non-existence results in the general case, we prove that for acyclic graphs a maximin share allocation always exists and can be found efficiently.Comment: 9 pages, long version of accepted IJCAI-17 pape

    The Fair Division of Hereditary Set Systems

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    We consider the fair division of indivisible items using the maximin shares measure. Recent work on the topic has focused on extending results beyond the class of additive valuation functions. In this spirit, we study the case where the items form an hereditary set system. We present a simple algorithm that allocates each agent a bundle of items whose value is at least 0.36670.3667 times the maximin share of the agent. This improves upon the current best known guarantee of 0.20.2 due to Ghodsi et al. The analysis of the algorithm is almost tight; we present an instance where the algorithm provides a guarantee of at most 0.37380.3738. We also show that the algorithm can be implemented in polynomial time given a valuation oracle for each agent.Comment: 22 pages, 1 figure, full version of WINE 2018 submissio

    Existence of EFX for Two Additive Valuations

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    Fair division of indivisible items is a well-studied topic in Economics and Computer Science.The objective is to allocate items to agents in a fair manner, where each agent has a valuation for each subset of items. Envy-freeness is one of the most widely studied notions of fairness. Since complete envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling one is envy-freeness up to any item (EFX), where no agent envies another agent after the removal of any single item from the other agent's bundle. However, despite significant efforts by many researchers for several years, it is known that a complete EFX allocation always exists only in limited cases. In this paper, we show that a complete EFX allocation always exists when each agent is of one of two given types, where agents of the same type have identical additive valuations. This is the first such existence result for non-identical valuations when there are any number of agents and items and no limit on the number of distinct values an agent can have for individual items. We give a constructive proof, in which we iteratively obtain a Pareto dominating (partial) EFX allocation from an existing partial EFX allocation.Comment: 14 pages, 2 figure

    On the complexity of Pareto-optimal and envy-free lotteries

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    We study the classic problem of dividing a collection of indivisible resources in a fair and efficient manner among a set of agents having varied preferences. Pareto optimality is a standard notion of economic efficiency, which states that it should be impossible to find an allocation that improves some agent's utility without reducing any other's. On the other hand, a fundamental notion of fairness in resource allocation settings is that of envy-freeness, which renders an allocation to be fair if every agent (weakly) prefers her own bundle over that of any other agent's bundle. Unfortunately, an envy-free allocation may not exist if we wish to divide a collection of indivisible items. Introducing randomness is a typical way of circumventing the non-existence of solutions, and therefore, allocation lotteries, i.e., distributions over allocations have been explored while relaxing the notion of fairness to ex-ante envy freeness. We consider a general fair division setting with nn agents and a family of admissible nn-partitions of an underlying set of items. Every agent is endowed with partition-based utilities, which specify her cardinal utility for each bundle of items in every admissible partition. In such fair division instances, Cole and Tao (2021) have proved that an ex-ante envy-free and Pareto-optimal allocation lottery is always guaranteed to exist. We strengthen their result while examining the computational complexity of the above total problem and establish its membership in the complexity class PPAD. Furthermore, for instances with a constant number of agents, we develop a polynomial-time algorithm to find an ex-ante envy-free and Pareto-optimal allocation lottery. On the negative side, we prove that maximizing social welfare over ex-ante envy-free and Pareto-optimal allocation lotteries is NP-hard.Comment: 22 page

    When Do Envy-Free Allocations Exist?

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    We consider a fair division setting in which mm indivisible items are to be allocated among nn agents, where the agents have additive utilities and the agents' utilities for individual items are independently sampled from a distribution. Previous work has shown that an envy-free allocation is likely to exist when m=Ω(nlog⁥n)m=\Omega(n\log n) but not when m=n+o(n)m=n+o(n), and left open the question of determining where the phase transition from non-existence to existence occurs. We show that, surprisingly, there is in fact no universal point of transition---instead, the transition is governed by the divisibility relation between mm and nn. On the one hand, if mm is divisible by nn, an envy-free allocation exists with high probability as long as m≄2nm\geq 2n. On the other hand, if mm is not "almost" divisible by nn, an envy-free allocation is unlikely to exist even when m=Θ(nlog⁥n/log⁥log⁥n)m=\Theta(n\log n/\log\log n).Comment: Appears in the 33rd AAAI Conference on Artificial Intelligence (AAAI), 201
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