31 research outputs found
The Good, the Bad and the Submodular: Fairly Allocating Mixed Manna Under Order-Neutral Submodular Preferences
We study the problem of fairly allocating indivisible goods (positively
valued items) and chores (negatively valued items) among agents with decreasing
marginal utilities over items. Our focus is on instances where all the agents
have simple preferences; specifically, we assume the marginal value of an item
can be either , or some positive integer . Under this assumption, we
present an efficient algorithm to compute leximin allocations for a broad class
of valuation functions we call order-neutral submodular valuations.
Order-neutral submodular valuations strictly contain the well-studied class of
additive valuations but are a strict subset of the class of submodular
valuations. We show that these leximin allocations are Lorenz dominating and
approximately proportional. We also show that, under further restriction to
additive valuations, these leximin allocations are approximately envy-free and
guarantee each agent their maxmin share. We complement this algorithmic result
with a lower bound showing that the problem of computing leximin allocations is
NP-hard when is a rational number
Fair Allocation of goods and chores -- Tutorial and Survey of Recent Results
Fair resource allocation is an important problem in many real-world
scenarios, where resources such as goods and chores must be allocated among
agents. In this survey, we delve into the intricacies of fair allocation,
focusing specifically on the challenges associated with indivisible resources.
We define fairness and efficiency within this context and thoroughly survey
existential results, algorithms, and approximations that satisfy various
fairness criteria, including envyfreeness, proportionality, MMS, and their
relaxations. Additionally, we discuss algorithms that achieve fairness and
efficiency, such as Pareto Optimality and Utilitarian Welfare. We also study
the computational complexity of these algorithms, the likelihood of finding
fair allocations, and the price of fairness for each fairness notion. We also
cover mixed instances of indivisible and divisible items and investigate
different valuation and allocation settings. By summarizing the
state-of-the-art research, this survey provides valuable insights into fair
resource allocation of indivisible goods and chores, highlighting computational
complexities, fairness guarantees, and trade-offs between fairness and
efficiency. It serves as a foundation for future advancements in this vital
field
On Approximate Envy-Freeness for Indivisible Chores and Mixed Resources
We study the fair allocation of undesirable indivisible items, or chores. While the case of desirable indivisible items (or goods) is extensively studied, with many results known for different notions of fairness, less is known about the fair division of chores. We study envy-free allocation of chores and make three contributions. First, we show that determining the existence of an envy-free allocation is NP-complete even in the simple case when agents have binary additive valuations. Second, we provide a polynomial-time algorithm for computing an allocation that satisfies envy-freeness up to one chore (EF1), correcting a claim in the existing literature. A modification of our algorithm can be used to compute an EF1 allocation for doubly monotone instances (where each agent can partition the set of items into objective goods and objective chores). Our third result applies to a mixed resources model consisting of indivisible items and a divisible, undesirable heterogeneous resource (i.e., a bad cake). We show that there always exists an allocation that satisfies envy-freeness for mixed resources (EFM) in this setting, complementing a recent result of Bei et al. [Bei et al., 2021] for indivisible goods and divisible cake
Algorithms for Competitive Division of Chores
We study the problem of allocating divisible bads (chores) among multiple
agents with additive utilities, when money transfers are not allowed. The
competitive rule is known to be the best mechanism for goods with additive
utilities and was recently extended to chores by Bogomolnaia et al (2017). For
both goods and chores, the rule produces Pareto optimal and envy-free
allocations. In the case of goods, the outcome of the competitive rule can be
easily computed. Competitive allocations solve the Eisenberg-Gale convex
program; hence the outcome is unique and can be approximately found by standard
gradient methods. An exact algorithm that runs in polynomial time in the number
of agents and goods was given by Orlin.
In the case of chores, the competitive rule does not solve any convex
optimization problem; instead, competitive allocations correspond to local
minima, local maxima, and saddle points of the Nash Social Welfare on the
Pareto frontier of the set of feasible utilities. The rule becomes multivalued
and none of the standard methods can be applied to compute its outcome.
In this paper, we show that all the outcomes of the competitive rule for
chores can be computed in strongly polynomial time if either the number of
agents or the number of chores is fixed. The approach is based on a combination
of three ideas: all consumption graphs of Pareto optimal allocations can be
listed in polynomial time; for a given consumption graph, a candidate for a
competitive allocation can be constructed via explicit formula; and a given
allocation can be checked for being competitive using a maximum flow
computation as in Devanur et al (2002).
Our algorithm immediately gives an approximately-fair allocation of
indivisible chores by the rounding technique of Barman and Krishnamurthy
(2018).Comment: 38 pages, 4 figure