15,798 research outputs found
Asymptotic Existence of Proportionally Fair Allocations
Fair division has long been an important problem in the economics literature.
In this note, we consider the existence of proportionally fair allocations of
indivisible goods, i.e., allocations of indivisible goods in which every agent
gets at least her proportionally fair share according to her own utility
function. We show that when utilities are additive and utilities for individual
goods are drawn independently at random from a distribution, proportionally
fair allocations exist with high probability if the number of goods is a
multiple of the number of agents or if the number of goods grows asymptotically
faster than the number of agents
Fair Division with a Secretive Agent
We study classic fair-division problems in a partial information setting.
This paper respectively addresses fair division of rent, cake, and indivisible
goods among agents with cardinal preferences. We will show that, for all of
these settings and under appropriate valuations, a fair (or an approximately
fair) division among n agents can be efficiently computed using only the
valuations of n-1 agents. The nth (secretive) agent can make an arbitrary
selection after the division has been proposed and, irrespective of her choice,
the computed division will admit an overall fair allocation.
For the rent-division setting we prove that the (well-behaved) utilities of
n-1 agents suffice to find a rent division among n rooms such that, for every
possible room selection of the secretive agent, there exists an allocation (of
the remaining n-1 rooms among the n-1 agents) which ensures overall envy
freeness (fairness). We complement this existential result by developing a
polynomial-time algorithm that finds such a fair rent division under
quasilinear utilities.
In this partial information setting, we also develop efficient algorithms to
compute allocations that are envy-free up to one good (EF1) and
epsilon-approximate envy free. These two notions of fairness are applicable in
the context of indivisible goods and divisible goods (cake cutting),
respectively. This work also addresses fairness in terms of proportionality and
maximin shares. Our key result here is an efficient algorithm that, even with a
secretive agent, finds a 1/19-approximate maximin fair allocation (of
indivisible goods) under submodular valuations of the non-secretive agents.
One of the main technical contributions of this paper is the development of
novel connections between different fair-division paradigms, e.g., we use our
existential results for envy-free rent-division to develop an efficient EF1
algorithm.Comment: 27 page
Fair division of indivisible goods under risk
International audienceWe consider the problem of fairly allocating a set of m indivisible objects to n agents having additive preferences over them. In this paper we propose an extension of this classical problem, where each object can possibly be in bad condition (\textite.g broken), in which case its actual value is zero. We assume that the central authority in charge of allocating the objects does not know beforehand the objects conditions, but only has probabilistic information. The aim of this work is to propose a formal model of this problem, to adapt some classical fairness criteria to this extended setting, and to introduce several approaches to compute optimal allocations for small instances as well as suboptimal good allocations for real-world inspired allocation problems of realistic size
Communication Complexity of Discrete Fair Division
We initiate the study of the communication complexity of fair division with
indivisible goods. We focus on some of the most well-studied fairness notions
(envy-freeness, proportionality, and approximations thereof) and valuation
classes (submodular, subadditive and unrestricted). Within these parameters,
our results completely resolve whether the communication complexity of
computing a fair allocation (or determining that none exist) is polynomial or
exponential (in the number of goods), for every combination of fairness notion,
valuation class, and number of players, for both deterministic and randomized
protocols.Comment: Accepted to SODA 201
Chore division on a graph
The paper considers fair allocation of indivisible nondisposable items that
generate disutility (chores). We assume that these items are placed in the
vertices of a graph and each agent's share has to form a connected subgraph of
this graph. Although a similar model has been investigated before for goods, we
show that the goods and chores settings are inherently different. In
particular, it is impossible to derive the solution of the chores instance from
the solution of its naturally associated fair division instance. We consider
three common fair division solution concepts, namely proportionality,
envy-freeness and equitability, and two individual disutility aggregation
functions: additive and maximum based. We show that deciding the existence of a
fair allocation is hard even if the underlying graph is a path or a star. We
also present some efficiently solvable special cases for these graph
topologies
Fair Division of Mixed Divisible and Indivisible Goods
We study the problem of fair division when the resources contain both
divisible and indivisible goods. Classic fairness notions such as envy-freeness
(EF) and envy-freeness up to one good (EF1) cannot be directly applied to the
mixed goods setting. In this work, we propose a new fairness notion
envy-freeness for mixed goods (EFM), which is a direct generalization of both
EF and EF1 to the mixed goods setting. We prove that an EFM allocation always
exists for any number of agents. We also propose efficient algorithms to
compute an EFM allocation for two agents and for agents with piecewise
linear valuations over the divisible goods. Finally, we relax the envy-free
requirement, instead asking for -envy-freeness for mixed goods
(-EFM), and present an algorithm that finds an -EFM
allocation in time polynomial in the number of agents, the number of
indivisible goods, and .Comment: Appears in the 34th AAAI Conference on Artificial Intelligence
(AAAI), 202
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
On the Proximity of Markets with Integral Equilibria
We study Fisher markets that admit equilibria wherein each good is integrally
assigned to some agent. While strong existence and computational guarantees are
known for equilibria of Fisher markets with additive valuations, such
equilibria, in general, assign goods fractionally to agents. Hence, Fisher
markets are not directly applicable in the context of indivisible goods. In
this work we show that one can always bypass this hurdle and, up to a bounded
change in agents' budgets, obtain markets that admit an integral equilibrium.
We refer to such markets as pure markets and show that, for any given Fisher
market (with additive valuations), one can efficiently compute a "near-by,"
pure market with an accompanying integral equilibrium.
Our work on pure markets leads to novel algorithmic results for fair division
of indivisible goods. Prior work in discrete fair division has shown that,
under additive valuations, there always exist allocations that simultaneously
achieve the seemingly incompatible properties of fairness and efficiency; here
fairness refers to envy-freeness up to one good (EF1) and efficiency
corresponds to Pareto efficiency. However, polynomial-time algorithms are not
known for finding such allocations. Considering relaxations of proportionality
and EF1, respectively, as our notions of fairness, we show that fair and Pareto
efficient allocations can be computed in strongly polynomial time.Comment: 17 page
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