354 research outputs found
Fairly Allocating Contiguous Blocks of Indivisible Items
In this paper, we study the classic problem of fairly allocating indivisible
items with the extra feature that the items lie on a line. Our goal is to find
a fair allocation that is contiguous, meaning that the bundle of each agent
forms a contiguous block on the line. While allocations satisfying the
classical fairness notions of proportionality, envy-freeness, and equitability
are not guaranteed to exist even without the contiguity requirement, we show
the existence of contiguous allocations satisfying approximate versions of
these notions that do not degrade as the number of agents or items increases.
We also study the efficiency loss of contiguous allocations due to fairness
constraints.Comment: Appears in the 10th International Symposium on Algorithmic Game
Theory (SAGT), 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
Pareto-Optimal Allocation of Indivisible Goods with Connectivity Constraints
We study the problem of allocating indivisible items to agents with additive
valuations, under the additional constraint that bundles must be connected in
an underlying item graph. Previous work has considered the existence and
complexity of fair allocations. We study the problem of finding an allocation
that is Pareto-optimal. While it is easy to find an efficient allocation when
the underlying graph is a path or a star, the problem is NP-hard for many other
graph topologies, even for trees of bounded pathwidth or of maximum degree 3.
We show that on a path, there are instances where no Pareto-optimal allocation
satisfies envy-freeness up to one good, and that it is NP-hard to decide
whether such an allocation exists, even for binary valuations. We also show
that, for a path, it is NP-hard to find a Pareto-optimal allocation that
satisfies maximin share, but show that a moving-knife algorithm can find such
an allocation when agents have binary valuations that have a non-nested
interval structure.Comment: 21 pages, full version of paper at AAAI-201
Approximate Maximin Shares for Groups of Agents
We investigate the problem of fairly allocating indivisible goods among
interested agents using the concept of maximin share. Procaccia and Wang showed
that while an allocation that gives every agent at least her maximin share does
not necessarily exist, one that gives every agent at least of her share
always does. In this paper, we consider the more general setting where we
allocate the goods to groups of agents. The agents in each group share the same
set of goods even though they may have conflicting preferences. For two groups,
we characterize the cardinality of the groups for which a constant factor
approximation of the maximin share is possible regardless of the number of
goods. We also show settings where an approximation is possible or impossible
when there are several groups.Comment: To appear in the 10th International Symposium on Algorithmic Game
Theory (SAGT), 201
Almost Envy-Free Allocations with Connected Bundles
We study the existence of allocations of indivisible goods that are envy-free up to one good (EF1), under the additional constraint that each bundle needs to be connected in an underlying item graph G. When the items are arranged in a path, we show that EF1 allocations are guaranteed to exist for arbitrary monotonic utility functions over bundles, provided that either there are at most four agents, or there are any number of agents but they all have identical utility functions. Our existence proofs are based on classical arguments from the divisible cake-cutting setting, and involve discrete analogues of cut-and-choose, of Stromquist\u27s moving-knife protocol, and of the Su-Simmons argument based on Sperner\u27s lemma. Sperner\u27s lemma can also be used to show that on a path, an EF2 allocation exists for any number of agents. Except for the results using Sperner\u27s lemma, all of our procedures can be implemented by efficient algorithms. Our positive results for paths imply the existence of connected EF1 or EF2 allocations whenever G is traceable, i.e., contains a Hamiltonian path. For the case of two agents, we completely characterize the class of graphs G that guarantee the existence of EF1 allocations as the class of graphs whose biconnected components are arranged in a path. This class is strictly larger than the class of traceable graphs; one can check in linear time whether a graph belongs to this class, and if so return an EF1 allocation
The Distribution of Ownership in an Apartment Ownership or Condominium Scheme
This is a preprint of an article that appeared in: 2002 Georgia Journal of International and Comparative Law, pp. 101-37
Fairly Allocating Many Goods with Few Queries
We investigate the query complexity of the fair allocation of indivisible
goods. For two agents with arbitrary monotonic valuations, we design an
algorithm that computes an allocation satisfying envy-freeness up to one good
(EF1), a relaxation of envy-freeness, using a logarithmic number of queries. We
show that the logarithmic query complexity bound also holds for three agents
with additive valuations. These results suggest that it is possible to fairly
allocate goods in practice even when the number of goods is extremely large. By
contrast, we prove that computing an allocation satisfying envy-freeness and
another of its relaxations, envy-freeness up to any good (EFX), requires a
linear number of queries even when there are only two agents with identical
additive valuations
Democratic Fair Allocation of Indivisible Goods
We study the problem of fairly allocating indivisible goods to groups of
agents. Agents in the same group share the same set of goods even though they
may have different preferences. Previous work has focused on unanimous
fairness, in which all agents in each group must agree that their group's share
is fair. Under this strict requirement, fair allocations exist only for small
groups. We introduce the concept of democratic fairness, which aims to satisfy
a certain fraction of the agents in each group. This concept is better suited
to large groups such as cities or countries. We present protocols for
democratic fair allocation among two or more arbitrarily large groups of agents
with monotonic, additive, or binary valuations. For two groups with arbitrary
monotonic valuations, we give an efficient protocol that guarantees
envy-freeness up to one good for at least of the agents in each group,
and prove that the fraction is optimal. We also present other protocols
that make weaker fairness guarantees to more agents in each group, or to more
groups. Our protocols combine techniques from different fields, including
combinatorial game theory, cake cutting, and voting.Comment: Appears in the 27th International Joint Conference on Artificial
Intelligence and the 23rd European Conference on Artificial Intelligence
(IJCAI-ECAI), 201
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