14 research outputs found
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
Fair and Efficient Allocations under Subadditive Valuations
We study the problem of allocating a set of indivisible goods among agents
with subadditive valuations in a fair and efficient manner. Envy-Freeness up to
any good (EFX) is the most compelling notion of fairness in the context of
indivisible goods. Although the existence of EFX is not known beyond the simple
case of two agents with subadditive valuations, some good approximations of EFX
are known to exist, namely -EFX allocation and EFX allocations
with bounded charity.
Nash welfare (the geometric mean of agents' valuations) is one of the most
commonly used measures of efficiency. In case of additive valuations, an
allocation that maximizes Nash welfare also satisfies fairness properties like
Envy-Free up to one good (EF1). Although there is substantial work on
approximating Nash welfare when agents have additive valuations, very little is
known when agents have subadditive valuations. In this paper, we design a
polynomial-time algorithm that outputs an allocation that satisfies either of
the two approximations of EFX as well as achieves an
approximation to the Nash welfare. Our result also improves the current
best-known approximation of and to
Nash welfare when agents have submodular and subadditive valuations,
respectively.
Furthermore, our technique also gives an approximation to a
family of welfare measures, -mean of valuations for ,
thereby also matching asymptotically the current best known approximation ratio
for special cases like while also retaining the fairness
properties
Computing Welfare-Maximizing Fair Allocations of Indivisible Goods
We study the computational complexity of computing allocations that are both
fair and maximize the utilitarian social welfare, i.e., the sum of agents'
utilities. We focus on two tractable fairness concepts: envy-freeness up to one
item (EF1) and proportionality up to one item (PROP1). In particular, we
consider the following two computational problems: (1) Among the
utilitarian-maximal allocations, decide whether there exists one that is also
fair according to either EF1 or PROP1; (2) among the fair allocations, compute
one that maximizes the utilitarian welfare. We show that both problems are
strongly NP-hard when the number of agents is variable, and remain NP-hard for
a fixed number of agents greater than two. For the special case of two agents,
we find that problem (1) is polynomial-time solvable, while problem (2) remains
NP-hard. Finally, with a fixed number of agents, we design
pseudopolynomial-time algorithms for both problems
Efficient Fair Division with Minimal Sharing
A collection of objects, some of which are good and some are bad, is to be
divided fairly among agents with different tastes, modeled by additive
utility-functions. If the objects cannot be shared, so that each of them must
be entirely allocated to a single agent, then a fair division may not exist.
What is the smallest number of objects that must be shared between two or more
agents in order to attain a fair and efficient division? We focus on
Pareto-optimal, envy-free and/or proportional allocations. We show that, for a
generic instance of the problem -- all instances except of a zero-measure set
of degenerate problems -- a fair Pareto-optimal division with the smallest
possible number of shared objects can be found in polynomial time, assuming
that the number of agents is fixed. The problem becomes computationally hard
for degenerate instances, where agents' valuations are aligned for many
objects.Comment: Add experiments with Spliddit.org dat
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