1,386 research outputs found
Envy-free cake division without assuming the players prefer nonempty pieces
Consider players having preferences over the connected pieces of a cake,
identified with the interval . A classical theorem, found independently
by Stromquist and by Woodall in 1980, ensures that, under mild conditions, it
is possible to divide the cake into connected pieces and assign these
pieces to the players in an envy-free manner, i.e, such that no player strictly
prefers a piece that has not been assigned to her. One of these conditions,
considered as crucial, is that no player is happy with an empty piece. We prove
that, even if this condition is not satisfied, it is still possible to get such
a division when is a prime number or is equal to . When is at most
, this has been previously proved by Erel Segal-Halevi, who conjectured that
the result holds for any . The main step in our proof is a new combinatorial
lemma in topology, close to a conjecture by Segal-Halevi and which is
reminiscent of the celebrated Sperner lemma: instead of restricting the labels
that can appear on each face of the simplex, the lemma considers labelings that
enjoy a certain symmetry on the boundary
Competitive division of a mixed manna
A mixed manna contains goods (that everyone likes) and bads (that everyone dislikes),
as well as items that are goods to some agents, but bads or satiated to others.
If all items are goods and utility functions are homogeneous of degree 1 and concave
(and monotone), the competitive division maximizes the Nash product of utilities
(Gale–Eisenberg): hence it is welfarist (determined by the set of feasible utility profiles),
unique, continuous, and easy to compute.
We show that the competitive division of a mixed manna is still welfarist. If the zero
utility profile is Pareto dominated, the competitive profile is strictly positive and still
uniquely maximizes the product of utilities. If the zero profile is unfeasible (for instance,
if all items are bads), the competitive profiles are strictly negative and are the
critical points of the product of disutilities on the efficiency frontier. The latter allows
for multiple competitive utility profiles, from which no single-valued selection can be
continuous or resource monotonic.
Thus the implementation of competitive fairness under linear preferences in interactive
platforms like SPLIDDIT will be more difficult when the manna contains bads
that overwhelm the goods
Cutting a Cake Fairly for Groups Revisited
Cake cutting is a classic fair division problem, with the cake serving as a
metaphor for a heterogeneous divisible resource. Recently, it was shown that
for any number of players with arbitrary preferences over a cake, it is
possible to partition the players into groups of any desired size and divide
the cake among the groups so that each group receives a single contiguous piece
and every player is envy-free. For two groups, we characterize the group sizes
for which such an assignment can be computed by a finite algorithm, showing
that the task is possible exactly when one of the groups is a singleton. We
also establish an analogous existence result for chore division, and show that
the result does not hold for a mixed cake
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
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
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
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