11 research outputs found
Towards Optimal Subsidy Bounds for Envy-freeable Allocations
We study the fair division of indivisible items with subsidies among
agents, where the absolute marginal valuation of each item is at most one.
Under monotone valuations (where each item is a good), Brustle et al. (2020)
demonstrated that a maximum subsidy of and a total subsidy of
are sufficient to guarantee the existence of an envy-freeable
allocation. In this paper, we improve upon these bounds, even in a wider model.
Namely, we show that, given an EF1 allocation, we can compute in polynomial
time an envy-free allocation with a subsidy of at most per agent and a
total subsidy of at most . Moreover, we present further improved
bounds for monotone valuations.Comment: 14page
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
Maximin Fairness with Mixed Divisible and Indivisible Goods
We study fair resource allocation when the resources contain a mixture of
divisible and indivisible goods, focusing on the well-studied fairness notion
of maximin share fairness (MMS). With only indivisible goods, a full MMS
allocation may not exist, but a constant multiplicative approximate allocation
always does. We analyze how the MMS approximation guarantee would be affected
when the resources to be allocated also contain divisible goods. In particular,
we show that the worst-case MMS approximation guarantee with mixed goods is no
worse than that with only indivisible goods. However, there exist problem
instances to which adding some divisible resources would strictly decrease the
MMS approximation ratio of the instance. On the algorithmic front, we propose a
constructive algorithm that will always produce an -MMS allocation for
any number of agents, where takes values between and and is
a monotone increasing function determined by how agents value the divisible
goods relative to their MMS values.Comment: Appears in the 35th AAAI Conference on Artificial Intelligence
(AAAI), 202
Fairly Allocating Goods in Parallel
We initiate the study of parallel algorithms for fairly allocating
indivisible goods among agents with additive preferences. We give fast parallel
algorithms for various fundamental problems, such as finding a Pareto Optimal
and EF1 allocation under restricted additive valuations, finding an EF1
allocation for up to three agents, and finding an envy-free allocation with
subsidies. On the flip side, we show that fast parallel algorithms are unlikely
to exist (formally, -hard) for the problem of computing Round-Robin EF1
allocations
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
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
On Interim Envy-Free Allocation Lotteries
With very few exceptions, recent research in fair division has mostly focused on deterministic allocations. Deviating from this trend, we study the fairness notion of interim envy-freeness (iEF) for lotteries over allocations, which serves as a sweet spot between the too stringent notion of ex-post envy-freeness and the very weak notion of ex-ante envy-freeness. iEF is a natural generalization of envy-freeness to random allocations in the sense that a deterministic envy-free allocation is iEF (when viewed as a degenerate lottery). It is also certainly meaningful as it allows for a richer solution space, which includes solutions that are provably better than envy-freeness according to several criteria. Our analysis relates iEF to other fairness notions as well, and reveals tradeoffs between iEF and efficiency. Even though several of our results apply to general fair division problems, we are particularly interested in instances with equal numbers of agents and items where allocations are perfect matchings of the items to the agents. Envy-freeness can be trivially decided and (when it can be achieved, it) implies full efficiency in this setting. Although computing iEF allocations in matching allocation instances is considerably more challenging, we show how to compute them in polynomial time, while also maximizing several efficiency objectives. Our algorithms use the ellipsoid method for linear programming and efficient solutions to a novel variant of the bipartite matching problem as a separation oracle. We also study the extension of interim envy-freeness notion when payments to or from the agents are allowed. We present a series of results on two optimization problems, including a generalization of the classical rent division problem to random allocations using interim envy-freeness as the solution concept