209 research outputs found
Approximating Maximin Share Allocations
We study the problem of fair allocation of M indivisible items among N agents using the popular notion of maximin share as our measure of fairness. The maximin share of an agent is the largest value she can guarantee herself if she is allowed to choose a partition of the items into N bundles (one for each agent), on the condition that she receives her least preferred bundle. A maximin share allocation provides each agent a bundle worth at least their maximin share. While it is known that such an allocation need not exist [Procaccia and Wang, 2014; Kurokawa et al., 2016], a series of work [Procaccia and Wang, 2014; David Kurokawa et al., 2018; Amanatidis et al., 2017; Barman and Krishna Murthy, 2017] provided 2/3 approximation algorithms in which each agent receives a bundle worth at least 2/3 times their maximin share. Recently, [Ghodsi et al., 2018] improved the approximation guarantee to 3/4. Prior works utilize intricate algorithms, with an exception of [Barman and Krishna Murthy, 2017] which is a simple greedy solution but relies on sophisticated analysis techniques. In this paper, we propose an alternative 2/3 maximin share approximation which offers both a simple algorithm and straightforward analysis. In contrast to other algorithms, our approach allows for a simple and intuitive understanding of why it works
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
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
Computing large market equilibria using abstractions
Computing market equilibria is an important practical problem for market
design (e.g. fair division, item allocation). However, computing equilibria
requires large amounts of information (e.g. all valuations for all buyers for
all items) and compute power. We consider ameliorating these issues by applying
a method used for solving complex games: constructing a coarsened abstraction
of a given market, solving for the equilibrium in the abstraction, and lifting
the prices and allocations back to the original market. We show how to bound
important quantities such as regret, envy, Nash social welfare, Pareto
optimality, and maximin share when the abstracted prices and allocations are
used in place of the real equilibrium. We then study two abstraction methods of
interest for practitioners: 1) filling in unknown valuations using techniques
from matrix completion, 2) reducing the problem size by aggregating groups of
buyers/items into smaller numbers of representative buyers/items and solving
for equilibrium in this coarsened market. We find that in real data
allocations/prices that are relatively close to equilibria can be computed from
even very coarse abstractions
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
Multi-agent Online Scheduling: MMS Allocations for Indivisible Items
We consider the problem of fairly allocating a sequence of indivisible items
that arrive online in an arbitrary order to a group of n agents with additive
normalized valuation functions. We consider both the allocation of goods and
chores and propose algorithms for approximating maximin share (MMS)
allocations. When agents have identical valuation functions the problem
coincides with the semi-online machine covering problem (when items are goods)
and load balancing problem (when items are chores), for both of which optimal
competitive ratios have been achieved. In this paper, we consider the case when
agents have general additive valuation functions. For the allocation of goods,
we show that no competitive algorithm exists even when there are only three
agents and propose an optimal 0.5-competitive algorithm for the case of two
agents. For the allocation of chores, we propose a (2-1/n)-competitive
algorithm for n>=3 agents and a square root of 2 (approximately
1.414)-competitive algorithm for two agents. Additionally, we show that no
algorithm can do better than 15/11 (approximately 1.364)-competitive for two
agents.Comment: 29 pages, 1 figure (to appear in ICML 2023
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