6 research outputs found
Envy-free Relaxations for Goods, Chores, and Mixed Items
In fair division problems, we are given a set of items and a set
of agents with individual preferences, and the goal is to find an
allocation of items among agents so that each agent finds the allocation fair.
There are several established fairness concepts and envy-freeness is one of the
most extensively studied ones. However envy-free allocations do not always
exist when items are indivisible and this has motivated relaxations of
envy-freeness: envy-freeness up to one item (EF1) and envy-freeness up to any
item (EFX) are two well-studied relaxations. We consider the problem of finding
EF1 and EFX allocations for utility functions that are not necessarily
monotone, and propose four possible extensions of different strength to this
setting.
In particular, we present a polynomial-time algorithm for finding an EF1
allocation for two agents with arbitrary utility functions. An example is given
showing that EFX allocations need not exist for two agents with non-monotone,
non-additive, identical utility functions. However, when all agents have
monotone (not necessarily additive) identical utility functions, we prove that
an EFX allocation of chores always exists. As a step toward understanding the
general case, we discuss two subclasses of utility functions: Boolean utilities
that are -valued functions, and negative Boolean utilities that are
-valued functions. For the latter, we give a polynomial time
algorithm that finds an EFX allocation when the utility functions are
identical.Comment: 21 pages, 1 figur
Fair Allocation of goods and chores -- Tutorial and Survey of Recent Results
Fair resource allocation is an important problem in many real-world
scenarios, where resources such as goods and chores must be allocated among
agents. In this survey, we delve into the intricacies of fair allocation,
focusing specifically on the challenges associated with indivisible resources.
We define fairness and efficiency within this context and thoroughly survey
existential results, algorithms, and approximations that satisfy various
fairness criteria, including envyfreeness, proportionality, MMS, and their
relaxations. Additionally, we discuss algorithms that achieve fairness and
efficiency, such as Pareto Optimality and Utilitarian Welfare. We also study
the computational complexity of these algorithms, the likelihood of finding
fair allocations, and the price of fairness for each fairness notion. We also
cover mixed instances of indivisible and divisible items and investigate
different valuation and allocation settings. By summarizing the
state-of-the-art research, this survey provides valuable insights into fair
resource allocation of indivisible goods and chores, highlighting computational
complexities, fairness guarantees, and trade-offs between fairness and
efficiency. It serves as a foundation for future advancements in this vital
field
Maximum Nash Welfare and Other Stories About EFX
We consider the classic problem of fairly allocating indivisible goods among agents with additive valuation functions and explore the connection between two prominent fairness notions: maximum Nash welfare (MNW) and envy-freeness up to any good (EFX). We establish that an MNW allocation is always EFX as long as there are at most two possible values for the goods, whereas this implication is no longer true for three or more distinct values. As a notable consequence, this proves the existence of EFX allocations for these restricted valuation functions. While the efficient computation of an MNW allocation for two possible values remains an open problem, we present a novel algorithm for directly constructing EFX allocations in this setting. Finally, we study the question of whether an MNW allocation implies any EFX guarantee for general additive valuation functions under a natural new interpretation of approximate EFX allocations
Nash Social Welfare in Selfish and Online Load Balancing
In load balancing problems there is a set of clients, each wishing to select
a resource from a set of permissible ones, in order to execute a certain task.
Each resource has a latency function, which depends on its workload, and a
client's cost is the completion time of her chosen resource. Two fundamental
variants of load balancing problems are {\em selfish load balancing} (aka. {\em
load balancing games}), where clients are non-cooperative selfish players aimed
at minimizing their own cost solely, and {\em online load balancing}, where
clients appear online and have to be irrevocably assigned to a resource without
any knowledge about future requests. We revisit both selfish and online load
balancing under the objective of minimizing the {\em Nash Social Welfare},
i.e., the geometric mean of the clients' costs. To the best of our knowledge,
despite being a celebrated welfare estimator in many social contexts, the Nash
Social Welfare has not been considered so far as a benchmarking quality measure
in load balancing problems. We provide tight bounds on the price of anarchy of
pure Nash equilibria and on the competitive ratio of the greedy algorithm under
very general latency functions, including polynomial ones. For this particular
class, we also prove that the greedy strategy is optimal as it matches the
performance of any possible online algorithm