442 research outputs found
The Good, the Bad and the Submodular: Fairly Allocating Mixed Manna Under Order-Neutral Submodular Preferences
We study the problem of fairly allocating indivisible goods (positively
valued items) and chores (negatively valued items) among agents with decreasing
marginal utilities over items. Our focus is on instances where all the agents
have simple preferences; specifically, we assume the marginal value of an item
can be either , or some positive integer . Under this assumption, we
present an efficient algorithm to compute leximin allocations for a broad class
of valuation functions we call order-neutral submodular valuations.
Order-neutral submodular valuations strictly contain the well-studied class of
additive valuations but are a strict subset of the class of submodular
valuations. We show that these leximin allocations are Lorenz dominating and
approximately proportional. We also show that, under further restriction to
additive valuations, these leximin allocations are approximately envy-free and
guarantee each agent their maxmin share. We complement this algorithmic result
with a lower bound showing that the problem of computing leximin allocations is
NP-hard when is a rational number
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
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
Distorted optimal transport
Classic optimal transport theory is built on minimizing the expected cost
between two given distributions. We propose the framework of distorted optimal
transport by minimizing a distorted expected cost. This new formulation is
motivated by concrete problems in decision theory, robust optimization, and
risk management, and it has many distinct features compared to the classic
theory. We choose simple cost functions and study different distortion
functions and their implications on the optimal transport plan. We show that on
the real line, the comonotonic coupling is optimal for the distorted optimal
transport problem when the distortion function is convex and the cost function
is submodular and monotone. Some forms of duality and uniqueness results are
provided. For inverse-S-shaped distortion functions and linear cost, we obtain
the unique form of optimal coupling for all marginal distributions, which turns
out to have an interesting ``first comonotonic, then counter-monotonic"
dependence structure; for S-shaped distortion functions a similar structure is
obtained. Our results highlight several challenges and features in distorted
optimal transport, offering a new mathematical bridge between the fields of
probability, decision theory, and risk management
A Tight Competitive Ratio for Online Submodular Welfare Maximization
In this paper we consider the online Submodular Welfare (SW) problem. In this
problem we are given bidders each equipped with a general (not necessarily
monotone) submodular utility and items that arrive online. The goal is to
assign each item, once it arrives, to a bidder or discard it, while maximizing
the sum of utilities. When an adversary determines the items' arrival order we
present a simple randomized algorithm that achieves a tight competitive ratio
of \nicefrac{1}{4}. The algorithm is a specialization of an algorithm due to
[Harshaw-Kazemi-Feldman-Karbasi MOR`22], who presented the previously best
known competitive ratio of to the problem. When
the items' arrival order is uniformly random, we present a competitive ratio of
, improving the previously known \nicefrac{1}{4} guarantee.
Our approach for the latter result is based on a better analysis of the
(offline) Residual Random Greedy (RRG) algorithm of
[Buchbinder-Feldman-Naor-Schwartz SODA`14], which we believe might be of
independent interest
Weighted Fair Division with Matroid-Rank Valuations: Monotonicity and Strategyproofness
We study the problem of fairly allocating indivisible goods to agents with
weights corresponding to their entitlements. Previous work has shown that, when
agents have binary additive valuations, the maximum weighted Nash welfare rule
is resource-, population-, and weight-monotone, satisfies
group-strategyproofness, and can be implemented in polynomial time. We
generalize these results to the class of weighted additive welfarist rules with
concave functions and agents with matroid-rank (also known as binary
submodular) valuations.Comment: Appears in the 16th International Symposium on Algorithmic Game
Theory (SAGT), 202
Cut Sparsification and Succinct Representation of Submodular Hypergraphs
In cut sparsification, all cuts of a hypergraph are approximated
within factor by a small hypergraph . This widely applied
method was generalized recently to a setting where the cost of cutting each
is provided by a splitting function, . This
generalization is called a submodular hypergraph when the functions
are submodular, and it arises in machine learning,
combinatorial optimization, and algorithmic game theory. Previous work focused
on the setting where is a reweighted sub-hypergraph of , and measured
size by the number of hyperedges in . We study such sparsification, and
also a more general notion of representing succinctly, where size is
measured in bits.
In the sparsification setting, where size is the number of hyperedges, we
present three results: (i) all submodular hypergraphs admit sparsifiers of size
polynomial in ; (ii) monotone-submodular hypergraphs admit sparsifiers
of size ; and (iii) we propose a new parameter, called
spread, to obtain even smaller sparsifiers in some cases.
In the succinct-representation setting, we show that a natural family of
splitting functions admits a succinct representation of much smaller size than
via reweighted subgraphs (almost by factor ). This large gap is surprising
because for graphs, the most succinct representation is attained by reweighted
subgraphs. Along the way, we introduce the notion of deformation, where
is decomposed into a sum of functions of small description, and we provide
upper and lower bounds for deformation of common splitting functions
Incentive Ratios for Fairly Allocating Indivisible Goods: Simple Mechanisms Prevail
We study the problem of fairly allocating indivisible goods among strategic
agents. Amanatidis et al. show that truthfulness is incompatible with any
meaningful fairness notions. Thus we adopt the notion of incentive ratio, which
is defined as the ratio between the largest possible utility that an agent can
gain by manipulation and his utility in honest behavior under a given
mechanism. We select four of the most fundamental mechanisms in the literature
on discrete fair division, which are Round-Robin, a cut-and-choose mechanism of
Plaut and Roughgarden, Maximum-Nash-Welfare and Envy-Graph Procedure, and
obtain extensive results regarding the incentive ratios of them and their
variants.
For Round-Robin, we establish the incentive ratio of for additive and
subadditive cancelable valuations, the unbounded incentive ratio for cancelable
valuations, and the incentive ratios of and for
submodular and XOS valuations, respectively. Moreover, the incentive ratio is
unbounded for a variant that provides the -approximate maximum social
welfare guarantee. For the algorithm of Plaut and Roughgarden, the incentive
ratio is either unbounded or with lexicographic tie-breaking and is
with welfare maximizing tie-breaking. This separation exhibits the essential
role of tie-breaking rules in the design of mechanisms with low incentive
ratios. For Maximum-Nash-Welfare, the incentive ratio is unbounded.
Furthermore, the unboundedness can be bypassed by restricting agents to have a
strictly positive value for each good. For Envy-Graph Procedure, both of the
two possible ways of implementation lead to an unbounded incentive ratio.
Finally, we complement our results with a proof that the incentive ratio of
every mechanism satisfying envy-freeness up to one good is at least ,
and thus is larger than by a constant
Nonbossy Mechanisms: Mechanism Design Robust to Secondary Goals
We study mechanism design when agents may have hidden secondary goals which
will manifest as non-trivial preferences among outcomes for which their primary
utility is the same. We show that in such cases, a mechanism is robust against
strategic manipulation if and only if it is not only incentive-compatible, but
also nonbossy -- a well-studied property in the context of matching and
allocation mechanisms. We give complete characterizations of
incentive-compatible and nonbossy mechanisms in various settings, including
auctions with single-parameter agents and public decision settings where all
agents share a common outcome. In particular, we show that in the single-item
setting, a mechanism is incentive-compatible, individually rational, and
nonbossy if and only if it is a sequential posted-price mechanism. In contrast,
we show that in more general single-parameter environments, there exist
mechanisms satisfying our characterization that significantly outperform
sequential posted-price mechanisms in terms of revenue or efficiency (sometimes
by an exponential factor)
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