42 research outputs found
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
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
Existence of EFX for Two Additive Valuations
Fair division of indivisible items is a well-studied topic in Economics and
Computer Science.The objective is to allocate items to agents in a fair manner,
where each agent has a valuation for each subset of items. Envy-freeness is one
of the most widely studied notions of fairness. Since complete envy-free
allocations do not always exist when items are indivisible, several relaxations
have been considered. Among them, possibly the most compelling one is
envy-freeness up to any item (EFX), where no agent envies another agent after
the removal of any single item from the other agent's bundle. However, despite
significant efforts by many researchers for several years, it is known that a
complete EFX allocation always exists only in limited cases. In this paper, we
show that a complete EFX allocation always exists when each agent is of one of
two given types, where agents of the same type have identical additive
valuations. This is the first such existence result for non-identical
valuations when there are any number of agents and items and no limit on the
number of distinct values an agent can have for individual items. We give a
constructive proof, in which we iteratively obtain a Pareto dominating
(partial) EFX allocation from an existing partial EFX allocation.Comment: 14 pages, 2 figure
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
On the Existence of Competitive Equilibrium with Chores
We study the chore division problem in the classic Arrow-Debreu exchange setting, where a set of agents want to divide their divisible chores (bads) to minimize their disutilities (costs). We assume that agents have linear disutility functions. Like the setting with goods, a division based on competitive equilibrium is regarded as one of the best mechanisms for bads. Equilibrium existence for goods has been extensively studied, resulting in a simple, polynomial-time verifiable, necessary and sufficient condition. However, dividing bads has not received a similar extensive study even though it is as relevant as dividing goods in day-to-day life.
In this paper, we show that the problem of checking whether an equilibrium exists in chore division is NP-complete, which is in sharp contrast to the case of goods. Further, we derive a simple, polynomial-time verifiable, sufficient condition for existence. Our fixed-point formulation to show existence makes novel use of both Kakutani and Brouwer fixed-point theorems, the latter nested inside the former, to avoid the undefined demand issue specific to bads
Tight Approximation Algorithms for p-Mean Welfare Under Subadditive Valuations
We develop polynomial-time algorithms for the fair and efficient allocation of indivisible goods among n agents that have subadditive valuations over the goods. We first consider the Nash social welfare as our objective and design a polynomial-time algorithm that, in the value oracle model, finds an 8n-approximation to the Nash optimal allocation. Subadditive valuations include XOS (fractionally subadditive) and submodular valuations as special cases. Our result, even for the special case of submodular valuations, improves upon the previously best known O(n log n)-approximation ratio of Garg et al. (2020).
More generally, we study maximization of p-mean welfare. The p-mean welfare is parameterized by an exponent term p ? (-?, 1] and encompasses a range of welfare functions, such as social welfare (p = 1), Nash social welfare (p ? 0), and egalitarian welfare (p ? -?). We give an algorithm that, for subadditive valuations and any given p ? (-?, 1], computes (in the value oracle model and in polynomial time) an allocation with p-mean welfare at least 1/(8n) times the optimal.
Further, we show that our approximation guarantees are essentially tight for XOS and, hence, subadditive valuations. We adapt a result of Dobzinski et al. (2010) to show that, under XOS valuations, an O (n^{1-?}) approximation for the p-mean welfare for any p ? (-?,1] (including the Nash social welfare) requires exponentially many value queries; here, ? > 0 is any fixed constant
Nash-Bargaining-Based Models for Matching Markets: One-Sided and Two-Sided; Fisher and Arrow-Debreu
This paper addresses two deficiencies of models in the area of matching-based market design. The first arises from the recent realization that the most prominent solution that uses cardinal utilities, namely the Hylland-Zeckhauser (HZ) mechanism [Hylland and Zeckhauser, 1979], is intractable; computation of even an approximate equilibrium is PPAD-complete [Vazirani and Yannakakis, 2021; Chen et al., 2021]. The second is the extreme paucity of models that use cardinal utilities, in sharp contrast with general equilibrium theory.
Our paper addresses both these issues by proposing Nash-bargaining-based matching market models. Since the Nash bargaining solution is captured by a convex program, efficiency follow; in addition, it possesses a number of desirable game-theoretic properties. Our approach yields a rich collection of models: for one-sided as well as two-sided markets, for Fisher as well as Arrow-Debreu settings, and for a wide range of utility functions, all the way from linear to Leontief.
We also give very fast implementations for these models which solve large instances, with n = 2000, in one hour on a PC, even for a two-sided matching market. A number of new ideas were needed, beyond the standard methods, to obtain these implementations