109 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
Communication Complexity of Discrete Fair Division
We initiate the study of the communication complexity of fair division with
indivisible goods. We focus on some of the most well-studied fairness notions
(envy-freeness, proportionality, and approximations thereof) and valuation
classes (submodular, subadditive and unrestricted). Within these parameters,
our results completely resolve whether the communication complexity of
computing a fair allocation (or determining that none exist) is polynomial or
exponential (in the number of goods), for every combination of fairness notion,
valuation class, and number of players, for both deterministic and randomized
protocols.Comment: Accepted to SODA 201
Welfare and Revenue Guarantees for Competitive Bundling Equilibrium
We study equilibria of markets with heterogeneous indivisible goods and
consumers with combinatorial preferences. It is well known that a
competitive equilibrium is not guaranteed to exist when valuations are not
gross substitutes. Given the widespread use of bundling in real-life markets,
we study its role as a stabilizing and coordinating device by considering the
notion of \emph{competitive bundling equilibrium}: a competitive equilibrium
over the market induced by partitioning the goods for sale into fixed bundles.
Compared to other equilibrium concepts involving bundles, this notion has the
advantage of simulatneous succinctness ( prices) and market clearance.
Our first set of results concern welfare guarantees. We show that in markets
where consumers care only about the number of goods they receive (known as
multi-unit or homogeneous markets), even in the presence of complementarities,
there always exists a competitive bundling equilibrium that guarantees a
logarithmic fraction of the optimal welfare, and this guarantee is tight. We
also establish non-trivial welfare guarantees for general markets, two-consumer
markets, and markets where the consumer valuations are additive up to a fixed
budget (budget-additive).
Our second set of results concern revenue guarantees. Motivated by the fact
that the revenue extracted in a standard competitive equilibrium may be zero
(even with simple unit-demand consumers), we show that for natural subclasses
of gross substitutes valuations, there always exists a competitive bundling
equilibrium that extracts a logarithmic fraction of the optimal welfare, and
this guarantee is tight. The notion of competitive bundling equilibrium can
thus be useful even in markets which possess a standard competitive
equilibrium
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
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