106 research outputs found
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
Combinatorial Auctions Do Need Modest Interaction
We study the necessity of interaction for obtaining efficient allocations in
subadditive combinatorial auctions. This problem was originally introduced by
Dobzinski, Nisan, and Oren (STOC'14) as the following simple market scenario:
items are to be allocated among bidders in a distributed setting where
bidders valuations are private and hence communication is needed to obtain an
efficient allocation. The communication happens in rounds: in each round, each
bidder, simultaneously with others, broadcasts a message to all parties
involved and the central planner computes an allocation solely based on the
communicated messages. Dobzinski et.al. showed that no non-interactive
(-round) protocol with polynomial communication (in the number of items and
bidders) can achieve approximation ratio better than ,
while for any , there exists -round protocols that achieve
approximation with polynomial
communication; in particular, rounds of interaction suffice to
obtain an (almost) efficient allocation.
A natural question at this point is to identify the "right" level of
interaction (i.e., number of rounds) necessary to obtain an efficient
allocation. In this paper, we resolve this question by providing an almost
tight round-approximation tradeoff for this problem: we show that for any , any -round protocol that uses polynomial communication can only
approximate the social welfare up to a factor of . This in particular implies that
rounds of interaction are necessary for
obtaining any efficient allocation in these markets. Our work builds on the
recent multi-party round-elimination technique of Alon, Nisan, Raz, and
Weinstein (FOCS'15) and settles an open question posed by Dobzinski et.al. and
Alon et. al
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
Implementation in Advised Strategies: Welfare Guarantees from Posted-Price Mechanisms When Demand Queries Are NP-Hard
State-of-the-art posted-price mechanisms for submodular bidders with
items achieve approximation guarantees of [Assadi and
Singla, 2019]. Their truthfulness, however, requires bidders to compute an
NP-hard demand-query. Some computational complexity of this form is
unavoidable, as it is NP-hard for truthful mechanisms to guarantee even an
-approximation for any [Dobzinski and
Vondr\'ak, 2016]. Together, these establish a stark distinction between
computationally-efficient and communication-efficient truthful mechanisms.
We show that this distinction disappears with a mild relaxation of
truthfulness, which we term implementation in advised strategies, and that has
been previously studied in relation to "Implementation in Undominated
Strategies" [Babaioff et al, 2009]. Specifically, advice maps a tentative
strategy either to that same strategy itself, or one that dominates it. We say
that a player follows advice as long as they never play actions which are
dominated by advice. A poly-time mechanism guarantees an -approximation
in implementation in advised strategies if there exists poly-time advice for
each player such that an -approximation is achieved whenever all
players follow advice. Using an appropriate bicriterion notion of approximate
demand queries (which can be computed in poly-time), we establish that (a
slight modification of) the [Assadi and Singla, 2019] mechanism achieves the
same -approximation in implementation in advised
strategies
Simple Mechanisms for a Subadditive Buyer and Applications to Revenue Monotonicity
We study the revenue maximization problem of a seller with n heterogeneous
items for sale to a single buyer whose valuation function for sets of items is
unknown and drawn from some distribution D. We show that if D is a distribution
over subadditive valuations with independent items, then the better of pricing
each item separately or pricing only the grand bundle achieves a
constant-factor approximation to the revenue of the optimal mechanism. This
includes buyers who are k-demand, additive up to a matroid constraint, or
additive up to constraints of any downwards-closed set system (and whose values
for the individual items are sampled independently), as well as buyers who are
fractionally subadditive with item multipliers drawn independently. Our proof
makes use of the core-tail decomposition framework developed in prior work
showing similar results for the significantly simpler class of additive buyers
[LY13, BILW14].
In the second part of the paper, we develop a connection between
approximately optimal simple mechanisms and approximate revenue monotonicity
with respect to buyers' valuations. Revenue non-monotonicity is the phenomenon
that sometimes strictly increasing buyers' values for every set can strictly
decrease the revenue of the optimal mechanism [HR12]. Using our main result, we
derive a bound on how bad this degradation can be (and dub such a bound a proof
of approximate revenue monotonicity); we further show that better bounds on
approximate monotonicity imply a better analysis of our simple mechanisms.Comment: Updated title and body to version included in TEAC. Adapted Theorem
5.2 to accommodate \eta-BIC auctions (versus exactly BIC
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
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