775 research outputs found
Coverage, Matching, and Beyond: New Results on Budgeted Mechanism Design
We study a type of reverse (procurement) auction problems in the presence of
budget constraints. The general algorithmic problem is to purchase a set of
resources, which come at a cost, so as not to exceed a given budget and at the
same time maximize a given valuation function. This framework captures the
budgeted version of several well known optimization problems, and when the
resources are owned by strategic agents the goal is to design truthful and
budget feasible mechanisms, i.e. elicit the true cost of the resources and
ensure the payments of the mechanism do not exceed the budget. Budget
feasibility introduces more challenges in mechanism design, and we study
instantiations of this problem for certain classes of submodular and XOS
valuation functions. We first obtain mechanisms with an improved approximation
ratio for weighted coverage valuations, a special class of submodular functions
that has already attracted attention in previous works. We then provide a
general scheme for designing randomized and deterministic polynomial time
mechanisms for a class of XOS problems. This class contains problems whose
feasible set forms an independence system (a more general structure than
matroids), and some representative problems include, among others, finding
maximum weighted matchings, maximum weighted matroid members, and maximum
weighted 3D-matchings. For most of these problems, only randomized mechanisms
with very high approximation ratios were known prior to our results
Optimal Auctions vs. Anonymous Pricing: Beyond Linear Utility
The revenue optimal mechanism for selling a single item to agents with
independent but non-identically distributed values is complex for agents with
linear utility (Myerson,1981) and has no closed-form characterization for
agents with non-linear utility (cf. Alaei et al., 2012). Nonetheless, for
linear utility agents satisfying a natural regularity property, Alaei et al.
(2018) showed that simply posting an anonymous price is an e-approximation. We
give a parameterization of the regularity property that extends to agents with
non-linear utility and show that the approximation bound of anonymous pricing
for regular agents approximately extends to agents that satisfy this
approximate regularity property. We apply this approximation framework to prove
that anonymous pricing is a constant approximation to the revenue optimal
single-item auction for agents with public-budget utility, private-budget
utility, and (a special case of) risk-averse utility.Comment: Appeared at EC 201
Budget Feasible Mechanism Design: From Prior-Free to Bayesian
Budget feasible mechanism design studies procurement combinatorial auctions
where the sellers have private costs to produce items, and the
buyer(auctioneer) aims to maximize a social valuation function on subsets of
items, under the budget constraint on the total payment. One of the most
important questions in the field is "which valuation domains admit truthful
budget feasible mechanisms with `small' approximations (compared to the social
optimum)?" Singer showed that additive and submodular functions have such
constant approximations. Recently, Dobzinski, Papadimitriou, and Singer gave an
O(log^2 n)-approximation mechanism for subadditive functions; they also
remarked that: "A fundamental question is whether, regardless of computational
constraints, a constant-factor budget feasible mechanism exists for subadditive
functions."
We address this question from two viewpoints: prior-free worst case analysis
and Bayesian analysis. For the prior-free framework, we use an LP that
describes the fractional cover of the valuation function; it is also connected
to the concept of approximate core in cooperative game theory. We provide an
O(I)-approximation mechanism for subadditive functions, via the worst case
integrality gap I of LP. This implies an O(log n)-approximation for subadditive
valuations, O(1)-approximation for XOS valuations, and for valuations with a
constant I. XOS valuations are an important class of functions that lie between
submodular and subadditive classes. We give another polynomial time O(log
n/loglog n) sub-logarithmic approximation mechanism for subadditive valuations.
For the Bayesian framework, we provide a constant approximation mechanism for
all subadditive functions, using the above prior-free mechanism for XOS
valuations as a subroutine. Our mechanism allows correlations in the
distribution of private information and is universally truthful.Comment: to appear in STOC 201
Lottery pricing equilibria
We extend the notion of Combinatorial Walrasian Equilibrium, as defined by Feldman et al. [2013], to settings with budgets. When agents have budgets, the maximum social welfare as traditionally defined is not a suitable benchmark since it is overly optimistic. This motivated the liquid welfare of [Dobzinski and Paes Leme 2014] as an alternative. Observing that no combinatorial Walrasian equilibrium guarantees a non-zero fraction of the maximum liquid welfare in the absence of randomization, we instead work with randomized allocations and extend the notions of liquid welfare and Combinatorial Walrasian Equilibrium accordingly. Our generalization of the Combinatorial Walrasian Equilibrium prices lotteries over bundles of items rather than bundles, and we term it a lottery pricing equilibrium. Our results are two-fold. First, we exhibit an efficient algorithm which turns a randomized allocation with liquid expected welfare W into a lottery pricing equilibrium with liquid expected welfare 3-√5/2 W (≈ 0.3819-W). Next, given access to a demand oracle and an α-approximate oblivious rounding algorithm for the configuration linear program for the welfare maximization problem, we show how to efficiently compute a randomized allocation which is (a) supported on polynomially-many deterministic allocations and (b) obtains [nearly] an α fraction of the optimal liquid expected welfare. In the case of subadditive valuations, combining both results yields an efficient algorithm which computes a lottery pricing equilibrium obtaining a constant fraction of the optimal liquid expected welfare. © Copyright 2016 ACM
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