2,258 research outputs found
Budget feasible mechanisms on matroids
Motivated by many practical applications, in this paper we study budget feasible mechanisms where the goal is to procure independent sets from matroids. More specifically, we are given a matroid =(,) where each ground (indivisible) element is a selfish agent. The cost of each element (i.e., for selling the item or performing a service) is only known to the element itself. There is a buyer with a budget having additive valuations over the set of elements E. The goal is to design an incentive compatible (truthful) budget feasible mechanism which procures an independent set of the matroid under the given budget that yields the largest value possible to the buyer. Our result is a deterministic, polynomial-time, individually rational, truthful and budget feasible mechanism with 4-approximation to the optimal independent set. Then, we extend our mechanism to the setting of matroid intersections in which the goal is to procure common independent sets from multiple matroids. We show that, given a polynomial time deterministic blackbox that returns -approximation solutions to the matroid intersection problem, there exists a deterministic, polynomial time, individually rational, truthful and budget feasible mechanism with (3+1) -approximation to the optimal common independent set
Truthful Multi-unit Procurements with Budgets
We study procurement games where each seller supplies multiple units of his
item, with a cost per unit known only to him. The buyer can purchase any number
of units from each seller, values different combinations of the items
differently, and has a budget for his total payment.
For a special class of procurement games, the {\em bounded knapsack} problem,
we show that no universally truthful budget-feasible mechanism can approximate
the optimal value of the buyer within , where is the total number of
units of all items available. We then construct a polynomial-time mechanism
that gives a -approximation for procurement games with {\em concave
additive valuations}, which include bounded knapsack as a special case. Our
mechanism is thus optimal up to a constant factor. Moreover, for the bounded
knapsack problem, given the well-known FPTAS, our results imply there is a
provable gap between the optimization domain and the mechanism design domain.
Finally, for procurement games with {\em sub-additive valuations}, we
construct a universally truthful budget-feasible mechanism that gives an
-approximation in polynomial time with a
demand oracle.Comment: To appear at WINE 201
Revenue Maximization and Ex-Post Budget Constraints
We consider the problem of a revenue-maximizing seller with m items for sale
to n additive bidders with hard budget constraints, assuming that the seller
has some prior distribution over bidder values and budgets. The prior may be
correlated across items and budgets of the same bidder, but is assumed
independent across bidders. We target mechanisms that are Bayesian Incentive
Compatible, but that are ex-post Individually Rational and ex-post budget
respecting. Virtually no such mechanisms are known that satisfy all these
conditions and guarantee any revenue approximation, even with just a single
item. We provide a computationally efficient mechanism that is a
-approximation with respect to all BIC, ex-post IR, and ex-post budget
respecting mechanisms. Note that the problem is NP-hard to approximate better
than a factor of 16/15, even in the case where the prior is a point mass
\cite{ChakrabartyGoel}. We further characterize the optimal mechanism in this
setting, showing that it can be interpreted as a distribution over virtual
welfare maximizers.
We prove our results by making use of a black-box reduction from mechanism to
algorithm design developed by \cite{CaiDW13b}. Our main technical contribution
is a computationally efficient -approximation algorithm for the algorithmic
problem that results by an application of their framework to this problem. The
algorithmic problem has a mixed-sign objective and is NP-hard to optimize
exactly, so it is surprising that a computationally efficient approximation is
possible at all. In the case of a single item (), the algorithmic problem
can be solved exactly via exhaustive search, leading to a computationally
efficient exact algorithm and a stronger characterization of the optimal
mechanism as a distribution over virtual value maximizers
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
Composable and Efficient Mechanisms
We initiate the study of efficient mechanism design with guaranteed good
properties even when players participate in multiple different mechanisms
simultaneously or sequentially. We define the class of smooth mechanisms,
related to smooth games defined by Roughgarden, that can be thought of as
mechanisms that generate approximately market clearing prices. We show that
smooth mechanisms result in high quality outcome in equilibrium both in the
full information setting and in the Bayesian setting with uncertainty about
participants, as well as in learning outcomes. Our main result is to show that
such mechanisms compose well: smoothness locally at each mechanism implies
efficiency globally.
For mechanisms where good performance requires that bidders do not bid above
their value, we identify the notion of a weakly smooth mechanism. Weakly smooth
mechanisms, such as the Vickrey auction, are approximately efficient under the
no-overbidding assumption. Similar to smooth mechanisms, weakly smooth
mechanisms behave well in composition, and have high quality outcome in
equilibrium (assuming no overbidding) both in the full information setting and
in the Bayesian setting, as well as in learning outcomes.
In most of the paper we assume participants have quasi-linear valuations. We
also extend some of our results to settings where participants have budget
constraints
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
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