8 research outputs found
A Bridge between Liquid and Social Welfare in Combinatorial Auctions with Submodular Bidders
We study incentive compatible mechanisms for Combinatorial Auctions where the
bidders have submodular (or XOS) valuations and are budget-constrained. Our
objective is to maximize the \emph{liquid welfare}, a notion of efficiency for
budget-constrained bidders introduced by Dobzinski and Paes Leme (2014). We
show that some of the known truthful mechanisms that best-approximate the
social welfare for Combinatorial Auctions with submodular bidders through
demand query oracles can be adapted, so that they retain truthfulness and
achieve asymptotically the same approximation guarantees for the liquid
welfare. More specifically, for the problem of optimizing the liquid welfare in
Combinatorial Auctions with submodular bidders, we obtain a universally
truthful randomized -approximate mechanism, where is the number
of items, by adapting the mechanism of Krysta and V\"ocking (2012).
Additionally, motivated by large market assumptions often used in mechanism
design, we introduce a notion of competitive markets and show that in such
markets, liquid welfare can be approximated within a constant factor by a
randomized universally truthful mechanism. Finally, in the Bayesian setting, we
obtain a truthful -approximate mechanism for the case where bidder
valuations are generated as independent samples from a known distribution, by
adapting the results of Feldman, Gravin and Lucier (2014).Comment: AAAI-1
On Simultaneous Two-player Combinatorial Auctions
We consider the following communication problem: Alice and Bob each have some
valuation functions and over subsets of items,
and their goal is to partition the items into in a way that
maximizes the welfare, . We study both the allocation
problem, which asks for a welfare-maximizing partition and the decision
problem, which asks whether or not there exists a partition guaranteeing
certain welfare, for binary XOS valuations. For interactive protocols with
communication, a tight 3/4-approximation is known for both
[Fei06,DS06].
For interactive protocols, the allocation problem is provably harder than the
decision problem: any solution to the allocation problem implies a solution to
the decision problem with one additional round and additional bits of
communication via a trivial reduction. Surprisingly, the allocation problem is
provably easier for simultaneous protocols. Specifically, we show:
1) There exists a simultaneous, randomized protocol with polynomial
communication that selects a partition whose expected welfare is at least
of the optimum. This matches the guarantee of the best interactive, randomized
protocol with polynomial communication.
2) For all , any simultaneous, randomized protocol that
decides whether the welfare of the optimal partition is or correctly with probability requires
exponential communication. This provides a separation between the attainable
approximation guarantees via interactive () versus simultaneous () protocols with polynomial communication.
In other words, this trivial reduction from decision to allocation problems
provably requires the extra round of communication
Separating the Communication Complexity of Truthful and Non-Truthful Combinatorial Auctions
We provide the first separation in the approximation guarantee achievable by
truthful and non-truthful combinatorial auctions with polynomial communication.
Specifically, we prove that any truthful mechanism guaranteeing a
-approximation for two buyers with XOS
valuations over items requires
communication, whereas a non-truthful algorithm by Dobzinski and Schapira [SODA
2006] and Feige [2009] is already known to achieve a
-approximation in communication.
We obtain our separation by proving that any {simultaneous} protocol ({not}
necessarily truthful) which guarantees a
-approximation requires communication
. The taxation complexity framework of
Dobzinski [FOCS 2016] extends this lower bound to all truthful mechanisms
(including interactive truthful mechanisms)
Computational Efficiency Requires Simple Taxation
We characterize the communication complexity of truthful mechanisms. Our
departure point is the well known taxation principle. The taxation principle
asserts that every truthful mechanism can be interpreted as follows: every
player is presented with a menu that consists of a price for each bundle (the
prices depend only on the valuations of the other players). Each player is
allocated a bundle that maximizes his profit according to this menu. We define
the taxation complexity of a truthful mechanism to be the logarithm of the
maximum number of menus that may be presented to a player.
Our main finding is that in general the taxation complexity essentially
equals the communication complexity. The proof consists of two main steps.
First, we prove that for rich enough domains the taxation complexity is at most
the communication complexity. We then show that the taxation complexity is much
smaller than the communication complexity only in "pathological" cases and
provide a formal description of these extreme cases.
Next, we study mechanisms that access the valuations via value queries only.
In this setting we establish that the menu complexity -- a notion that was
already studied in several different contexts -- characterizes the number of
value queries that the mechanism makes in exactly the same way that the
taxation complexity characterizes the communication complexity.
Our approach yields several applications, including strengthening the
solution concept with low communication overhead, fast computation of prices,
and hardness of approximation by computationally efficient truthful mechanisms