1,007 research outputs found
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
Economic Efficiency Requires Interaction
We study the necessity of interaction between individuals for obtaining
approximately efficient allocations. The role of interaction in markets has
received significant attention in economic thinking, e.g. in Hayek's 1945
classic paper.
We consider this problem in the framework of simultaneous communication
complexity. We analyze the amount of simultaneous communication required for
achieving an approximately efficient allocation. In particular, we consider two
settings: combinatorial auctions with unit demand bidders (bipartite matching)
and combinatorial auctions with subadditive bidders. For both settings we first
show that non-interactive systems have enormous communication costs relative to
interactive ones. On the other hand, we show that limited interaction enables
us to find approximately efficient allocations
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
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
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
Draft Auctions
We introduce draft auctions, which is a sequential auction format where at
each iteration players bid for the right to buy items at a fixed price. We show
that draft auctions offer an exponential improvement in social welfare at
equilibrium over sequential item auctions where predetermined items are
auctioned at each time step. Specifically, we show that for any subadditive
valuation the social welfare at equilibrium is an -approximation
to the optimal social welfare, where is the number of items. We also
provide tighter approximation results for several subclasses. Our welfare
guarantees hold for Bayes-Nash equilibria and for no-regret learning outcomes,
via the smooth-mechanism framework. Of independent interest, our techniques
show that in a combinatorial auction setting, efficiency guarantees of a
mechanism via smoothness for a very restricted class of cardinality valuations,
extend with a small degradation, to subadditive valuations, the largest
complement-free class of valuations. Variants of draft auctions have been used
in practice and have been experimentally shown to outperform other auctions.
Our results provide a theoretical justification
Tight Bounds for the Price of Anarchy of Simultaneous First Price Auctions
We study the Price of Anarchy of simultaneous first-price auctions for buyers
with submodular and subadditive valuations. The current best upper bounds for
the Bayesian Price of Anarchy of these auctions are e/(e-1) [Syrgkanis and
Tardos 2013] and 2 [Feldman et al. 2013], respectively. We provide matching
lower bounds for both cases even for the case of full information and for mixed
Nash equilibria via an explicit construction.
We present an alternative proof of the upper bound of e/(e-1) for first-price
auctions with fractionally subadditive valuations which reveals the worst-case
price distribution, that is used as a building block for the matching lower
bound construction.
We generalize our results to a general class of item bidding auctions that we
call bid-dependent auctions (including first-price auctions and all-pay
auctions) where the winner is always the highest bidder and each bidder's
payment depends only on his own bid.
Finally, we apply our techniques to discriminatory price multi-unit auctions.
We complement the results of [de Keijzer et al. 2013] for the case of
subadditive valuations, by providing a matching lower bound of 2. For the case
of submodular valuations, we provide a lower bound of 1.109. For the same class
of valuations, we were able to reproduce the upper bound of e/(e-1) using our
non-smooth approach.Comment: 37 pages, 5 figures, ACM Transactions on Economics and Computatio
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