37 research outputs found
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
On the Inefficiency of the Uniform Price Auction
We present our results on Uniform Price Auctions, one of the standard
sealed-bid multi-unit auction formats, for selling multiple identical units of
a single good to multi-demand bidders. Contrary to the truthful and
economically efficient multi-unit Vickrey auction, the Uniform Price Auction
encourages strategic bidding and is socially inefficient in general. The
uniform pricing rule is, however, widely popular by its appeal to the natural
anticipation, that identical items should be identically priced. In this work
we study equilibria of the Uniform Price Auction for bidders with (symmetric)
submodular valuation functions, over the number of units that they win. We
investigate pure Nash equilibria of the auction in undominated strategies; we
produce a characterization of these equilibria that allows us to prove that a
fraction 1-1/e of the optimum social welfare is always recovered in undominated
pure Nash equilibrium -- and this bound is essentially tight. Subsequently, we
study the auction under the incomplete information setting and prove a bound of
4-2/k on the economic inefficiency of (mixed) Bayes Nash equilibria that are
supported by undominated strategies.Comment: Additions and Improvements upon SAGT 2012 results (and minor
corrections on the previous version
A Unifying Hierarchy of Valuations with Complements and Substitutes
We introduce a new hierarchy over monotone set functions, that we refer to as
(Maximum over Positive Hypergraphs). Levels of the hierarchy
correspond to the degree of complementarity in a given function. The highest
level of the hierarchy, - (where is the total number of
items) captures all monotone functions. The lowest level, -,
captures all monotone submodular functions, and more generally, the class of
functions known as . Every monotone function that has a positive
hypergraph representation of rank (in the sense defined by Abraham,
Babaioff, Dughmi and Roughgarden [EC 2012]) is in -. Every
monotone function that has supermodular degree (in the sense defined by
Feige and Izsak [ITCS 2013]) is in -. In both cases, the
converse direction does not hold, even in an approximate sense. We present
additional results that demonstrate the expressiveness power of
-.
One can obtain good approximation ratios for some natural optimization
problems, provided that functions are required to lie in low levels of the
hierarchy. We present two such applications. One shows that the
maximum welfare problem can be approximated within a ratio of if all
players hold valuation functions in -. The other is an upper
bound of on the price of anarchy of simultaneous first price auctions.
Being in - can be shown to involve two requirements -- one
is monotonicity and the other is a certain requirement that we refer to as
(Positive Lower Envelope). Removing the monotonicity
requirement, one obtains the hierarchy over all non-negative
set functions (whether monotone or not), which can be fertile ground for
further research
On the Complexity of Computing an Equilibrium in Combinatorial Auctions
We study combinatorial auctions where each item is sold separately but
simultaneously via a second price auction. We ask whether it is possible to
efficiently compute in this game a pure Nash equilibrium with social welfare
close to the optimal one.
We show that when the valuations of the bidders are submodular, in many
interesting settings (e.g., constant number of bidders, budget additive
bidders) computing an equilibrium with good welfare is essentially as easy as
computing, completely ignoring incentives issues, an allocation with good
welfare. On the other hand, for subadditive valuations, we show that computing
an equilibrium requires exponential communication. Finally, for XOS (a.k.a.
fractionally subadditive) valuations, we show that if there exists an efficient
algorithm that finds an equilibrium, it must use techniques that are very
different from our current ones
On the Efficiency of All-Pay Mechanisms
We study the inefficiency of mixed equilibria, expressed as the price of
anarchy, of all-pay auctions in three different environments: combinatorial,
multi-unit and single-item auctions. First, we consider item-bidding
combinatorial auctions where m all-pay auctions run in parallel, one for each
good. For fractionally subadditive valuations, we strengthen the upper bound
from 2 [Syrgkanis and Tardos STOC'13] to 1.82 by proving some structural
properties that characterize the mixed Nash equilibria of the game. Next, we
design an all-pay mechanism with a randomized allocation rule for the multi-
unit auction. We show that, for bidders with submodular valuations, the
mechanism admits a unique, 75% efficient, pure Nash equilibrium. The efficiency
of this mechanism outperforms all the known bounds on the price of anarchy of
mechanisms used for multi-unit auctions. Finally, we analyze single-item
all-pay auctions motivated by their connection to contests and show tight
bounds on the price of anarchy of social welfare, revenue and maximum bid.Comment: 26 pages, 2 figures, European Symposium on Algorithms(ESA) 201