13,704 research outputs found
Optimal Competitive Auctions
We study the design of truthful auctions for selling identical items in
unlimited supply (e.g., digital goods) to n unit demand buyers. This classic
problem stands out from profit-maximizing auction design literature as it
requires no probabilistic assumptions on buyers' valuations and employs the
framework of competitive analysis. Our objective is to optimize the worst-case
performance of an auction, measured by the ratio between a given benchmark and
revenue generated by the auction.
We establish a sufficient and necessary condition that characterizes
competitive ratios for all monotone benchmarks. The characterization identifies
the worst-case distribution of instances and reveals intrinsic relations
between competitive ratios and benchmarks in the competitive analysis. With the
characterization at hand, we show optimal competitive auctions for two natural
benchmarks.
The most well-studied benchmark measures the
envy-free optimal revenue where at least two buyers win. Goldberg et al. [13]
showed a sequence of lower bounds on the competitive ratio for each number of
buyers n. They conjectured that all these bounds are tight. We show that
optimal competitive auctions match these bounds. Thus, we confirm the
conjecture and settle a central open problem in the design of digital goods
auctions. As one more application we examine another economically meaningful
benchmark, which measures the optimal revenue across all limited-supply Vickrey
auctions. We identify the optimal competitive ratios to be
for each number of buyers n, that is as
approaches infinity
Core-competitive Auctions
One of the major drawbacks of the celebrated VCG auction is its low (or zero)
revenue even when the agents have high value for the goods and a {\em
competitive} outcome could have generated a significant revenue. A competitive
outcome is one for which it is impossible for the seller and a subset of buyers
to `block' the auction by defecting and negotiating an outcome with higher
payoffs for themselves. This corresponds to the well-known concept of {\em
core} in cooperative game theory.
In particular, VCG revenue is known to be not competitive when the goods
being sold have complementarities. A bottleneck here is an impossibility result
showing that there is no auction that simultaneously achieves competitive
prices (a core outcome) and incentive-compatibility.
In this paper we try to overcome the above impossibility result by asking the
following natural question: is it possible to design an incentive-compatible
auction whose revenue is comparable (even if less) to a competitive outcome?
Towards this, we define a notion of {\em core-competitive} auctions. We say
that an incentive-compatible auction is -core-competitive if its
revenue is at least fraction of the minimum revenue of a
core-outcome. We study the Text-and-Image setting. In this setting, there is an
ad slot which can be filled with either a single image ad or text ads. We
design an core-competitive randomized auction and an
competitive deterministic auction for the Text-and-Image
setting. We also show that both factors are tight
On some methods of construction of invariant normalizations of lightlike hypersurfaces
The authors study the geometry of lightlike hypersurfaces on
pseudo-Riemannian manifolds of Lorentzian signature. Such
hypersurfaces are of interest in general relativity since they can be models of
different types of physical horizons. For a lightlike hypersurface of general type and for some special lightlike hypersurfaces (namely,
for totally umbilical and belonging to a manifold of constant
curvature), in a third-order neighborhood of a point , the authors
construct invariant normalizations intrinsically connected with the geometry of
and investigate affine connections induced by these normalizations. For
this construction, they used relative and absolute invariants defined by the
first and second fundamental forms of . The authors show that if , their methods allow to construct three invariant normalizations and affine
connections intrinsically connected with the geometry of . Such a
construction is given in the present paper for the first time. The authors also
consider the fibration of isotropic geodesics of and investigate their
singular points and singular submanifolds.Comment: LaTeX, 25 page
Conformal and Grassmann structures
The main results on the theory of conformal and almost Grassmann structures
are presented. The common properties of these structures and also the
differences between them are outlined. In particular, the structure groups of
these structures and their differential prolongations are found. A complete
system of geometric objects of the almost Grassmann structure totally defining
its geometric structure is determined. The vanishing of these objects
determines a locally Grassmann manifold. It is proved that the integrable
almost Grassmann structures are locally Grassmann. The criteria of
semiintegrability of almost Grassmann structures is proved in invariant form.Comment: LaTeX, 25 page
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