26,566 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
Generalised Reichenbachian common cause systems
The principle of the common cause claims that if an improbable coincidence has occurred, there must exist a common cause. This is generally taken to mean that positive correlations between non-causally related events should disappear when conditioning on the action of some underlying common cause. The extended interpretation of the principle, by contrast, urges that common causes should be called for in order to explain positive deviations between the estimated correlation of two events and the expected value of their correlation. The aim of this paper is to provide the extended reading of the principle with a general probabilistic model, capturing the simultaneous action of a system of multiple common causes. To this end, two distinct models are elaborated, and the necessary and sufficient conditions for their existence are determined
Involutions on the Algebra of Physical Observables From Reality Conditions
Some aspects of the algebraic quantization programme proposed by Ashtekar are
revisited in this article. It is proved that, for systems with first-class
constraints, the involution introduced on the algebra of quantum operators via
reality conditions can never be projected unambiguously to the algebra of
physical observables, ie, of quantum observables modulo constraints. It is
nevertheless shown that, under sufficiently general assumptions, one can still
induce an involution on the algebra of physical observables from reality
conditions, though the involution obtained depends on the choice of particular
representatives for the equivalence classes of quantum observables and this
implies an additional ambiguity in the quantization procedure suggested by
Ashtekar.Comment: 19 pages, latex, no figure
Generalised Reichenbachian Common Cause Systems
The principle of the common cause claims that if an improbable coincidence
has occurred, there must exist a common cause. This is generally taken to mean
that positive correlations between non-causally related events should disappear
when conditioning on the action of some underlying common cause. The extended
interpretation of the principle, by contrast, urges that common causes should
be called for in order to explain positive deviations between the estimated
correlation of two events and the expected value of their correlation. The aim
of this paper is to provide the extended reading of the principle with a
general probabilistic model, capturing the simultaneous action of a system of
multiple common causes. To this end, two distinct models are elaborated, and
the necessary and sufficient conditions for their existence are determined
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