234 research outputs found
Revealed cardinal preference
I prove that as long as we allow the marginal utility for money (lambda) to
vary between purchases (similarly to the budget) then the quasi-linear and
the ordinal budget-constrained models rationalize the same data. However, we know that lambda is approximately constant. I provide a simple constructive proof for the necessary and sufficient condition for the constant lambda rationalization, which I argue should replace the Generalized Axiom of
Revealed Preference in empirical studies of consumer behavior.
'Go Cardinals!'
It is the minimal requirement of any scientifi c theory that it is consistent with
the data it is trying to explain. In the case of (Hicksian) consumer theory it was
revealed preference -introduced by Samuelson (1938,1948) - that provided an
empirical test to satisfy this need. At that time most of economic reasoning was
done in terms of a competitive general equilibrium, a concept abstract enough
so that it can be built on the ordinal preferences over baskets of goods - even if
the extremely specialized ones of Arrow and Debreu. However, starting in the
sixties, economics has moved beyond the 'invisible hand' explanation of how
-even competitive- markets operate. A seemingly unavoidable step of this
'revolution' was that ever since, most economic research has been carried out
in a partial equilibrium context. Now, the partial equilibrium approach does
not mean that the rest of the markets are ignored, rather that they are held
constant. In other words, there is a special commodity -call it money - that
reflects the trade-offs of moving purchasing power across markets. As a result,
the basic building block of consumer behavior in partial equilibrium is no longer
the consumer's preferences over goods, rather her valuation of them, in terms
of money. This new paradigm necessitates a new theory of revealed preference
Quantum correlations from local amplitudes and the resolution of the Einstein-Podolsky-Rosen nonlocality puzzle
The Einstein-Podolsky-Rosen nonlocality puzzle has been recognized as one of
the most important unresolved issues in the foundational aspects of quantum
mechanics. We show that the problem is resolved if the quantum correlations are
calculated directly from local quantities which preserve the phase information
in the quantum system. We assume strict locality for the probability amplitudes
instead of local realism for the outcomes, and calculate an amplitude
correlation function.Then the experimentally observed correlation of outcomes
is calculated from the square of the amplitude correlation function. Locality
of amplitudes implies that the measurement on one particle does not collapse
the companion particle to a definite state. Apart from resolving the EPR
puzzle, this approach shows that the physical interpretation of apparently
`nonlocal' effects like quantum teleportation and entanglement swapping are
different from what is usually assumed. Bell type measurements do not change
distant states. Yet the correlations are correctly reproduced, when measured,
if complex probability amplitudes are treated as the basic local quantities. As
examples we discuss the quantum correlations of two-particle maximally
entangled states and the three-particle GHZ entangled state.Comment: Std. Latex, 11 pages, 1 table. Prepared for presentation at the
International Conference on Quantum Optics, ICQO'2000, Minsk, Belaru
Constructive updating/downdating of oblique projectors: a generalization of the Gram-Schmidt process
A generalization of the Gram-Schmidt procedure is achieved by providing
equations for updating and downdating oblique projectors. The work is motivated
by the problem of adaptive signal representation outside the orthogonal basis
setting. The proposed techniques are shown to be relevant to the problem of
discriminating signals produced by different phenomena when the order of the
signal model needs to be adjusted.Comment: As it will appear in Journal of Physics A: Mathematical and
Theoretical (2007
Testing Consumer Rationality using Perfect Graphs and Oriented Discs
Given a consumer data-set, the axioms of revealed preference proffer a binary
test for rational behaviour. A natural (non-binary) measure of the degree of
rationality exhibited by the consumer is the minimum number of data points
whose removal induces a rationalisable data-set.We study the computational
complexity of the resultant consumer rationality problem in this paper. This
problem is, in the worst case, equivalent (in terms of approximation) to the
directed feedback vertex set problem. Our main result is to obtain an exact
threshold on the number of commodities that separates easy cases and hard
cases. Specifically, for two-commodity markets the consumer rationality problem
is polynomial time solvable; we prove this via a reduction to the vertex cover
problem on perfect graphs. For three-commodity markets, however, the problem is
NP-complete; we prove thisusing a reduction from planar 3-SAT that is based
upon oriented-disc drawings
The Combinatorial World (of Auctions) According to GARP
Revealed preference techniques are used to test whether a data set is
compatible with rational behaviour. They are also incorporated as constraints
in mechanism design to encourage truthful behaviour in applications such as
combinatorial auctions. In the auction setting, we present an efficient
combinatorial algorithm to find a virtual valuation function with the optimal
(additive) rationality guarantee. Moreover, we show that there exists such a
valuation function that both is individually rational and is minimum (that is,
it is component-wise dominated by any other individually rational, virtual
valuation function that approximately fits the data). Similarly, given upper
bound constraints on the valuation function, we show how to fit the maximum
virtual valuation function with the optimal additive rationality guarantee. In
practice, revealed preference bidding constraints are very demanding. We
explain how approximate rationality can be used to create relaxed revealed
preference constraints in an auction. We then show how combinatorial methods
can be used to implement these relaxed constraints. Worst/best-case welfare
guarantees that result from the use of such mechanisms can be quantified via
the minimum/maximum virtual valuation function
Spin-dependent Bohm trajectories associated with an electronic transition in hydrogen
The Bohm causal theory of quantum mechanics with spin-dependence is used to
determine electron trajectories when a hydrogen atom is subjected to
(semi-classical) radiation. The transition between the 1s ground state and the
2p0 state is examined. It is found that transitions can be identified along
Bohm trajectories. The trajectories lie on invariant hyperboloid surfaces of
revolution in R^3. The energy along the trajectories is also discussed in
relation to the hydrogen energy eigenvalues.Comment: 18 pages, 8 figure
Weylâs gauge argument
The standard U(1) âgauge principleâ or âgauge argumentâ produces an exact potential A=dλ and a vanishing field F=ddλ=0. Weyl has his own gauge argument, which is sketchy, archaic and hard to follow; but at least it produces an inexact potential A and a nonvanishing field F=dAâ 0. I attempt a reconstruction
Nonlocality, Bell's Ansatz and Probability
Quantum Mechanics lacks an intuitive interpretation, which is the cause of a
generally formalistic approach to its use. This in turn has led to a certain
insensitivity to the actual meaning of many words used in its description and
interpretation. Herein, we analyze carefully the possible mathematical meanings
of those terms used in analysis of EPR's contention, that Quantum Mechanics is
incomplete, as well as Bell's work descendant therefrom. As a result, many
inconsistencies and errors in contemporary discussions of nonlocality, as well
as in Bell's Ansatz with respect to the laws of probability, are identified.
Evading these errors precludes serious conflicts between Quantum Mechanics and
both Special Relativity and Philosophy.Comment: 8&1/2 pages revtex; v2: many corrections, clairifications &
extentions, all small; v3: editorial scru
A realist view of the canonical EPRB experiment based on quantum theory and its consequences
A realist view of the Einstein-Podolsky-Rosen-Bohm experiment with spins
based on quantum theory is presented. This view implies that there is no action
at a distance. It also implies that the measurement result A (B) for particle 1
(2) depends on both magnet angles, and hence the probability of obtaining the
result A (B) also depends on both magnet angles. In light of these realist
implications, it is clear that what is wrong at least with local realistic
theory is not the locality or no action-at-a-distance assumption itself but
rather the formal implementation of that assumption.Comment: To be published in the Journal of Optics B as part of the
proceeedings of the Garda 2001 Workshop on Mysteries, Puzzles and Paradoxes
in Quantum Mechanic
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