9 research outputs found
No-Signalling Is Equivalent To Free Choice of Measurements
No-Signalling is a fundamental constraint on the probabilistic predictions
made by physical theories. It is usually justified in terms of the constraints
imposed by special relativity. However, this justification is not as clear-cut
as is usually supposed. We shall give a different perspective on this condition
by showing an equivalence between No-Signalling and Lambda Independence, or
"free choice of measurements", a condition on hidden-variable theories which is
needed to make no-go theorems such as Bell's theorem non-trivial. More
precisely, we shall show that a probability table describing measurement
outcomes is No-Signalling if and only if it can be realized by a
Lambda-Independent hidden-variable theory of a particular canonical form, in
which the hidden variables correspond to non-contextual deterministic
predictions of measurement outcomes. The key proviso which avoids contradiction
with Bell's theorem is that we consider hidden-variable theories with signed
probability measures over the hidden variables - i.e. negative probabilities.
Negative probabilities have often been discussed in the literature on quantum
mechanics. We use a result proved previously in "The Sheaf-theoretic Structure
of Locality and Contextuality" by Abramsky and Brandenburger, which shows that
they give rise to, and indeed characterize, the entire class of No-Signalling
behaviours. In the present paper, we put this result in a broader context,
which reveals the surprising consequence that the No-Signalling condition is
equivalent to the apparently completely different notion of free choice of
measurements.Comment: In Proceedings QPL 2013, arXiv:1412.791
Characterizations of smooth ambiguity based on continuous and discrete data
In the Anscombe-Aumann setup, we provide conditions for a collection of observations to be consistent with a well-known class of smooth ambiguity preferences (Klibanoff P, Marinacci M, Mukerji S (2005) A smooth model of decision making under ambiguity. Econometrica 73(6): 1849-1892.). Each observation is assumed to take the form of an equivalence between an uncertain act and a certain outcome. We provide three results that describe these conditions for data sets of different cardinality. Our findings uncover surprising links between the smooth ambiguity model and classic mathematical results in complex and functional analysis
Subjective Contingencies and Limited Bayesian Updating
We depart from Savage’s (1954) common state space assumption and introduce a model that allows for a subjective understanding of uncertainty. Within the revealed preference paradigm, we uniquely identify the agent’s subjective state space via her preferences conditional on incoming information. According to our representation, the agent’s subjective contingencies are coarser than the analyst’s states; she uses an additively separable utility with respect to her set of contingencies; and she adopts an updating rule that follows the Bayesian spirit but is limited by her perception of uncertainty. We illustrate our theory with an application to the Confirmatory Bias
Preferences with grades of indecisiveness
International audienceDeparting from the traditional approach of modeling indecisiveness based on the weakening of the completeness axiom, we introduce the notion of graded preferences: The agent is characterized by a binary relation over (ordered) pairs of alternatives, which allows her to express her inclination to prefer one alternative over another and her confidence in the relative superiority of the indicated alternative. In the classical Anscombe–Aumann framework, we derive a representation of a graded preference by a measure of the set of beliefs that rank one option better than the other. Our model is a refinement of Bewley's [6] model of Knightian uncertainty: It is based on the same object of representation — the set of beliefs — but provides more information about how the agent compares alternatives