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Time functions as utilities
, 2009
"... Every time function on spacetime gives a (continuous) total preordering of the spacetime events which respects the notion of causal precedence. The problem of the existence of a (semi)time function on spacetime and the problem of recovering the causal structure starting from the set of time funct ..."
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Cited by 7 (5 self)
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Every time function on spacetime gives a (continuous) total preordering of the spacetime events which respects the notion of causal precedence. The problem of the existence of a (semi)time function on spacetime and the problem of recovering the causal structure starting from the set of time functions are studied. It is pointed out that these problems have an analog in the field of microeconomics known as utility theory. In a chronological spacetime the semitime functions correspond to the utilities for the chronological relation, while in a Kcausal (stably causal) spacetime the time functions correspond to the utilities for the K + relation (Seifert’s relation). By exploiting this analogy, we are able to import some mathematical results, most notably Peleg’s and Levin’s theorems, to the spacetime framework. As a consequence, we prove that a Kcausal (i.e. stably causal) spacetime admits a time function and that the time or temporal functions can be used to recover the K + (or Seifert) relation which indeed turns out to be the intersection of the time or temporal orderings. This result tells us in which circumstances it is possible to recover the chronological or causal relation starting from the set of time or temporal functions allowed by the spacetime. Moreover, it is proved that a chronological spacetime in which the closure of the causal relation is transitive (for instance a reflective spacetime) admits a semitime function. Along the way a new proof avoiding smoothing techniques is given that the existence of a time function implies stable causality, and a new short proof of the equivalence between Kcausality and stable causality is given which takes advantage of Levin’s theorem and smoothing techniques.
Topological conditions for the representation of preorders by continuous utilities
, 2012
"... We remove the Hausdorff condition from Levin’s theorem on the representation of preorders by families of continuous utilities. We compare some alternative topological assumptions in a Levin’s type theorem, and show that they are equivalent to a Polish space assumption. ..."
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Cited by 3 (3 self)
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We remove the Hausdorff condition from Levin’s theorem on the representation of preorders by families of continuous utilities. We compare some alternative topological assumptions in a Levin’s type theorem, and show that they are equivalent to a Polish space assumption.
Essential Supremum with respect to a random partial order. Preprint available at hal.archivesouvertes.fr/hal00608856
"... Abstract Inspired by the theory of financial markets with transaction costs, we study a concept of essential supremum in the framework where a random partial In contrast to the classical definition, we define the essential supremum as a subset of random variables satisfying some natural properties. ..."
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Cited by 2 (2 self)
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Abstract Inspired by the theory of financial markets with transaction costs, we study a concept of essential supremum in the framework where a random partial In contrast to the classical definition, we define the essential supremum as a subset of random variables satisfying some natural properties. Applications of the introduced notion to a hedging problem under transaction costs and setvalued dynamic risk measures are given.
Conditional Preference Orders and their Numeri cal Representations
, 2014
"... This work provides an axiomatic framework to the concept of conditional preference orders based on conditional sets. Conditional numerical representations of such preference orders are introduced and a conditional version of the theorems of Debreu about the existence of such numerical representati ..."
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This work provides an axiomatic framework to the concept of conditional preference orders based on conditional sets. Conditional numerical representations of such preference orders are introduced and a conditional version of the theorems of Debreu about the existence of such numerical representations is given. The continuous representations follow from a conditional version of Debreu’s Gap Lemma the proof of which is free of any measurable selection arguments but is derived from the existence of a conditional axiom of choice.
A Unified Approach to Revealed Preference Theory: The Case of Rational Choice
, 2014
"... The theoretical literature on (nonrandom) choice largely follows the route of Richter (1966) by working in abstract environments and by stipulating that we see all choices of an agent from a given feasible set. On the other hand, empirical work on consumption choice using revealed preference analys ..."
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The theoretical literature on (nonrandom) choice largely follows the route of Richter (1966) by working in abstract environments and by stipulating that we see all choices of an agent from a given feasible set. On the other hand, empirical work on consumption choice using revealed preference analysis follows the approach of Afriat (1967), which assumes that we observe only one (and not necessarily all) of the potential choices of an agent. These two approaches are structurally di¤erent and are treated in the literature in isolation from each other. This paper introduces a framework in which both approaches can be formulated in tandem. We prove a rationalizability theorem in this framework that simultaneously generalizes the results of Afriat and Richter. This approach also gives a new, tight version of Afriats Theorem and a continuous version of Richters Theorem, and leads to a number of novel observations for the theory of consumer demand.