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34
Towards a unified theory of imprecise probability
 Int. J. Approx. Reasoning
, 2000
"... Belief functions, possibility measures and Choquet capacities of order 2, which are special kinds of coherent upper or lower probability, are amongst the most popular mathematical models for uncertainty and partial ignorance. I give examples to show that these models are not sufficiently general to ..."
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Cited by 64 (0 self)
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Belief functions, possibility measures and Choquet capacities of order 2, which are special kinds of coherent upper or lower probability, are amongst the most popular mathematical models for uncertainty and partial ignorance. I give examples to show that these models are not sufficiently general to represent some common types of uncertainty. Coherent lower previsions and sets of probability measures are considerably more general but they may not be sufficiently informative for some purposes. I discuss two other models for uncertainty, involving sets of desirable gambles and partial preference orderings. These are more informative and more general than the previous models, and they may provide a suitable mathematical setting for a unified theory of imprecise probability.
Expected Utility Theory without the Completeness Axiom
 JOURNAL OF ECONOMIC THEORY, VOLUME 115, ISSUE 1
, 2004
"... We study the problem of obtaining an expected utility representation for a potentially incomplete preference relation over lotteriesby meansof a set of von Neumann–Morgenstern utility functions. It is shown that, when the prize space is a compact metric space, a preference relation admits such a mul ..."
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Cited by 48 (8 self)
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We study the problem of obtaining an expected utility representation for a potentially incomplete preference relation over lotteriesby meansof a set of von Neumann–Morgenstern utility functions. It is shown that, when the prize space is a compact metric space, a preference relation admits such a multiutility representation provided that it satisfies the standard axioms of expected utility theory. Moreover, the representing set of utilities is unique in a welldefined sense.
The Value of Using Imprecise Probabilities in Engineering Design.
 Journal of Mechanical Design,
, 2006
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Anchored Preference Relations
, 2003
"... This paper explores the implications of a simple and intuitive restriction on reference dependent preferences assuming the status quo serves as the reference point. The condition imposed rules out situations in which a decision maker has a choice between two prospects, selects one, subsequently chan ..."
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Cited by 25 (1 self)
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This paper explores the implications of a simple and intuitive restriction on reference dependent preferences assuming the status quo serves as the reference point. The condition imposed rules out situations in which a decision maker has a choice between two prospects, selects one, subsequently changes her mind and selects the other – even if the change is costly. It is shown that a surprising number of models in a riskless and risky setting violate this behavioral assumption, including Cumulative Prospect Theory as well as any theory exhibiting local nonsatiation in which all reference dependent indifference surfaces are smooth. It is also shown that one can construct simple alternative models that do satisfy the condition, axiomatically derived or otherwise. These alternative theories take the form of maxmin representations over a set of expected (or Choquetexpected) utility differences, where utility difference is measured between the prospect evaluated and the reference point.
A contrast between two decision rules for use with (convex) sets of probabilities: ΓMaximin versus Eadmissibilty.
, 2002
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Extensions of Expected Utility Theory and Some Limitations of Pairwise Comparisons
 In Proceedings of the Third ISIPTA (JM
, 2003
"... We contrast three decision rules that extend Expected Utility to contexts where a convex set of probabilities is used to depict uncertainty: #Maximin, Maximality, and Eadmissibility. The rules extend Expected Utility theory as they require that an option is inadmissible if there is another that ..."
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Cited by 15 (2 self)
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We contrast three decision rules that extend Expected Utility to contexts where a convex set of probabilities is used to depict uncertainty: #Maximin, Maximality, and Eadmissibility. The rules extend Expected Utility theory as they require that an option is inadmissible if there is another that carries greater expected utility for each probability in a (closed) convex set. If the convex set is a singleton, then each rule agrees with maximizing expected utility. We show that, even when the option set is convex, this pairwise comparison between acts may fail to identify those acts which are Bayes for some probability in a convex set that is not closed. This limitation affects two of the decision rules but not Eadmissibility, which is not a pairwise decision rule. Eadmissibility can be used to distinguish between two convex sets of probabilities that intersect all the same supporting hyperplanes.
Shape of Incomplete Preferences.
 Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications, Carleton Scienti c Proceedings in Informatics
, 2003
"... Abstract Incomplete preferences provide the epistemic foundation for models of imprecise subjective probabilities and utilities that are used in robust Bayesian analysis and in theories of bounded rationality. This paper presents a simple axiomatization of incomplete preferences and characterizes t ..."
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Cited by 13 (2 self)
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Abstract Incomplete preferences provide the epistemic foundation for models of imprecise subjective probabilities and utilities that are used in robust Bayesian analysis and in theories of bounded rationality. This paper presents a simple axiomatization of incomplete preferences and characterizes the shape of their representing sets of probabilities and utilities. Deletion of the completeness assumption from the axiom system of Anscombe and Aumann yields preferences represented by a convex set of statedependent expected utilities, of which at least one must be a probability/utility pair. A strengthening of the stateindependence axiom is needed to obtain a representation purely in terms of a set of probability/utility pairs.
Objective and subjective expected utility with incomplete preferences, working paper
, 2010
"... Abstract This paper extends the subjective expected utility model of decision making under uncertainty to include incomplete beliefs and tastes. The main results are two axiomatizations of the multiprior expected multiutility representations of preference relation under uncertainty. The paper als ..."
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Cited by 12 (3 self)
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Abstract This paper extends the subjective expected utility model of decision making under uncertainty to include incomplete beliefs and tastes. The main results are two axiomatizations of the multiprior expected multiutility representations of preference relation under uncertainty. The paper also introduces new axiomatizations of Knightian uncertainty and expected multiutility model with complete beliefs.