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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 multi-prior expected multi-utility representations of preference relation under uncertainty. The paper also introduces new axiomatizations of Knightian uncertainty and expected multi-utility model with complete beliefs.

We investigate the classical Anscombe-Aumann model of decision-making under uncertainty without assuming the completeness axiom. We deduce the exact nature of preferences in this setup, and give a simple example that shows why one cannot use state-independent utility functions to represent such preferences (even for those that are separable and monotonic). In turn, we distinguish between the dual traits of “indecisiveness in beliefs ” and “indecisiveness in tastes.” The former is captured by the Knightian Uncertainty model while the latter by the single-prior expected multi-utility model. We show how these dual models would jointly obtain on the basis of an ambiguity aversion property, and then show how each model may be deduced from this result in isolation. As an application, we provide a new characterization of the Anscombe-Aumann representation of a preference relation in which we obtain the completeness and monotonicity hypotheses as consequences of a standard ambiguity aversion axiom and completeness of risk preferences alone.

. Anscombe and Aumann showed that if one accepts the existence of a physical randomizing device such as a roulette wheel then Savage's derivation of subjective expected utility can be considerably simplified. They, however, invoked compound gambles to define their axioms. We demonstrate that the subjective expected utility derivation can be further simplified and need not invoke compound gambles. Our simplification is obtained by closely following the steps by which probabilities and utilities are revealed. KEYWORDS: subjective expected utility, revealed preference, decision analysis, subjective probability, multistage gambles Journal of Economic Literature Classification Number D81 * The support for this research was provided in part by the Decision, Risk, and Management Science branch of the National Science Foundation. 2 1. INTRODUCTION The most well-known justification for subjective expected utility theory (SEU) was provided by Savage (1954). Savage's hallmark contribution wa...

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This is a study of the representations of subjective expected utility preferences that admits state-dependent incompleteness, and subjective expected utility preferences displaying noncomparability of acts from distinct sources. The notions familiar events and sources are defined and characterized. The relation greater familiarity on sources and increasing familiairity of a source are also defined and characterized.

Abstract. I show that there is a common order-theoretic structure underlying many of the models for representing beliefs in the literature. After identifying this structure, and studying it in some detail, I argue that it is useful. On the one hand, it can be used to study the relationships between several models for representing beliefs, and I show in particular that the model based on classical propositional logic can be embedded in that based on the theory of coherent lower previsions. On the other hand, it can be used to generalise the coherentist study of belief dynamics (belief expansion and revision) by using an abstract order-theoretic definition of the belief spaces where the dynamics of expansion and revision take place. Interestingly, many of the existing results for expansion and revision in the context of classical propositional logic can still be proven in this much more abstract setting, and therefore remain valid for many other belief models, such as those based on imprecise probabilities.

The Theory of Adaptive Utility for sequential decision making under uncertainty provides a generalisation of the standard Bayesian approach, permitting initial utility uncertainty and the learning of preferences. In this paper we discuss the motivation for the use of Adaptive Utility and comment on how such a setting entails a different optimal decision strategy from that which assumes known utility. Various possibilities for the learning of preferences are considered, and we provide an example of how the theory may be used in cases where the assumption of a known utility function is too restrictive. 1

The state-preference framework for modeling choice under uncertainty, in which objects of choice are allocations of wealth or commodities across states of the world, is a natural one for modeling “smooth” ambiguity-averse preferences. It does not require reference to objective probabilities, personalistic consequences, or counterfactual acts, and it allows for state-dependence of utility and unobservable background risk. The decision maker’s local beliefs are encoded in her risk neutral probabilities (her relative marginal rates of substitution between states) and her local risk preferences are encoded in the matrix of derivatives of the risk neutral probabilities. This matrix plays a central but generally unappreciated role in the modeling of risk attitudes in the state-preference framework. It can be computed by inverting a bordered Slutsky matrix and vice versa, it generalizes the Arrow-Pratt measure for approximating local risk premia, and its structure reveals whether the decision maker’s risk preferences are ambiguity-averse as well as risk averse. Two versions of the smoothambiguity model are analyzed — the source-dependent risk aversion model and the second-order uncertainty (KMM) model — and it is shown that in both cases the overall premium for local uncertainty can be decomposed as the sum of a risk premium and an ambiguity premium.

Decision making with imprecise probabilities and utilities by means of statistical preference and stochastic dominance