Choquet Expected Utility Representation of Preferences on Generalized Lotteries (original) (raw)
Expected Utility for Probabilistic Prospects and the Common Ratio Property
We prove the existence of an expected utility function for preferences over probabilistic prospects satisfying Strict Monotonicity, Indifference, the Common Ration Property, Substitution and Reducibility of Extreme Prospects. The example in Rubinstein (1988) that is inconsistent with the existence of a von Neumann-Morgenstern for preferences over probabilistic prospects, violates the Common Ratio Property. Subsequently, we prove the existence of expected utility functions with piecewise linear Bernoulli utility functions for preferences that are piece-wise linear. For this case a weaker version of the Indifference Assumption that is used in the earlier existence theorems is sufficient. We also state analogous results for probabilistic lotteries. We do not require any compound prospects or mixture spaces to prove any of our results. In the second last section of this paper, we “argue” that the observations related to Allais paradox, do not constitute a violation of expected utility maximization by individuals, but is a likely manifestation of individuals assigning (experiment or menu-dependent?) subjective probabilities to events which disagree with their objective probabilities.
On expected and von Neumann-Morgenstern utility functions
2010
In this note we analyze the relationship between the properties of von Neumann-Morgenstern utility functions and expected utility functions. More precisely, we investigate which of the regularity and concavity assumptions usually imposed on the latter transfer to the former and vice versa. In particular we obtain that, in order for the expected utility functions to fulfill such classical properties, it
Dual Representations of Cardinal Preferences
SSRN Electronic Journal, 2008
Given a set of possible vector outcomes and the set of lotteries over it, we define sets of (a) von Neumann-Morgenstern representations of preferences over the lotteries, (b) mappings that yield the certainty equivalent outcomes corresponding to a lottery, (c) mappings that yield the risk premia corresponding to a lottery, (d) mappings that yield the acceptance set of lotteries corresponding to an outcome, and (e) vector-valued functions that yield generalized Arrow-Pratt coefficients corresponding to an outcome. Our main results establish bijections between these sets of mappings for very general specifications of outcome spaces, lotteries and preferences. As corollaries of these results, we derive analogous dual representations of risk averse preferences. Some applications to financial theory illustrate the potential uses of our results. Finally, we provide criteria for comparing the risk aversion of preferences in terms of the dual representations.
Axiomatising Incomplete Preferences through Sets of Desirable Gambles
Journal of Artificial Intelligence Research
We establish the equivalence of two very general theories: the first is the decision-theoretic formalisation of incomplete preferences based on the mixture independence axiom; the second is the theory of coherent sets of desirable gambles (bounded variables) developed in the context of imprecise probability and extended here to vector-valued gambles. Such an equivalence allows us to analyse the theory of incomplete preferences from the point of view of desirability. Among other things, this leads us to uncover an unexpected and clarifying relation: that the notion of `state independence'---the traditional assumption that we can have separate models for beliefs (probabilities) and values (utilities)---coincides with that of `strong independence' in imprecise probability; this connection leads us also to propose much weaker, and arguably more realistic, notions of state independence. Then we simplify the treatment of complete beliefs and values by putting them on a more equal ...
Orderings and Probability Functionals Consistent with Preferences
Applied Mathematical Finance, 2009
This paper unifies the classical theory of stochastic dominance and investor preferences with the recent literature on risk measures applied to the choice problem faced by investors. First we summarize the main stochastic dominance rules used in the finance literature. Then we discuss the connection with the theory of integral stochastic orders and we introduce orderings consistent with investors' preferences. Thus, we classify them, distinguishing several categories of orderings associated with different classes of investors. Finally we show how we can use risk measures and orderings consistent with some preferences to determine the investors' optimal choices.
2018
We extend de Finetti's (1974) theory of coherence to apply also to unbounded random variables. We show that for random variables with mandated infinite prevision, such as for the St. Petersburg gamble, coherence precludes indifference between equivalent random quantities. That is, we demonstrate when the prevision of the difference between two such equivalent random variables must be positive. This result conflicts with the usual approach to theories of Subjective Expected Utility, where preference is defined over lotteries. In addition, we explore similar results for unbounded variables when their previsions, though finite, exceed their expected values, as is permitted within de Finetti's theory. In such cases, the decision maker's coherent preferences over random quantities is not even a function of probability and utility. One upshot of these findings is to explain further the differences between Savage's theory (1954), which requires bounded utility for non-simpl...
Functional Characterizations of Basic Properties of Utility Representations
Monatshefte für Mathematik, 2002
Consider uncertain alternatives for which an event has two consequences (binary gambles,``gambles'' for short) and over them an operation of joint receipt which need not be closed and may be non-commutative. The two structures are linked by a distributivity property called segregation and a preference order. Utility functions order nonnegative numbers to consequences and gambles. Utility representations describe how the utility of a gamble depends on the utilities of consequences and on the``weight'' of the event (a number in [0,1], depending on the event). Functional characterizations give necessary and suf®cient conditions, often in form of functional equations, for certain properties of representations. We ®rst give a functional characterization of the often postulated event commutativity stating that two events can be interchanged in special composite gambles where one outcome is a consequence but the other is itself a gamble. A utility representation is separable if it is multiplicative for gambles with one consequence having 0 utility. We give three more speci®c characterizations of separable representations by segregation, by homogeneity and event commutativity, and by homogeneity and segregation, and show that in the last case event commutativity follows.
Preference for equivalent random variables: A price for unbounded utilities
Journal of Mathematical Economics, 2009
When real-valued utilities for outcomes are bounded, or when all variables are simple, it is consistent with expected utility to have preferences defined over probability distributions or lotteries. That is, under such circumstances two variables with a common probability distribution over outcomes -equivalent variables -occupy the same place in a preference ordering. However, if strict preference respects uniform, strict dominance in outcomes between variables, and if indifference between two variables entails indifference between their difference and the status quo, then preferences over rich sets of unbounded variables, such as variables used in the St. Petersburg paradox, cannot preserve indifference between all pairs of equivalent variables. In such circumstances, preference is not a function only of probability and utility for outcomes. Then the preference ordering is not defined in terms of lotteries. Let T be independent of Y and let it have the uniform distribution on the interval [0,1]. Partition the interval [0,1] into the subintervals I 0 =[0, ], and I 1 ,…, I 2 m 1, where the last 2 m 1 intervals are all of equal length, 2 -m . Define the events B i ={T I i } for i=0,1,…,2 m 1 .