Expected utility theory without the completeness axiom

We study the problem of obtaining an expected utility representation for a potentially incomplete preference relation over lotteries by means of 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 multi-utility representation provided that it satisfies the standard axioms of expected utility theory. Moreover, the representing set of utilities is unique in a well-defined sense.     

Salience and social choice

The axioms of expected utility and discounted utility theory have been tested extensively. In contrast, the axioms of social welfare functions have only been tested in a few questionnaire studies involving choices between hypothetical income distributions. In a controlled experiment with 100 subjects placed in the role of social planners, we test five fundamental properties of social welfare functions to determine the efficacy of traditional social choice models in predicting social planner allocations when presented with choice sets designed to test the axioms of the theory. We find that three properties of the standard social welfare functions tested are systematically violated, producing an Allais paradox, a common ratio effect, and a framing effect in social choice. We find support for scale invariance and a preference for tail-increasing transfers. Our experiment also enables us to test a model of salience-based social choice which predicts the systematic deviations and highlights the close relationship between these anomalies and the classical paradoxes for risk and time.

     

Revealed favorability,indirect utility,and direct utility

This paper studies the axiomatic foundations of the indirect utility function, based upon a revealed preference approach. A chain of comparisons of budgets is regarded as giving a relation on the price-income space (a revealed favorability relation) rather than a relation on the commodity space (a revealed preference relation). The weak and strong axioms of revealed favorability are introduced by analogy with the weak and strong axioms of revealed preference, and the relationship between the former two axioms is investigated. The indirect and direct utility functions are then derived on the basis of the strong axiom of revealed favorability. Neither the continuity property of the demand function nor the convex property of its range is required for the approach taken in the paper.     

Is the expected utility maximization hypothesis refutable by finite market data: The case of two observations

It is shown that with only two observations on the asset demand functions of an investor, it is not possible to refute the hypothesis of expected utility maximization on the part of the investor as long as the axioms of revealed preference are satisfied.     

Bliss,Catastrophe, and Rational Policy

Lotteries with infinite expected utility are inconsistent with the axioms of expected utility theory. To rule them out, either the set of permissible lotteries must be restricted (to exclude, at a minimum, “fat-tailed” distributions such as that underlying the St. Petersburg Paradox and power laws that are popular in models of climate change), or the utility function must be bounded. This note explores the second approach and proposes a number of tractable specifications leading to utility functions that are bounded both from above and below. This property is intimately related to that of increasing relative risk aversion as first hypothesized by Arrow (1965).     

A theory of anticipated utility

A new theory of cardinal utility, with an associated set of axioms, is presented. It is a generalization of the von Neumann-Morgenstern expected utility theory, which permits the analysis of phenomena associated with the distortion of subjective probability.     

Nest-monotonic two-stage acts and exponential probability capacities

This paper examines conditions for Choquet expected utility (CEU) to satisfy both the reduction of two-stage acts and the recursion axioms, which are taken for granted in economics. A key idea of this paper is to consider nest-monotonic two-stage acts, which share their rankings of states with those of their reduced one-stage acts. Our main theorem shows that the axioms, one of which is restricted to nest-monotonic two-stage acts, and consequentialism are satisfied if and only if the preference is exponential CEU, which is such that the probability capacity is an exponential transformation of a probability measure. This result indicates that within a specified range of decision problems, exponential CEU is the only form of CEU that derives indifference to the timing of information resolution. Furthermore, the relation between first- and second-stage exponential CEU is characterized both by the f*-Bayesian updating rule and by comonotonic dynamic consistency. Conditions to establish the law of iterated expectation for CEU are also discussed.     

Reference-dependent subjective expected utility

A reference-dependent generalisation of subjective expected utility theory is presented. In this theory, preferences between acts depend both on final outcomes and on reference points (which may be uncertain acts). It is characterised by a set of axioms in a Savage-style framework. A restricted form of the theory separates attitudes to end states (encoded in a ‘satisfaction function’) from attitudes to gains and losses of satisfaction. Given weak additional assumptions, the restricted theory excludes cycles of choice, explains observed disparities between willingness-to-pay and willingness-to-accept valuations of lotteries, and predicts preference reversal.     

A Model of Procedural Decision Making in the Presence of Risk*

We introduce a procedural model of risky choice in which an individual is endowed with a core preference relation that may be highly incomplete. She can, however, derive further rankings of alternatives from her core preferences by means of a procedure based on the independence axiom. We find that the preferences that are generated from an initial set of rankings according to this procedure can be represented by means of a set of von Neumann–Morgenstern utility functions, thereby allowing for incompleteness of preference relations. The proposed theory also yields new characterizations of the stochastic dominance orderings.     

Choice, preference and rationalizability in pareto’s theory of economic behavior

Pareto initiated the shift of economic theory from utility/preference to choice in order to fill a gap between theoretical and empirical economics. His suggestions for an empirically-oriented theory of choice were to be developed decades later in the literature on revealed preference and on the conditions of equivalence between preference-based and choice-based axiomatic structures. In the process, however, substantial departures from Pareto’s implicit design of the situation of choice were introduced. In this paper, the conditions for the rationalizability of choice are re-defined so that they fit the type of situation that Pareto had in mind. The result is that the main consistency axioms of standard choice theory have to give way to a different combination of axioms that concern decisiveness as well as consistency. (JEL:B13, D01, D11).     

A reformulation of the theory of demand by compensated demand functions

We present a formal approach to consumer demand by compensated demand functions. In accordance with the integrability theory or with the theory of revealed preference, we do not require the existence of a utility function, but we do assume certain hypotheses concerning functions describing rational behavior. In view of their properties, these functions can be interpreted as compensated demand functions. According to traditional neoclassical consumer theory, Shephard's lemma and the symmetry and negative semidefiniteness of the Slutsky-Hicks matrix can be shown. We shall also see that a convex, continuous, and monotonic preference ordering, which is representable by income compensation functions, can be introduced. It can also be shown that the existence of a compensated equilibrium can be derived within this approach by compensated demand functions. In order to obtain the existence of a compensated equilibrium under less stringent conditions we finally generalize the axioms assuming that a compensated demand correspondence is given.     

Elicitation of Subjective Probabilities when Preferences are State-Dependent

This paper presents an experimental design that enables the elicitation of subjective probabilities of decision makers' whose preferences satisfy the axioms of expected utility theory and are state-dependent.     

Imprecision as an account of violations of independence and betweenness

In an earlier paper we put forward a model of imprecise preferences which accounted for various forms of preference reversal. In this paper we show that the same model can also explain the best-known violations of expected utility theory's axioms of independence and betweenness. It appears that a simple model of imprecise preferences can account for a broader range of anomalies than many of the more elaborate alternative theories developed to date.     

Risk–risk versus standard gamble procedures for measuring health state utilities

Contingent valuation (CV) has been widely used to measure the potential benefits derived from different policy decisions. However, doubt now exists about the validity of the CV method and alternative approaches to benefit valuation have been proposed. The paper reports on the results of a study which was designed to test the viability of two of the most prominent of the alternatives: the risk–risk (RR) and standard gamble (SG) approaches. If individual preferences are consistent with the axioms of von Neumann–Morgenstern expected utility theory (EUT) then the two methods should generate the same interval scales for any given set of health states. However, the results show that SG utilities are substantially higher than RR ones, thus casting doubt on these axioms. The paper discusses alternatives to EUT which might better expalin the discrepancies found. It also considers whether the results might be explained in terms of status-quo bias and/or by the relative difficulty of RR questions. The results presented may have important implications for other areas of applied research in which there exists uncertainty about outcomes.     

Valuing life: Experimental evidence using sensitivity to rare events

Global environmental phenomena like climate change, major extinction events or flutype pandemics can have catastrophic consequences. By properly assessing the outcomes involved – especially those concerning human life – economic theory of choice under uncertainty is expected to help people take the best decision. However, the widely used expected utility theory values life in terms of the low probability of death someone would be willing to accept in order to receive extra payment. Common sense and experimental evidence refute this way of valuing life, and here we provide experimental evidence of people's unwillingness to accept a low probability of death, contrary to expected utility predictions. This work uses new axioms of choice defined by Chichilnisky (2000), especially an axiom that allows extreme responses to extreme events, and the choice criterion that they imply. The implied decision criteria are a combination of expected utility with extreme responses, and seem more consistent with observations.     

When is there state independence?

Whether a preference relation can be represented using state-independent utilities as opposed to state-dependent utilities may depend on which acts count as constant acts. This observation underlies an extension of Savage's expected utility theory to the state-dependent case that was proposed in this journal by Edi Karni. His result contains a condition requiring the existence of a set of acts which can play the role of constant acts and support a representation involving a state-independent utility function. This paper contains necessary and sufficient conditions on the preference relation for such a set of acts to exist. Results are obtained both for the Savage and the Anscombe and Aumann frameworks. Among the corollaries are representation theorems for state-dependent utilities. Relationships to Karni's work and extensions of the results are discussed.     

An axiomatic approach to choice under uncertainty with catastrophic risks

This paper analyses decision under uncertainty with catastrophic risks, and is motivated by problems emerging from global environmental risks. These are typically low-probability events with major irreversible consequences. For such risks, the Von Neumann–Morgenstern (NM) axioms for decision making under uncertainty are not appropriate, since they are shown here to be insensitive to low-probability events. The paper introduces an alternative set of axioms requiring sensitivity to both low- and large-probability events. Through a new representation theorem in functional analysis, the results characterize all the operators whose maximization leads to the fulfillment of these axioms. They involve a convex combination of expected utility and a criterion based on the desire to avoid low probability and potentially catastrophic events. It is shown that the new axioms help resolve the Allais paradox. Open questions about risk aversion, games under uncertainty and calculus of variations are discussed.     

Revisiting Savage in a conditional world

Summary. I present an axiomatization of subjective expected utility and Bayesian updating in a conditional decision problem. This result improves our understanding of the Bayesian standard from two perspectives: 1) it uses a set of axioms which are weak and intuitive; 2) it provides a formal proof to results on the relation between dynamic consistency, expected utility and Bayesian updating which have never been explicitly proved in a fully subjective framework. Received: December 1, 2000; revised version: February 26, 2001     

Subjective Expected Utility Theory with State-Dependent Preferences

This paper extends Savage′s subjective expected utility theory to include state-dependent preferences. The dependence of the decision maker′s preferences over consequences on the states of nature is represented by state-specific mappings of the set of consequences onto itself. Within this framework Savage′s postulates are reformulated and it is shown that there exist subjective expected utility representations of the preference relation over acts with unique, nonatomic, probability measure on the algebra of all events, and a state-dependent utility function over the set of consequences. Journal of Economic Literature Classification Number: D81.     

Relative risk aversion, relative prudence and comparative statics under uncertainty: The case of (μ, σ)‐preferences

From the expected‐utility approach, relative risk aversion being smaller than one and relative prudence being smaller than two emerge as preference restrictions that fully determine the optimal responses of decisions under uncertainty to certain shifts in probability distributions. We characterize the magnitudes of relative risk aversion and relative prudence in terms of the two‐parameter, mean‐standard deviation approach. We demonstrate that this characterization is instrumental in obtaining comparative static results in the two‐parameter setting. We further relate our findings to the results in the expected‐utility framework.