A Process Approach to the Utility for Gambling

This paper argues that any specific utility or disutility for gambling must be excluded from expected utility because such a theory is consequential while a pleasure or displeasure for gambling is a matter of process, not of consequences. A (dis)utility for gambling is modeled as a process utility which monotonically combines with expected utility restricted to consequences. This allows for a process (dis)utility for gambling to be revealed. As an illustration, the model shows how empirical observations in the Allais paradox can reveal a process disutility of gambling. A more general model of rational behavior combining processes and consequences is then proposed and discussed.     

Insurance buying gamblers

The puzzling coincidence of gambling and insurance has often been analysed by taking recourse to utility functions with convex and concave regions. In this paper we show that it may be optimal for utility maximizing risk seekers to engage in insurance and gambling activities simultaneously. A possible reason for this behavior is that these individuals try to take advantage of a moral hazard situation.     

Lottery Dependent Utility: a Reexamination

In order to accommodate empirically observed violations of the independence axiom of expected utility theory Becker and Sarin (1987) proposed their model of lottery dependent utility in which the utility of an outcome may depend on the lottery being evaluated. Although this dependence is intuitively very appealing and provides a simple functional form of the resulting decision criterion, lottery dependent utility has been nearly completely neglected in the recent literature on decision making under risk. The goal of this paper is to revive the lottery dependent utility model. Therefore, we derive first a sound axiomatic foundation of lottery dependent utility. Secondly, we develop a discontinuous variant of the model which can accommodate boundary effects and may lead to a lexicographic non-expected utility model. Both analyses are accompanied by considering some functional specifications which are in accordance with recent experimental results and may have significant applications in business and management science.     

A Further Examination of Cumulative Prospect Theory Parameterizations

Recent experimental studies have focused on fitting parameterized functional forms to cumulative prospect theory's weighting function. This paper examines the behavioral implications of the functional forms and the estimated parameters. We find that none of the parameterizations can simultaneously account for gambling on unlikely gains and the Allais paradox behavior or other strong choice patterns from experiments. Parameter estimates that lead to reasonable amounts of insurance and gambling behavior tend to also generate large risk premia. Taken as a whole, the analysis suggests that the functional forms proposed in the literature are not suitable for generalization to applied settings.     

Economic choice in generalized expected utility theory

Generalized expected utility models have enjoyed considerable success in explaining observed choices under uncertainty. However, there has been only limited progress in deriving comparative static results. This paper presents a general framework which permits the incorporation of a wide range of generalized expected utility models, but is sufficiently powerful to permit the derivation of comparative static results. The central idea is to represent preferences by the expected utility of a transformed probability distribution.     

Expected utility without utility

This paper advances an interpretation of Von Neumann-Morgenstern's expected utility model for preferences over lotteries which does not require the notion of a cardinal utility over prizes and can be phrased entirely in the language of probability. According to it, the expected utility of a lottery can be read as the probability that this lottery outperforms another given independent lottery. The implications of this interpretation for some topics and models in decision theory are considered.     

A Simple Tool for Qualitatively Testing,Quantitatively Measuring,and Normatively Justifying Savage's Subjective Expected Utility

This paper introduces a new preference condition that can be used to justify (or criticize) expected utility. The approach taken in this paper is an alternative to Savage's, and is accessible to readers without a mathematical background. It is based on a method for deriving comparisons of tradeoffs from ordinal preferences. Our condition simplifies previously-published tradeoff conditions, and at the same time provides more general and more powerful tools to specialists. The condition is more closely related to empirical methods for measuring utility than its predecessors. It provides a unifying tool for qualitatively testing, quantitatively measuring, and normatively justifying expected utility.     

Stochastic expected utility theory

This paper proposes a new decision theory of how individuals make random errors when they compute the expected utility of risky lotteries. When distorted by errors, the expected utility of a lottery never exceeds (falls below) the utility of the highest (lowest) outcome. This assumption implies that errors are likely to overvalue (undervalue) lotteries with expected utility close to the utility of the lowest (highest) outcome. Proposed theory explains many stylized empirical facts such as the fourfold pattern of risk attitudes, common consequence effect (Allais paradox), common ratio effect and violations of betweenness. Theory fits the data from ten well-known experimental studies at least as well as cumulative prospect theory.

The utility of gambling

A tiny utility of gambling is appended to an expected utility model for a risk-averse individual. It is shown that the model can explain small payoff gambles, large prize lotteries, and patterns of risk-seeking in the experimental evidence that are puzzling from the viewpoint of standard theory. At the same time, the model maintains expected utility theory's ability to explain insurance purchase, portfolio diversification, and other risk-averting behavior. The tiny utility of gambling could equally well be appended to models of risky choice other than the expected utility model.     

Reconciliation with the Utility of Chance by Elaborated Outcomes Destroys the Axiomatic Basis of Expected Utility Theory

Expected utility theory does not directly deal with the utility of chance. It has been suggested in the literature (Samuelson, 1952, Markowitz, 1959) that this can be remedied by an approach which explicitly models the emotional consequences which give rise to the utility of chance. We refer to this as the elaborated outcomes approach. It is argued that the elaborated outcomes approach destroys the possibility of deriving a representation theorem based on the usual axioms of expected utility theory. This is shown with the help of an example due to Markowitz. It turns out that the space of conceivable lotteries over elaborated outcomes is too narrow to permit the application of the axioms. Moreover it is shown that a representation theorem does not hold for the example.     

Revealed Preference and Expected Utility

This essay gives necessary and sufficient conditions for recovering expected utility from choice behavior in several popular models of uncertainty. In particular, these techniques handle a finite state model; a model for which the choice space consists of probability densities and the expected utility representation requires bounded, measurable utility; and a model for which the choice space consists of Borel probability measures and the expected utility representation requires bounded, continuous utility. The key result is the identification of the continuity condition necessary for the revelation of linear utility.     

Expected utility from additive utility on semigroups

In the present paper we study the framework of additive utility theory, obtaining new results derived from a concurrence of algebraic and topological techniques. Such techniques lean on the concept of a connected topological totally ordered semigroup. We achieve a general result concerning the existence of continuous and additive utility functions on completely preordered sets endowed with a binary operation ``+', not necessarily being commutative or associative. In the final part of the paper we get some applications to expected utility theory, and a representation theorem for a class of complete preorders on a quite general family of real mixture spaces.     

Local utility functions

In Machina's approach to generalised expected utility theory, decision makers maximise a choice functional which is smooth but not linear in the probabilities. When evaluating small changes, the choice functional can be approximated by the expectation of a local utility function. This local utility function is not however invariant under large changes in risk. This paper gives a simple explicit formula which can be used to write down the local utility functions of some common decision rules.     

Utility Functions for Wealth

We specify all utility functions on wealth implied by four special conditions on preferences between risky prospects in four theories of utility, under the presumption that preference increases in wealth. The theories are von Neumann-Morgenstern expected utility (EU), rank dependent utility (RDU), weighted linear utility (WLU), and skew-symmetric bilinear utility (SSBU). The special conditions are a weak version of risk neutrality, Pfanzagl's consistency axiom, Bell's one-switch condition, and a contextual uncertainty condition. Previous research has identified the functional forms for utility of wealth for all four conditions under EU, and for risk neutrality and Pfanzagl's consistency axiom under WLU and SSBU. The functional forms for the other condition-theory combinations are derived in this paper.     

The value of information in anticipated utility theory

A well-known property of expected utility theory is that the value of information is nonnegative. Given the widespread dissatisfaction with the expected utility hypothesis, a natural question to ask is whether competing theories of choice preserve this property. This article considers one widely discussed alternative to expected utility, anticipated utility theory. We show that, like expected utility, the anticipated value of perfect information is always nonnegative. The value of imperfect information, however, may be negative, though the precise valuation of information depends upon whether the reduction of compound lotteries axiom is used to derive the anticipated utility functional.I am indebted to Edi Karni, Peter Wakker, and an anonymous referee for helpful comments on earlier versions of this article. They are of course in no way responsible for errors or obscurities in the present version.     

Testing the Effects of Similarity on Risky Choice: Implications for Violations of Expected Utility

Our aim in this paper was to establish an empirical evaluation for similarity effects modeled by Rubinstein; Azipurua et al.; Leland; and Sileo. These tests are conducted through a sensitivity analysis of two well-known examples of expected utility (EU) independence violations. We found that subjective similarity reported by respondents was explained very well by objective measures suggested in the similarity literature. The empirical results of this analysis also show that: (1) the likelihood of selection for the riskier choice increases as the pair becomes more similar, (2) these choice patterns are consistent with well-known independence violations of expected utility, and (3) a significant proportion of individuals exhibit intransitive choice patterns predicted under similarity effects, but not allowed under generalized expected utility models for risky choice.     

On games under expected utility with rank dependent probabilities

Expected utility with rank dependent probabilities is a generalization of expected utility. If such preference representations are used for the payoffs in the mixed extension of a finite game, Nash equilibrium may fail to exist. Set-valued solutions, however, do exist even for those more general utility functions. But some set-valued solutions may have certain conceptual shortcomings. The paper thus proposes a new set-valued solution concept, called fixed sets under the best reply correspondence. All set-valued solution concepts are robust to perturbations of the expected utility hypothesis.     

The Likelihood Method for Decision under Uncertainty

This paper introduces the likelihood method for decision under uncertainty. The method allows the quantitative determination of subjective beliefs or decision weights without invoking additional separability conditions, and generalizes the Savage–de Finetti betting method. It is applied to a number of popular models for decision under uncertainty. In each case, preference foundations result from the requirement that no inconsistencies are to be revealed by the version of the likelihood method appropriate for the model considered. A unified treatment of subjective decision weights results for most of the decision models popular today. Savage’s derivation of subjective expected utility can now be generalized and simplified. In addition to the intuitive and empirical contributions of the likelihood method, we provide a number of technical contributions: We generalize Savage’s nonatomiticy condition (“P6”) and his assumption of (sigma) algebras of events, while fully maintaining his flexibility regarding the outcome set. Derivations of Choquet expected utility and probabilistic sophistication are generalized and simplified similarly. The likelihood method also reveals a common intuition underlying many other conditions for uncertainty, such as definitions of ambiguity aversion and pessimism.