The Utility of Gambling Reconsidered

The utility of gambling, which entails an intrinsic utility or disutility of risk, has been alluded to in the economics literature for over a century. This paper demonstrates that any utility of gambling almost unavoidably implies a violation of fundamental rationality properties, such as transitivity or stochastic dominance, for static choices between gambles. This result may explain why the utility of gambling, a phenomenon so widely discussed, has never been formalized in the economics literature. The model of this paper accommodates well-known deviations from expected utility, such as the Allais paradox and the coexistence of gambling and insurance, while minimally deviating from expected utility.     

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.     

An experimental evaluation of the descriptive validity of lottery-dependent utility theory

This article compares the performance of the expected utility (EU) and lottery-dependent expected utility (LDEU) models in predicting the actual choices of experimental subjects among risky options. In the process, we present two approaches for calibrating the LDEU model for an individual decision maker. The results indicate that while LDEU exhibits a higher potential for correctly predicting choice, the version of the model calibrated by indifference judgments does not outperform EU. We suggest a functional form for the parametric functions that defines the LDEU model, and discuss ways in which this function can be incorporated into choice-based assessment approaches to improve predictions.This research was supported in part by the Business Associates Fund at the Fuqua School of Business, Duke University.     

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.     

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.     

On utility functions

The von Neumann-Morgenstern (vNM) expected utility criterion is a particular case of a more general utility space structure. The general utility function can take care of the intrinsic utility or disutility of risk.An approximation to this general utility function is the proposed Constant Elasticities of Value and Risk (CEVR) function, which is easily applied in specific cases, for instance in psychometric studies of behaviors in the presence of certain or uncertain gains or losses.     

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.     

A foundation of Bayesian statistics (How to deal with fears)

A rational statistical decision maker whose preferences satisfy Savage's axioms will minimize a Bayesian risk function: the expectation with respect to a revealed (or subjective) probability distribution of a loss (or negative utility) function over the consequences of the statistical decision problem. However, the nice expected utility form of the Bayesian risk criterion is nothing but a representation of special preferences. The subjective probability is defined together with the utility (or loss) function and it is not possible, in general, to use a given loss function - say a quadratic loss - and to elicit independently a subjective distribution.I construct the Bayesian risk criterion with a set of five axioms, each with a simple mathematical implication. This construction clearly shows that the subjective probability that is revealed by a decider's preferences is nothing but a (Radon) measure equivalent to a linear functional (the criterion). The functions on which the criterion operates are expected utilities in the von Neumann-Morgenstern sense. It then becomes clear that the subjective distribution cannot be eliciteda priori, independently of the utility function on consequences.However, if one considers a statistical decision problem by itself, losses, defined by a given loss function, become the consequences of the decisions. It can be imagined that experienced statisticians are used to dealing with different losses and are able to compare them (i.e. have preferences, or fears over a set of possible losses). Using suitable axioms over these preferences, one can represent them by a (linear) criterion: this criterion is the expectation of losses with respect to a (revealed) distribution. It must be noted that such a distribution is a measure and need not be a 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.     

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.     

Dynamic Choice and NonExpected Utility

This paper explores how some widely studied classes of nonexpected utility models could be used in dynamic choice situations. A new "sequential consistency" condition is introduced for single-stage and multi-stage decision problems. Sequential consistency requires that if a decision maker has committed to a family of models (e.g., the multiple priors family, the rank-dependent family, or the betweenness family) then he use the same family throughout. Conditions are presented under which dynamic consistency, consequentialism, and sequential consistency can be simultaneously preserved for a nonexpected utility maximizer. An important class of applications concerns cases where the exact sequence of decisions and events, and thus the dynamic structure of the decision problem, is relevant to the decision maker. It is shown that for the multiple priors model, dynamic consistency, consequentialism, and sequential consistency can all be preserved. The result removes the argument that nonexpected utility models cannot be consistently used in dynamic choice situations. Rank-dependent and betweenness models can only be used in a restrictive manner, where deviation from expected utility is allowed in at most one stage.     

Choice under risk and the security factor: An axiomatic model

The particular attention paid by decision makers to the security level ensured by each decision under risk, which is responsible for the certainty effect, can be taken into account by weakening the independence and continuity axioms of expected utility theory. In the resulting model, preferences depend on: (i) the security level, (ii) the expected utility, offered by each decision. Choices are partially determined by security level comparison and completed by the maximization of a function, which express the existing tradeoffs between expected utility and security level, and is, at a given security level, an affine function of the expected utility. In the model, risk neutrality at a given security level implies risk aversion.     

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.     

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.     

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.     

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.     

Ranked-Weighted Utilities and Qualitative Convolution

For gambles—non-numerical consequences attached to uncertain chance events—analogues are proposed for the sum of independent random variables and their convolution. Joint receipt of gambles is the analogue of the sum of random variables. Because it has no unique expansion as a first-order gamble analogous to convolution, a definition of qualitative convolution is proposed. Assuming ranked, weighted-utility representations (RWU) over gains (and, separately, over losses, but not mixtures of both), conditions are given for the equivalence of joint receipt, qualitative convolution, and a utility expression like expected value. As background, some properties of RWU are developed.     

Reassessing the Testing of Generic Utility Models for Mixed Gambles

Miyamoto's generic utility theory (GUT) is a bilinear form that captures a diverse set of utility formulations. The present study, using mixed gambles, experimentally evaluates GUT in a fashion similar to the Chechile and Cooke and the Chechile and Butler studies, but employs a novel method of analysis. The reasons for the new method is to solve a fundamental flaw with the regression approacch used in the earlier experiments and to solve a problem of model overfit. Several participants from the earlier Chechile and Butler experiment are now recognized as being consistent with the GUT representation, but many are not. A new experiment, with actual economic consequences for the participants, does not support GUT. Suggestions are provided for subsequent research studies assessing the GUT class of models.