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.     

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.     

Buying and selling price for risky lotteries and expected utility theory with gambling wealth

I analyze two expected utility models which abandon the consequentialist assumption of terminal wealth positions. In the expected utility of gambling wealth model, in which initial wealth is allowed to be small, I show that a large WTA/WTP gap is possible and the (Rabin in Econometrica, 68(5), 1281–1292, 2000) paradox may be resolved. Within the same model the classical preference reversal which allows arbitrage is not possible, whereas preference reversal (involving buying prices in place of selling prices), which does not allow arbitrage, is possible. In the expected utility of wealth changes model, in which there is no initial wealth, I show that both a WTA/WTP gap as well as the classical preference reversal are possible due to loss aversion, both in its general as well as some specific forms.     

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.     

Resolving Rabin’s paradox

We present a theoretical model of Rabin’s famous calibration paradox that resolves confusions in the literature and that makes it possible to identify the causes of the paradox. Using suitable experimental stimuli, we show that the paradox truly violates expected utility and that it is caused by reference dependence. Rabin already showed that utility curvature alone cannot explain his paradox. We, more strongly, do not find any contribution of utility curvature to the explanation of the paradox. We find no contribution of probability weighting either. We conclude that Rabin’s paradox underscores the importance of reference dependence.

     

Decision analysis using lottery-dependent utility

In this article we show how the lottery-dependent expected utility (LDEU) model can be used in decision analysis. The LDEU model is an extension of the classical expected utility (EU) model and yet permits preference patterns that are infeasible in the EU model. We propose a framework for constructing decision trees in a particular way that permits us to use the principle of optimality and thus the divide and conquer strategy for analyzing complex problems using the LDEU model. Our approach may be applicable to some other nonlinear utility models as well. The result is that, if desired, decision analysis can be conducted without assuming the restrictive substitution principle/independence axiom.     

An Experimental Test of a General Class of Utility Models: Evidence for Context Dependency

Generic utility theory, a general axiomatization of utility principles developed by Miyamoto (1988, 1992), is discussed as a formulation that captures a large class of utility theories. Several general mathematical functions were used to specify further the scaling of utility within this class of models. The scaling parameters in the generic utility representation should remain invariant across gambling contexts, and this predicted invariance provided a means for testing the theory. Evidence is presented that the prediction of scaling-parameter invariance is violated. This failure is interpreted as a consequence of employing an absolute reference system for a problem that is context-sensitive.     

Risk aversion over finite domains

We investigate risk attitudes when the underlying domain of payoffs is finite and the payoffs are, in general, not numerical. In such cases, the traditional notions of absolute risk attitudes, that are designed for convex domains of numerical payoffs, are not applicable. We introduce comparative notions of weak and strong risk attitudes that remain applicable. We examine how they are characterized within the rank-dependent utility model, thus including expected utility as a special case. In particular, we characterize strong comparative risk aversion under rank-dependent utility. This is our main result. From this and other findings, we draw two novel conclusions. First, under expected utility, weak and strong comparative risk aversion are characterized by the same condition over finite domains. By contrast, such is not the case under non-expected utility. Second, under expected utility, weak (respectively: strong) comparative risk aversion is characterized by the same condition when the utility functions have finite range and when they have convex range (alternatively, when the payoffs are numerical and their domain is finite or convex, respectively). By contrast, such is not the case under non-expected utility. Thus, considering comparative risk aversion over finite domains leads to a better understanding of the divide between expected and non-expected utility, more generally, the structural properties of the main models of decision-making under risk.

     

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.     

Subjectively weighted linear utility

An axiomatized theory of nonlinear utility and subjective probability is presented in which assessed probabilities are allowed to depend on the consequences associated with events. The representation includes the expected utility model as a special case, but can accommodate the Ellsberg paradox and other types of ambiguity sensitive behavior, while retaining familiar properties of subjective probability, such as additivity for disjoint events and multiplication of conditional probabilities. It is an extension, to the states model of decision making under uncertainty, of Chew's weighted linear utility representation for decision making under risk.     

Interval scalability of rank-dependent utility

Luce and Narens (Journal of Mathematical Psychology, 29:1–72, 1985) showed that rank-dependent utility (RDU) is the most general interval scale utility model for binary lotteries. It can be easily established that this result cannot be generalized to lotteries with more than two outcomes. This article suggests several additional conditions to ensure RDU as the only utility model with the desired property of interval scalability in the general case. The related axiomatizations of some special cases of RDU of independent interest (the quantile utility, expected utility, and Yaari’s dual expected utility) are also given.     

Mean-risk decision analysis

Some decision theorists criticize expected utility decision analysis and propose mean-risk decision analysis as a replacement. They claim that expected utility decision analysis neglects attitudes toward risk whereas mean-risk decision analysis accords these attitudes their proper status. However mean-risk decision analysis and expected utility decision analysis are not incompatible, and it is advantageous for decision theory to develop each in a way that complements the other. Here I present a mean-risk rule that governs preferences among options and options given states. This mean-risk rule complements an expected utility rule that takes the utility of an option-state pair as the utility of the option given the state. I argue for the mean-risk rule using principles concerning basic intrinsic desires. The rule is comparative, but the last section offers some suggestions for its quantitative development.I am grateful for comments from my colleague, Henry E. Kyburg, Jr.     

Testing the Intransitivity Explanation of the Allais Paradox

This paper uses a two-dimensional version of a standard common consequence experiment to test the intransitivity explanation of Allais-paradox-type violations of expected utility theory. We compare the common consequence effect of two choice problems differing only with respect to whether alternatives are statistically correlated or independent. We framed the experiment so that intransitive preferences could explain violating behavior when alternatives are independent, but not when they are correlated. We found the same pattern of violation in the two cases. This is evidence against intransitivity as an explanation of the Allais Paradox. The question whether violations of expected utility are mainly due to intransitivity or to violation of independence is important since it is exactly on this issue the main new decision theories differ.     

Causality in the logic of decision

In recent years there has been an active debate between proponents of two different models of rational decision. One model is evidential decision theory, which is characterized by the fact that it holds the principle of maximizing expected utility to be appropriate whenever the states are probabilistically independent of the acts. The other model, causal decision theory, holds that the principle of maximizing expected utility is appropriate whenever the states are causally independent of the acts. The proponents of evidential decision theory include Richard Jeffrey and Ellery Eells, who claim that evidential decision theory has significant advantages over causal decision theory. In this paper I discuss the two main advantages which have been claimed for evidential decision theory, and show that in fact evidential decision theory does not possess either of these advantages.     

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.     

On matrix probabilities in nonarchimedean decision theory

In an earlier paper, we axiomatized a lexicographic expected utility model for preference in decision under uncertainty that is patterned on the models of Ramsey and Savage but omits their Archimedean axioms. Our model has the unusual feature that subjective probabilities are matrices that premultiply utility vectors in the lexicographic representation of preference between acts. Our purpose here is to analyze the model in relation to the Ramsey-Savage theory along with other models that have a lexicographic feature. A point of departure is Savage's postulate P4, whose purpose is to weakly order hisis more probable than relation on events. This postulate does not hold in our model and we therefore encounter incomparability between events. The paper explores the nature of incomparability, which can be widespread in high-dimensional situations. We include special cases of our model that retain a lexicographic component but also satisfy P4.