Anxiety and Decision Making with Delayed Resolution of Uncertainty

In many real-world gambles, a non-trivial amount of time passes before the uncertainty is resolved but after a choice is made. An individual may have a preference between gambles with identical probability distributions over final outcomes if they differ in the timing of resolution of uncertainty. In this domain, utility consists not only of the consumption of outcomes, but also the psychological utility induced by an unresolved gamble. We term this utility anxiety. Since a reflective decision maker may want to include anxiety explicitly in analysis of unresolved lotteries, a multiple-outcome model for evaluating lotteries with delayed resolution of uncertainty is developed. The result is a rank-dependent utility representation (e.g., Quiggin, 1982), in which period weighting functions are related iteratively. Substitution rules are proposed for evaluating compound temporal lotteries. The representation is appealing for a number of reasons. First, probability weights can be interpreted as the cognitive attention allocated to certain outcomes. Second, the model disaggregates strength of preference from temporal risk aversion and thus provides some insight into the old debate about the relationship between von Neumann–Morgenstern utility functions and strength of preference value functions.     

A simplified axiomatic approach to ambiguity aversion

This paper takes the Anscombe–Aumann framework with horse and roulette lotteries, and applies the Savage axioms to the horse lotteries and the von Neumann–Morgenstern axioms to the roulette lotteries. The resulting representation of preferences yields a subjective probability measure over states and two utility functions, one governing risk attitudes and one governing ambiguity attitudes. The model is able to accommodate the Ellsberg paradox and preferences for reductions in ambiguity.     

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.     

Preference for gradual resolution of uncertainty

Analyses of preference for the timing of uncertainty resolution usually assumes all uncertainty to resolve in one point in time. More realistically, uncertainty should be modelled to resolve gradually over time. Kreps and Porteus (1978) have introduced an axiomatically based model of time preference which can explain preferences for gradual uncertainty resolution. This paper presents an experimental test of the Kreps-Porteus model. We derive implications of the model relating preferences for gradual and one-time resolving lotteries. Our data do not support the Kreps-Porteus model but show that some of the behaviour observed may be explained by similarity heuristics.     

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.     

On the Representation of Beliefs by Probabilities

This paper explores two axiomatic structures of subjective expected utility assuming a finite state-space and state-dependent, connected, topological outcome-spaces. Building on the work of Karni and Schmeidler (1981) the analytical framework includes, in addition to the preference relation on acts, introspective preferences on hypothetical lotteries that are linked to the preference relation on acts by consistency axioms. The two models accommodate state-dependent preferences and yield subjective probabilities that correctly represent the decision-maker's beliefs. State-independent preferences are a special case.     

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.     

A Rule For Updating Ambiguous Beliefs

When preferences are such that there is no unique additive prior, the issue of which updating rule to use is of extreme importance. This paper presents an axiomatization of the rule which requires updating of all the priors by Bayes rule. The decision maker has conditional preferences over acts. It is assumed that preferences over acts conditional on event E happening, do not depend on lotteries received on E ^c, obey axioms which lead to maxmin expected utility representation with multiple priors, and have common induced preferences over lotteries. The paper shows that when all priors give positive probability to an event E, a certain coherence property between conditional and unconditional preferences is satisfied if and only if the set of subjective probability measures considered by the agent given E is obtained by updating all subjective prior probability measures using Bayes rule.     

Uncertainty,the bargaining problem,and the Nash-Zeuthen solution

The Zeuthen bargaining model occupies a prominent place among those theories of the bargaining process that have been formulated and expounded by economists. Its solution to the bargaining problem is essentially economic, since invariant utility functions based on economic factors alone determine the outcome. However, this paper shows that a necessary condition for reaching the Zeuthen solution (shown by Harsanyi to be mathematically equivalent to the game-theoretic solution of Nash's theory) is that bargainers initially take up positions on opposite sides of the outcome that maximizes their utility product. Whether utility functions are mutually known or unknown, inherent in the bargaining situation itself is the requirement that bargainers be at least initially uncertain as to each other's subsequent concession behaviour. With uncertainty, von Neumann-Morgenstern rationality implies that each bargainer would make an initial demand that maximizes the expected gain from holding fast. Therefore, even if Zeuthen's concession criterion should subsequently dictate concession behaviour, expected utility maximization within the context of subjective uncertainty may well yield initial demands that are inconsistent with reaching the Nash-Zeuthen solution. Finally, a general methodological conclusion that emerges from the analysis is that, since the bargaining process necessarily proceeds from a context of subjective uncertainty, greater emphasis needs to be placed on its role as a device for affecting expectations.     

Ellsberg paradox: Ambiguity and complexity aversions compared

We present a simple model where preferences with complexity aversion, rather than ambiguity aversion, resolve the Ellsberg paradox. We test our theory using laboratory experiments where subjects choose among lotteries that “range” from a simple risky lottery, through risky but more complex lotteries, to one similar to Ellsberg’s ambiguity urn. Our model ranks lotteries according to their complexity and makes different—at times contrasting—predictions than most models of ambiguity in response to manipulations of prizes. The results support that complexity aversion preferences play an important and separate role from beliefs with ambiguity aversion in explaining behavior under uncertainty.     

Effects of Uncertainty Aversion on the Call Option Market

This article examines the effects of uncertainty aversion in competitive call option markets using a partial equilibrium model with the Choquet-expected utility setup. We find that the trading volume of a call option is negatively affected by uncertainty aversion, whereas the price of the call is practically independent of it.      

Lottery qualities

The aim of this paper is to propose a model of decision-making for lotteries. Lottery qualities are the key concepts of the theory. Qualities allow the derivation of optimal decision-making processes and are taken explicitly into account for lottery evaluation. Our contribution explains the major violations of the expected utility theory for decisions on two-point lotteries and shows the necessity of giving explicit consideration to lottery qualities. Judged certainty equivalent and choice certainty equivalent concepts are discussed in detail along with the comparison of lotteries. Examples are provided by using different test results in the literature. JEL Classification D81     

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.     

A note on Savage's theorem with a finite number of states

This article gives a preference-based characterization of subjective expected utility for the general equilibrium model with a finite number of states. The characterization follows Savage (1954) as closely as possible but has to abandon his axiom (P6), atomlessness of events, since this requires an infinite state space. To introduce continuity we replace (P6) with a continuity assumption on the set of consequences and assume the preferences are smooth. Then we apply Savage's sure-thing principle and his state-independence axiom to get an additively separable utility representation. Finally, to separate subjective probabilities from basic tastes, we apply a new axiom, which states that for each pair of states the marginal rate of substitution is constant along the certainty line.     

On the Application of Multiattribute Utility Theory to Models of Choice

Ellsberg (The Quarterly Journal of Economics 75, 643–669 (1961); Risk, Ambiguity and Decision, Garland Publishing (2001)) argued that uncertainty is not reducible to risk. At the center of Ellsberg’s argument lies a thought experiment that has come to be known as the three-color example. It has been observed that a significant number of sophisticated decision makers violate the requirements of subjective expected utility theory when they are confronted with Ellsberg’s three-color example. More generally, such decision makers are in conflict with either the ordering assumption or the independence assumption of subjective expected utility theory. While a clear majority of the theoretical responses to these violations have advocated maintaining ordering while relaxing independence, a persistent minority has advocated abandoning the ordering assumption. The purpose of this paper is to consider a similar dilemma that exists within the context of multiattribute models, where it arises by considering indeterminacy in the weighting of attributes rather than indeterminacy in the determination of probabilities as in Ellsberg’s example.      

The Lottery and Income Inequality in the States*

Objective. Since the early 1970s, income inequality in the United States has increased dramatically. We examine the impact of state lotteries on income inequality in the American states from 1976–1995. Methods. We use cross‐sectional time‐series data to evaluate the effect of lotteries as well as those of other state tax policies, redistributive programs, and demographic factors on income inequality. Results. We find that state lotteries foster income concentration. Ceteris paribus, states with lotteries have higher levels of income inequality than those states without a lottery. We also find that additional demographic and policy factors have an impact on income inequality in the states. Conclusions. One of the most important policy‐oriented determinants of income inequality is the lottery and a significant portion of the increase in income inequality over our two‐decade time period is attributable to the increasing prevalence and popularity of state lotteries.     

Lifetime Uncertainty and Time Preference

Despite Fisher's (1930) psychological intuitions of and the formal treatment given by Yaari (1965, Review of Economic Studies 32, 137), the intertemporal model of choice is mainly a model with certain lifetime. The purpose of this paper is to reconsider this assumption, starting from a very simple two-period model of choice with lifetime uncertainty. We examine the comparative statics of the model at the first two orders and replace the concept of `pure time preference' by taking into account the subjective treatment of the probability of survival. This revised version was published online in June 2006 with corrections to the Cover Date.     

Intertemporal utility smoothing under uncertainty

This paper axiomatizes a recursive utility model that captures both intertemporal utility smoothing defined across time and ambiguity aversion defined over states. The resulting representation adapts Wakai (Econometrica 76:137–153, 2008) model of intertemporal utility smoothing as an aggregator function, where the utility of the certainty equivalent of future uncertainty is computed by Gilboa and Schmeidler (J Math Econ 18:141–153, 1989) multiple-priors utility. The model also permits the separation of intertemporal utility smoothing from ambiguity aversion.