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 influence of probabilities on the response mode bias in utility elicitation

The response mode bias, in which subjects exhibit different risk attitudes when assessing certainty equivalents versus indifference probabilities, is a well-known phenomenon in the assessment of utility functions. In this empirical study, we develop and apply a cardinal measure of risk attitudes to analyze not only the existence, but also the strength of this phenomenon. Since probability levels involved in decision problems are already known to have a strong impact on behavior, we use this approach to study the impact of probabilities on the extent of the response mode bias. We find that the direction in which probabilities influence measured risk aversion is the opposite in the certainty equivalence (CE) method versus in the probability equivalence (PE) method. Utilizing the CE elicitation approach leads to an increase of risk seeking for gambles involving high probabilities. For the PE method, subjects tend to behave risk averse with gambles of high probabilities. This behavior is reversed in the gain domain. This “tailwhip” effect is consistently replicated in several experiments, involving both loss and gain domains of lotteries.     

Expected discounted utility

Standard axioms of additively separable utility for choice over time and classic axioms of expected utility theory for choice under risk yield a generalized expected additively separable utility representation of risk-time preferences over probability distributions over sure streams of intertemporal outcomes. A dual approach is to use the analogues of the same axioms in a reversed order to obtain a generalized additively separable expected utility representation of time–risk preferences over intertemporal streams of probability distributions over sure outcomes. The paper proposes an additional axiom, which is called risk-time reversal, for obtaining a special case of the two representations—expected discounted utility. The axiom of risk-time reversal postulates that if a risky lottery over streams of sure intertemporal outcomes and an intertemporal stream of risky lotteries yield the same probability distribution of possible outcomes in every point in time then a decision-maker is indifferent between the two. This axiom is similar to assumption 2 “reversal of order in compound lotteries” in Anscombe and Aumann (Ann Math Stat 34(1):199–205, 1963, p. 201).

     

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.     

Comparing risk preferences over financial and environmental lotteries

This paper investigates whether preferences over environmental risks are best modeled using probability-weighted utility functions or can be reasonably approximated by expected utility (EU) or subjective EU models as is typically assumed. I elicit risk attitudes in the financial and environmental domains using multiple-price list experiment. I examine how subjects?? behavioral, attitudinal, and demographic characteristics affect their probability weighting functions first for financial risks, then for oil-spill risks. I find that most subjects tend to overweight extreme positive outcomes relative to expected utility in both the environmental and financial domains. Subjects are more likely to overemphasize low probability, extreme environmental outcomes than low probability, extreme financial outcomes, leading subjects to offer more support for mitigating environmental gambles than financial gambles with the same odds and equivalent outcomes. I conclude that EU models are likely to underestimate subjects?? willingness to pay for environmental cleanup programs or policies with uncertain outcomes.     

A variational model of preference under uncertainty

A familiar example devised by Daniel Ellsberg to highlight the effects of event ambiguity on preferences is transformed to separate aleatory uncertainty (chance) from epistemic uncertainty. The transformation leads to a lottery acts model whose states involve epistemic uncertainty; aleatory uncertainty enters into the statedependent lotteries. The model proposes von Neumann-Morgenstern utility for lotteries, additive subjective probability for states, and the use of across-states standard deviation weighted by a coefficient of aversion to variability to account for departures from Anscombe-Aumann subjective expected utility. Properties of the model are investigated and a partial axiomatization is provided.     

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.     

Rank-dependent,subjective expected-utility representations

Gambles are recursively generated from pure payoffs, events, and other gambles, and a preference order over them is assumed. Weighted average utility representations are studied that are strictly increasing in each payoff and for which the weights depend both on the events underlying the gamble and the preference ranking over the several component payoffs. Basically two results are derived: a characterization of monotonicity in terms of the weights, and an axiomatization of the representation. The latter rests on two important conditions: a decomposition of gambles into binary ones and a necessary commutativity condition on events in a particular class of binary gambles. A number of unsolved problems are cited.     

Hope: An empirical study of attitude toward the timing of uncertainty resolution

Hope is experienced when there is enjoyment in delaying the resolution of uncertainty. The main objective of this article is to identify the phenomenon of hope. In addition, we empirically test several axiomatic theories of temporal preferences which have implications for attitudes toward the timing of uncertainty resolution. Overall, the data support the extension of recursive expected utility specification to incorporate a weighted utility model of attitude toward future uncertainty. We find that the instances where hopefulness are more prevalent tend to be associated with a small probability of occurrence of a large gain. Interestingly, the degree of hopefulness is not correlated with risk attitude.     

The arguments of utility: Preference reversals in expected utility of income models

There is a debate in the literature about the arguments of utility in expected utility theory. Some implicitly assume utility is defined on final wealth whereas others argue it may be defined on initial wealth and income separately. I argue that making income and wealth separate arguments of utility has important implications that may not be widely recognized. A framework is presented that allows the unified treatment of expected utility models and anomalies. I show that expected utility of income models can predict framing induced preference reversals, a willingness to pay-willingness to accept gap for lotteries, and choice-value preference reversals. The main contribution is a theorem. It is proved that for all utility functions where initial wealth and income enter separately, either there will be preference reversals or preferences can be represented by a utility function defined on final wealth alone.     

Axiomatization of a Preference for Most Probable Winner

In binary choice between discrete outcome lotteries, an individual may prefer lottery L₁ to lottery L₂ when the probability that L₁ delivers a better outcome than L₂ is higher than the probability that L₂ delivers a better outcome than L₁. Such a preference can be rationalized by three standard axioms (solvability, convexity and symmetry) and one less standard axiom (a fanning-in). A preference for the most probable winner can be represented by a skew-symmetric bilinear utility function. Such a utility function has the structure of a regret theory when lottery outcomes are perceived as ordinal and the assumption of regret aversion is replaced with a preference for a win. The empirical evidence supporting the proposed system of axioms is discussed.     

When Allais meets Ulysses: Dynamic axioms and the common ratio effect

We report experimental findings about subjects’ behavior in dynamic decision problems involving multistage lotteries with different timings of resolution of uncertainty. Our within-subject design allows us to study violations of the independence and dynamic axioms: Dynamic Consistency, Consequentialism and Reduction of Compound Lotteries. We investigate the effects of changes in probability and outcome levels on the pattern of choices observed in the Common Ratio Effect (CRE) and in the Reverse Common Ratio Effect (RCRE) and on their dynamic counterparts. We find that the probability level plays an important role in violations of Reduction of Compound Lottery and Dynamic Consistency and the outcomes levels in violations of Consequentialism. Moreover, more than one quarter of our subjects satisfy the Independence axiom but violate two dynamic axioms. We thus suggest that there is a greater dissociation that might have been expected between preferences captured by dynamic axioms and those observed over single-stage lotteries.     

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.     

Probability Weighting in Choice under Risk: An Empirical Test

This paper reports a violation of rank-dependent utility with inverse S-shaped probability weighting for binary gambles. The paper starts with a violation of expected utility theory: one-stage gambles elicit systematically different utilities than theoretically equivalent two-stage gambles. This systematic disparity does not disappear, but becomes more pronounced after correction for inverse S-shaped probability weighting. The data are also inconsistent with configural weight theory and Machina's fanning out hypothesis. Possible explanations for the data are loss aversion and anchoring and insufficient adjustment.     

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.     

Maximizing expected utility is a survival criterion

The conventional expected utility of a lottery establishes a tight myopic lower bound on the probability of indefinitely avoiding economic ruin, the probability of survival. The analysis presented assumes that the decision maker is able to decline lotteries that are unacceptable in the expected utility sense. Application of the result to the assessment of utility functions is discussed.     

Investigating Risky Choices Over Losses Using Experimental Data

We conduct a battery of experiments in which agents make choices from several pairs of all-loss-lotteries. Using these choices, we estimate a representation of individual preferences over lotteries. We find statistically and economically significant departures from expected utility maximization for many subjects. We also estimate a preference representation based on summary statistics for behavior in the population of subjects, and again find departures from expected utility maximization. Our results suggest that public policies based on an expected utility approach could significantly underestimate preferences and willingness to pay for risk reduction.JEL Classification: C91, D81     

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.