Commonalities in Time and Ambiguity Aversion for Long-Term Risks*

Optimal protective responses to long-term risks depend on rational perceptions of ambiguous risks and uncertain time horizons. Our study examined the joint influence of uncertain delay and risk in an original sample of business owners and managers. We found that many subjects disliked uncertainty in the timing of an outcome, a reaction we term ``lottery timing risk aversion.' Such aversion to uncertain timing was positively related to aversion to ambiguous probabilities for lotteries involving storm damage risks. This association suggests that uncertainty may be processed similarly in both the risk and time dimensions.     

Recursive expected utility and the separation of attitudes towards risk and ambiguity: an experimental study

We use the multiple price list method and a recursive expected utility theory of smooth ambiguity to separate out attitude towards risk from that towards ambiguity. Based on this separation, we investigate if there are differences in agent behaviour under uncertainty over gain amounts vis-a-vis uncertainty over loss amounts. On an aggregate level, we find that (i) subjects are risk averse over gains and risk seeking over losses, displaying a “reflection effect” and (ii) they are ambiguity neutral over gains and are mildly ambiguity seeking over losses. Further analysis shows that on an individual level, and with respect to both risky and ambiguous prospects, there is limited incidence of a reflection effect where subjects are risk/ambiguity averse (seeking) in gains and seeking (averse) in losses, though this incidence is higher for ambiguous prospects. A very high proportion of such cases of reflection exhibit risk (ambiguity) aversion in gains and risk (ambiguity) seeking in losses, with the reverse effect being significantly present in the case of risk but almost absent in case of ambiguity. Our results suggest that reflection across gains and losses is not a stable individual characteristic, but depends upon whether the form of uncertainty is precise or ambiguous, since we rarely find an individual who exhibits reflection in both risky and ambiguous prospects. We also find that correlations between attitudes towards risk and ambiguity were domain dependent.      

Ambiguity aversion in the small and in the large for weighted linear utility

The widely observed preference for lotteries involving precise rather than vague of ambiguous probabilities is called ambiguity aversion. Ambiguity aversion cannot be predicted or explained by conventional expected utility models. For the subjectively weighted linear utility (SWLU) model, we define both probability and payoff premiums for ambiguity, and introduce alocal ambiguity aversion function a(u) that is proportional to these ambiguity premiums for small uncertainties. We show that one individual's ambiguity premiums areglobally larger than another's if and only if hisa(u) function is everywhere larger. Ambiguity aversion has been observed to increase 1) when the mean probability of gain increases and 2) when the mean probability of loss decreases. We show that such behavior is equivalent toa(u) increasing in both the gain and loss domains. Increasing ambiguity aversion also explains the observed excess of sellers' over buyers' prices for insurance against an ambiguous probability of loss.     

Ambiguity and the value of information

The value of information is studied in a non-expected utility model of ambiguity with second-order probabilities. Information that reduces ambiguity has a positive value for ambiguity-averse decision makers, and the value of information that resolves ambiguity increases with greater ambiguity and with greater ambiguity aversion. Since information that resolves risk is valuable, and must also resolve ambiguity, the value of such information for ambiguity averters increases with greater ambiguity and with greater ambiguity aversion.     

Estimating the demand for risky assets via the indirect expected utility function

This article obtains demand functions for risky assets without making a priori assumptions about the form of the utility function. In a simple portfolio model, the envelope theorem is applied to the indirect expected utility function to derive estimating equations. Tests for the existence of constant absolute or constant relative risk aversion are also developed. Empirical estimation of the demand for financial assets held by U.S. households for the period 1946–1985 indicates that aggregate household behavior is consistent with the existence of constant relative risk aversion, with the coefficient of risk aversion having a value of approximately 1.3.The authors gratefully acknowledge helpful comments from the editor and an anonymous referee.     

Effects of Outcome and Probabilistic Ambiguity on Managerial Choices

Information ambiguity is prevalent in organizations and likely influences management decisions. This study examines, given imprecise probabilities and outcomes, how managers make choices when they are provided with single-figure benchmarks. Seventy-nine MBA students completed two experiments. We found that, in a decision framed as a decision under certainty involving an ambiguous outcome, the majority of the subjects were ambiguity prone in the loss condition and switched to ambiguity aversion in the gain condition. However, in the presence of probabilistic ambiguity in a decision under risk, this expected switching pattern was shown only when the difference in riskiness between the two choice options (in the loss condition) was perceived to be relatively small. In a companion study, we used a written protocol approach to identify factors that affect decision makers' investment choices when faced with ambiguous outcomes. Protocols frequently mentioned that the ambiguous outcome option was risky, even in the case which was framed as a decision under certainty in the problem statement. In a decision under risk with ambiguous outcomes, the combination of probabilistic risk and outcome ambiguity was seen as even more risky.     

Risk aversion in bargaining: An experimental study

This paper reports the results of three experiments designed to test the predictions of the principal game-theoretic models of bargaining concerning the influence of risk aversion on bargaining outcomes. These models predict that risk aversion will be disadvantageous in bargaining except in situations in which potential agreements are lotteries with a positive probability of being worse than disagreement. The experimental results support the models' predictions. However, in the range of payoffs studied here, the effects due to risk aversion may be smaller than some of the focal point effects observed in previous experiments. Implications for further theoretical and experimental work are considered.     

Do ambiguity avoidance and the comparative ignorance hypothesis depend on people’s affective reactions?

Previous work has showed that people are averse to ambiguity and prefer to bet on known probabilities over unknown probabilities. There is also evidence that ambiguity aversion is stronger in comparative contexts rather than in non-comparative contexts. In the present paper we suggest that ambiguity aversion depends on people’s affective reactions and therefore the effect is more evident in comparative contexts because the comparison between the clear and ambiguous alternatives leads to more positive affective reactions toward the former rather than the latter. The present study extends the previous findings while, at the same time, supporting the “comparative ignorance hypothesis”.     

The heterogeneity of the value of statistical life: Introduction and overview

The refinement in worker fatality risk data used in hedonic wage studies and evidence from new stated preference studies have facilitated the exploration of the heterogeneity of the value of statistical life (VSL). Although the median VSL estimate for workers is $7–$7–8 million, the VSL varies considerably within the worker population. New estimates of the income elasticity of VSL are 1.0 or above, which are consistent with theoretical models linking VSL to the coefficient of relative risk aversion. The specific relationship between VSL and risk aversion is, however, more complex than previously understood. Age differences in VSL are substantial, with young children being accorded especially high VSL amounts. The public’s willingness to pay to reduce risks is reduced if those being protected are perceived as being blameworthy due to their responsibility for contributing to the risk.     

Uncertainty aversion and aversion to increasing uncertainty

According to the original Ellsberg (1961) examples there is uncertainty version if the decision maker prefers to bet on an urn of known composition rather than on an urn of unknown composition. According to another definition (Schmeidler, 1989), there is uncertainty aversion if any convex combination of two acts is preferred to the least favorable of these acts. We show that these two definitions differ: while the first one truly refers to uncertainty aversion, the second one refers to aversion to increasing uncertainty. Besides, with reference to Choquet Expected Utility theory, uncertainty aversion means that there exists the core of a capacity, while aversion to increasing uncertainty means that the capacity is convex. Consequently, aversion to increasing uncertainty implies uncertainty aversion, but the opposite does not hold. We also show that a completely analogous situation holds for the case of risk and we define a set of risk and uncertainty premiums according to the previous analysis.     

The “bomb” risk elicitation task

This paper presents the Bomb Risk Elicitation Task (BRET), an intuitive procedure aimed at measuring risk attitudes. Subjects decide how many boxes to collect out of 100, one of which contains a bomb. Earnings increase linearly with the number of boxes accumulated but are zero if the bomb is also collected. The BRET requires minimal numeracy skills, avoids truncation of the data, allows the precise estimation of both risk aversion and risk seeking, and is not affected by the degree of loss aversion or by violations of the Reduction Axiom. We validate the BRET, test its robustness in a large-scale experiment, and compare it to three popular risk elicitation tasks. Choices react significantly only to increased stakes, and are sensible to wealth effects. Our experiment rationalizes the gender gap that often characterizes choices under uncertainty by means of a higher loss rather than risk aversion.     

Measuring risk aversion with lists: a new bias

Various experimental procedures aimed at measuring individual risk aversion involve a list of pairs of alternative prospects. We first study the widely used method by Holt and Laury (Am Econ Rev 92(5):1644–1655, 2002), for which we find that the removal of some items from the lists yields a systematic decrease in risk aversion and scrambles the ranking of individuals by risk aversion. This bias, that we call embedding bias, is quite distinct from other confounds that have been previously observed in the use of the HL method. It may be related to empirical phenomena and theoretical developments where better prospects increase risk aversion. Nevertheless, we also find that the more recent elicitation method due to Abdellaoui et al. (Theory Decis 71:63–80, 2011), also based on lists but using only one and the same probability in the list, does not display any statistically significant bias when the corresponding items of the list are removed. Our results suggest that methods other than the popular HL one may be preferable for the measurement of risk aversion.     

Uncertain indemnity and the demand for insurance

This paper considers the demand for insurance in a model with uncertain indemnity. Uncertain indemnity tends to increase the demand for insurance for precautionary reasons, but it also tends to decrease the demand due to the risk created by indemnity uncertainty. When the coefficient of relative prudence is not too large, uncertain indemnity reduces the demand for insurance and partial coverage is optimal even at actuarially fair premiums. In addition, insurance may be an inferior good or a normal good, depending on the behavior of absolute risk aversion and the magnitude of the coefficient of relative risk aversion.     

Joint measurement of risk aversion,prudence, and temperance

Risk aversion—but also the higher-order risk preferences of prudence and temperance—are fundamental concepts in the study of economic decision making. We propose a method to jointly measure the intensity of risk aversion, prudence, and temperance. Our theoretical approach is to define risk compensations of different orders, and in an experiment we elicit these compensations with a price list technique. We find evidence for risk aversion, prudence, and temperance. These traits correlate within subjects. The compensations elicited for prudence are significantly larger than those for risk aversion and temperance. In contrast to commonly used utility functions, prospect theory can predict this behavioral pattern. In our experiment, risk-averse, risk-loving, and risk-neutral subjects are prudent. This supports a recent theoretical observation that prudence may be a more universal trait than previously realized.     

Bias and brains: Risk aversion and cognitive ability across real and hypothetical settings

I collect data on subjects’ risk attitudes using real and hypothetical risky choices. I also measure their cognitive ability using the cognitive reflective test (CRT). On average, measured risk preferences are not significantly different across real and hypothetical settings. However, cognitive ability is inversely related to risk aversion when choices are hypothetical, but it is unrelated when the choices are real. This interaction between cognitive ability and hypothetical setting is consistent with the notion that some individuals, specifically higher-ability individuals, may treat hypothetical choices as “puzzles,” and provides one potential explanation for why some studies find that subjects indicate that they are more tolerant of risk when they make hypothetical choices than when they make real choices.     

Risk aversion in expected intertemporal discounted utilities bandit problems

We consider a situation where an individual is facing an uncertain situation, but may costly alter his knowledge of the uncertainties. We study in this context how risk aversion may modify the individual search behavior. We consider a one-armed bandit problem (where one arm is safe and the other is risky) and study how the agent risk aversion can change the sequence of arms selected. The main result is that when the utility function is more concave, the agent has more chances to select the safe arm. We also discuss how search is affected by risk aversion.     

The risk aversion measure without the independence axiom

The risk premium (conveniently normalized) is defined as the measure of risk aversion. This measure does not require any relevant assumption in the theory of choice under uncertainty except the existence of a certainty equivalent. In particular, the independence axiom is not required. The measure of risk aversion of an action is provided not only for the case with one commodity and two consequences but also for the case with many commodities and consequences. The measure of mean risk aversion of all actions with given consequences is introduced and the local measure of risk aversion is obtained by making all these consequences approach the consequence under consideration. This measure is demonstrated to be zero when the von Neumann-Morgenstern utility function exists. In this case a measure of risk aversion of the second order is introduced, which turns out to be equal to the Arrow-Pratt absolute index when there is only one commodity and similar to the generalized measures proposed by several authors when there are many commodities and two consequences.Helpful comments by I. Gilboa and suggestions by the referee are gratefully acknowledged.     

Ambiguity aversion and the propensities for self-insurance and self-protection

Two models of ambiguity preferences that permit comparative statics analysis of greater ambiguity aversion yield definite predictions concerning propensities for self-insurance and self-protection: The levels of both activities that are optimal for an ambiguity-averse decision maker are higher in the presence of ambiguity than in its absence, and demands for both activities increase with greater ambiguity aversion. The reason is that, at levels optimal for one decision maker, an increase in either activity results in a mean-preserving contraction in the distribution of expected utility in the presence of ambiguity, which is valuable to anyone with the same risk preferences who is more ambiguity averse.     

On risk aversion,classical demand theory,and KM preferences

Building on Kihlstrom and Mirman (Journal of Economic Theory, 8(3), 361–388, 1974)’s formulation of risk aversion in the case of multidimensional utility functions, we study the effect of risk aversion on optimal behavior in a general consumer’s maximization problem under uncertainty. We completely characterize the relationship between changes in risk aversion and classical demand theory. We show that the effect of risk aversion on optimal behavior depends on the income and substitution effects. Moreover, the effect of risk aversion is determined not by the riskiness of the risky good, but rather the riskiness of the utility gamble associated with each decision.     

Giving in the face of risk

Decisions about how to share resources with others often need to be taken under uncertainty regarding its allocational consequences. Although risk preferences are likely important, existing research is silent about how social and risk preferences interact in such situations. In this paper we provide experimental evidence on this question. In a first experiment givers are not exposed to risk while beneficiaries’ final earnings may be larger or smaller than the allocation itself, depending on the realized state of the world. In a second experiment, risk affects the earnings of givers but not of beneficiaries. We find that individuals’ risk preferences are predictive for giving in both experiments. Increased risk exposure of beneficiaries tends to decrease giving whereas increased risk exposure of givers has no effect. We propose a simple non-linear generalization of a model allowing for other-regarding preferences, ex-post and ex-ante fairness, and risk aversion. We find some support for it in our data when risk is on the beneficiaries’ side but less so when risk is on the givers’ side. Our results point to the importance of the further development of models of social preferences that also incorporate risk preferences.