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.     

Confusion of Relative and Absolute Risk in Valuation

Subjects were less willing to pay for government medical insurance for diseases when the number of people who could not be cured was higher, holding constant the number who could be cured. In a second experiment, willingness to pay (from a hypothetical government windfall) for risk reduction was unaffected by whether the risk was described in terms of percentage or number of lives saved, even though subjects knew that the risks in question differed in prevalence. These results are consistent with the findings of Fetherstonhaugh et al., Jenni and Loewenstein, and others. I suggest that these results can be explained in terms of a general tendency to confuse proportions and differences, a confusion that is analogous to other confusions of quantitative dimensions in children, adults, the news media, and perhaps even the epidemiological literature.     

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.     

Experimental markets for insurance

This article extends the large amount of research on double-oral auction markets to hazards that produce only losses. We report results from a series of experiments in which subjects endowed with low-probability losses can pay a premium for insurance protection. Insurers specify the price at which they are willing to assume the risk of a loss. Insurance prices approach expected value for a large range of probabilities and loss amounts. Subjects seem to realize losses are statistically independent. Prices are not affected by ambiguity about the probability of loss.     

Risk taking after absolute and relative wealth changes: The role of reference point adaptation

We conducted an experiment to compare subjects’ attitudes toward risk before and after they experienced wealth changes induced by a real-effort task. We identified and estimated the subjects’ levels of reference point adaptation to absolute and relative wealth changes. We found that after experiencing a larger loss than others, the subjects did not completely adapt their reference points to the absolute wealth loss and the relative negative wealth gap, and thus significantly increased their risk-taking behavior. However, the subjects also did not adjust their attitudes toward risk after experiencing a smaller loss than others, a smaller gain than others, or a larger gain than others. This may be because they promptly adapted to wealth changes or because they did not adapt to wealth changes but the effects of absolute and relative wealth changes mostly offset each other.     

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.     

Dynamics in risk taking with a low-probability hazard

The experiment reported in this paper identifies the effects of experience on revealed attitudes toward risk. Subjects in the experiment encountered an uncertain risk of experiencing a negative income shock over multiple periods and were able to purchase insurance at the start of each period. Subjects engaged in greater risk taking, insuring less frequently, when faced with the same risk over multiple periods. Subjects weighted experienced outcomes proportionately, in a manner consistent with rational Bayesian inference and contrary to the theory that individuals exhibit recency bias. On the other hand, subjects assigned a greater weight to outcomes that directly impacted their earnings compared to observed outcomes that had no effect on income. Unexplained autocorrelation across subjects’ choices suggests that inertia also plays an important role in repeated risk settings. I explore the relevance of these findings to public policy aimed at influencing market outcomes in the presence of infrequent environmental hazards.     

Prospective reference theory: Toward an explanation of the paradoxes

This article develops a variant of the expected utility model termed prospective reference theory. Although the standard model occurs as a limiting case, the general approach is that individuals treat stated experimental probabilities as imperfect information. This model is applied to a wide variety of aberrant phenomena, including the Allais paradox, the overweighting of low-probability events, the existence of premiums for certain elimination of risks, and the representativeness heuristic. The prospective reference theory model predicts most of the observed behavioral patterns rather than being potentially reconcilable with such phenomena.Kenneth Arrow and Robert Viscusi provided helpful comments. A preliminary version of this article was presented at the 1987 AEA meetings.     

The impact of self-protection and self-insurance on individual response to risk

We develop four experimental markets to examine how individuals respond to risk: self-protection and self-insurance in both private and collective auctions. First, we find evidence that the mechanism used to reduce risk is important. Results indicate that the upper and lower bounds on value were elicited by the private self-protection and the collective self-insurance markets, respectively. Second, the robustness of these results declined with low-probability lotteries. We find further evidence that individuals overestimate the impact of low-probability events. Overestimation decreased, however, with repeated market exposure. Third, the four markets induced rapid value formation. Usually only one or two additional market trials were necessary before an individual's perception and valuation of reduced risk stabilized.     

Insurance demand and social comparison: An experimental analysis

This paper analyzes whether social comparison can explain the low take-up of disaster insurance usually reported in field studies. We argue that risks in the case of disasters are highly correlated between subjects whereas risks for which high insurance take-up can be observed (e.g. extended warranties or cell phone insurance) are typically idiosyncratic. We set up a simple model with social reference points and show that in the presence of inequality aversion social comparison makes insurance indeed less attractive if risks are correlated. In addition we conducted a simple experiment which confirms these theoretical results. The average willingness to pay for insurance is significantly higher for idiosyncratic than for correlated risks.     

Self-protection against repeated low probability risks

This paper reports the results of experiments designed to test the effect of experience on preferences for self-protection against low and high probability losses. Subjects gained experience by repeatedly making choices about whether or not to invest in a protective activity and then observing the result of their choice. Half of the subjects faced a low probability risk while the other half faced a higher probability risk with the severity of loss scaled down to hold expected value constant. Protection was more common against the high probability risk. Despite receiving full information about the risks in advance, most subjects made choices in response to prior outcomes. This led to a great deal of experimentation when losses were common (the high probability risk) but very little experimentation when losses were infrequent (the low probability risk).     

For whom does social comparison induce risk-taking?

A ‘bonus culture’ among financial traders has been blamed for the excessive risk-taking in the run-up to the latest financial crisis. I show that when individuals are more social gain seeking than social loss averse (i.e. gloating is stronger than envy), social comparison predicts more risk-taking as well as a preference for negatively correlated gambles. Testing these two joint propositions in a laboratory experiment, I find that preference for positively or negatively correlated outcomes is highly correlated with risk-taking in a social risky investment task. While only a third of subjects prefer negatively correlated outcomes in a peer comparison setting, in line with relatively stronger social gain seeking, those subjects invest on average 50 % more in a risky gamble in their peer comparison setting than a reference group that made the same decision in an isolated individual setting. Subjects with a preference for positively correlated outcomes, in line with relatively stronger social loss aversion, do not show a higher propensity to invest in a risky gamble compared to the individual reference group.     

Observing different orders of risk aversion

A decision maker's attitude towards risk is said to be of orderi, i=1, 2, if for every given riskē with expected value zero, the risk premium the decision maker is willing to pay to avoid the risktē goes witht to zero at the same order ast ⁱ. This article presents an experiment testing the order of decision makers' attitudes toward risk. Its major result is that both attitudes exist, each in significant proportions. Moreover, two classes of first-order behavior are defined. The rank-dependent model (Quiggin, 1982) belongs to one, the disappointment aversion model (Gul, 1991) to the other. We show that only the first of these two classes appears among our subjects.     

Do administrators have the same priorities for risk reductions as the general public?

A stated preference survey was used to investigate the potential discrepancy between the priorities of public administrators and the general public regarding risk reductions. Both groups of respondents were asked to assume the role of a public policy-maker and choose between different public safety projects. We investigate differences in three areas: (i) large vs. small accidents, (ii) actual vs. subjective risk, and (iii) the trade-off between avoiding fatalities and serious injuries for different age groups and accidents. We find only minor differences between the responses of administrators and the general public, the most important of which is the difference in priorities between reducing the risk of many small or one large accident. In this area the most common response from the general public is that they prefer avoiding many small accidents rather than one large accident while among the administrators there is almost an equal split between the two options.     

Time preferences and risk aversion: Tests on domain differences

Understanding how individuals discount and evaluate the risks of environmental outcomes is a prime component in designing effective environmental policy. We use an incentivized experimental design to investigate whether subjects’ time preferences and risk aversion across the monetary and environmental domains differ. We find that subjects’ time preferences are not significantly different across the two domains. In contrast, subjects exhibit a higher degree of risk aversion in the environmental domain. Furthermore, we corroborate earlier results, documenting that women are more risk averse than men in the monetary domain, and show this finding to also hold in the environmental domain.     

Terrorism and Probability Neglect

When strong emotions are involved, people tend to focus on the badness of the outcome, rather than on the probability that the outcome will occur. The resulting “probability neglect” helps to explain excessive reactions to low-probability risks of catastrophe. Terrorists show a working knowledge of probability neglect, producing public fear that might greatly exceed the discounted harm. As a result of probability neglect, people often are far more concerned about the risks of terrorism than about statistically larger risks that they confront in ordinary life. In the context of terrorism and analogous risks, the legal system frequently responds to probability neglect, resulting in regulation that might be unjustified or even counterproductive. But public fear is itself a cost, and it is associated with many other costs, in the form of “ripple effects” produced by fear. As a normative matter, government should reduce even unjustified fear, if the benefits of the response can be shown to outweigh the costs.     

Heterogeneity in life-duration preferences: Are risky recreationists really more risk loving?

We conduct multiple price list experiments that elicit life duration risk preferences from amateur auto racers, technical rock climbers, SCUBA divers, and a student control group. We posit a preference function that allows for risk aversion and probability weighting. We are particularly interested in whether the behavior of risk takers, such as risky recreationists or smokers, is best explained by a risk-tolerant utility function or if immunity to possibility bias arising from overweighting of low probabilities is a more important motivator of the choice to engage in risky activities. We find that amateur auto racers are more rational than either students or other risky recreationists because they are less likely to overemphasize low-probability events. Women, older subjects, and rock climbers are more susceptible to possibility bias than others, making them likely to overinvest in disease treatments that have a low probability of success.     

Analysis of Intertemporal Choice: A New Framework and Experimental Results

This paper reports the results of a series of experiments examining intertemporal choice. The paper makes three contributions: First, it presents a new analytic device, the intertemporal choice triangle, which is analogous to the Marschak--Machina choice triangle used in the analysis of choice under risk. Second, we have developed a new experimental design based on the intertemporal choice triangle which allows subjects greater flexibility in making choices, and which allows the researcher to make more subtle inferences, than are possible with designs previously employed. Subjects are able to create their most-preferred outcome in each choice situation by choosing a constrained linear combination of two extreme options. Third, our results show that while subjects do not typically maximize present value, they are significantly influenced break by present value considerations. We refer to this finding as it present value-seeking behavior. We find only weak evidence of several previously documented intertemporal choice anomalies in our framework.     

Horizon Length and Portfolio Risk

In this paper, we compare the attitude towards current risk of two expected-utility-maximizing investors who are identical except that the first investor will live longer than the second one. It is often suggested that the young investor should take more risks than the old investor. We consider as a benchmark the case of complete markets with a zero risk-free rate. We show that a necessary and sufficient condition to assure that younger is riskier is that the Arrow-Pratt index of absolute tolerance (T) be convex. If we allow for a positive risk-free rate, the necessary and sufficient condition is T convex, plus T(0) = 0. It extends the well-known result that rational investors can behave myopically if and only if the utility function exhibits constant relative risk aversion.