Inferring beliefs as subjectively imprecise probabilities

We propose a method for estimating subjective beliefs, viewed as a subjective probability distribution. The key insight is to characterize beliefs as a parameter to be estimated from observed choices in a well-defined experimental task and to estimate that parameter as a random coefficient. The experimental task consists of a series of standard lottery choices in which the subject is assumed to use conventional risk attitudes to select one lottery or the other and then a series of betting choices in which the subject is presented with a range of bookies offering odds on the outcome of some event that the subject has a belief over. Knowledge of the risk attitudes of subjects conditions the inferences about subjective beliefs. Maximum simulated likelihood methods are used to estimate a structural model in which subjects employ subjective beliefs to make bets. We present evidence that some subjective probabilities are indeed best characterized as probability distributions with non-zero variance.     

Elicitation of Subjective Probabilities when the Initial Endowment is Unobservable

In the framework of subjective expected utility theory we develop two distinct procedures for the elicitation of a person's subjective probabilities when the initial endowment is random and unobservable. Procedures of the first kind rely on the boundedness of the utility function to elicit the person's subjective probabilities directly. Procedures of the second kind infer the person's initial endowment from his choice behavior prior to the application of probability elicitation.     

An Experimental Comparison of Induced and Elicited Beliefs

Understanding choice under risk requires knowledge of beliefs and preferences. A variety of methods have been proposed to elicit peoples beliefs. The efficacy of alternative methods, however, has not been rigorously documented. Herein we use an experiment to test whether an induced probability can be recovered using an elicitation mechanism based on peoples predictions about a random event. We are unable to recover the induced belief. Instead, the estimated belief is systematically biased in a way that is consistent with anecdotal evidence in the economics, psychology, and statistics literature: people seem to overestimate low and underestimate high probabilities.     

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.     

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.     

The impossibility of experimental elicitation of subjective probabilities

We show that if decision makers may have stakes in certain events then the experimental elicitation of their subjective probabilities of these events is impossible.We thank Eyal Sulganik for helpful comments on an earlier draft of this paper.     

Information,risk perceptions,and smoking choices of youth

Conventional wisdom maintains that youths take risks because they underestimate probabilities of harm. Presumably if they knew the true probabilities, they would behave differently. We used the National Longitudinal Survey of Youth 1997 to assess whether differences between subjective and objective probabilities that an adverse outcome to self will occur are systematically related to a harmful behavior, initiating smoking. We find that youths are generally pessimistic about probabilities of their own deaths and being violent crime victims. After smoking initiation, youths increase subjective probabilities of death by more than the objective increase in mortality risk, implying recognition of potential harms. Virtually all 12–14 year-olds know that smoking causes heart disease. The minority who believe that smoking causes AIDS are less likely to become smokers; i.e., risk misperceptions deter rather than cause smoking initiation. Messages designed to deter smoking initiation should stress other disadvantages of smoking than just probabilities of harm.     

Reflections on gains and losses: A 2 × 2 × 7 experiment

What determines risk attraction or aversion? We experimentally examine three factors: the gain-loss dichotomy, the probabilities (0.2 vs. 0.8), and the money at risk (7 amounts). We find that the majority display risk attraction for small amounts of money, and risk aversion for larger amounts. Yet the frequency of risk attraction varies according to the gain-loss dichotomy and the probabilities. Kahneman and Tversky studied gain-loss reflections. We submit that a reflection can be decomposed into a translation and a probability switch. We find significant translation and switch effects, which are of comparable magnitude, a result that is equidistant from the diverging two popular views inspired by Prospect Theory: the gain-loss asymmetry, and the fourfold pattern.     

Probability perceptions and preventive health care

We study the effect of perceptions in comparison with more objective measures of risk on individuals’ decisions to decline or accept risk reducing interventions such as flu shots, mammograms, and aspirin for the prevention of heart disease. In particular, we elicit individuals’ subjective probabilities of risk, with and without the interventions, and compare these perceptions to individually predicted risk based on epidemiological models. Respondents, especially women, appear to be aware of some of the qualitative relationships between risk factors and probabilities. However, on average they have very poor perceptions of the absolute probability levels as reported in the epidemiological literature. Perceptions of the level of risk are less accurate if a respondent is female and has poor numeracy skills. We find that perceived probabilities significantly affect the subsequent take-up rate of flu shots, mammograms, and aspirin, even after controlling for individually predicted risk using epidemiological models.     

What can multiple price lists really tell us about risk preferences?

Multiple price lists have emerged as a simple and popular method for eliciting risk preferences. Despite their popularity, a key downside of multiple price lists has not been widely recognized — namely that the approach is unlikely to generate sufficient information to accurately identify different dimensions of risk preferences. The most popular theories of decision making under risk posit that preferences for risk are driven by a combination of two factors: the curvature of the utility function and the extent to which probabilities are weighted non-linearly. In this paper, we show that the widely used multiple price list introduced by Holt and Laury (The American Economic Review 92(5) 1644–1655 2002) is likely more accurate at eliciting the shape of the probability weighting function, and we construct a different multiple price list that is likely more accurate at eliciting the shape of the utility function. We show that by combining information from different multiple price lists, greater predictive performance can be achieved.     

Error Propagation in the Elicitation of Utility and Probability Weighting Functions

Elicitation methods in decision-making under risk allow us to infer the utilities of outcomes as well as the probability weights from the observed preferences of an individual. An optimally efficient elicitation method is proposed, which takes the inevitable distortion of preferences by random errors into account and minimizes the effect of such errors on the inferred utility and probability weighting functions. Under mild assumptions, the optimally efficient method for eliciting utilities and probability weights is the following three-stage procedure. First, a probability is elicited whose subjective weight is one half. Second, the utility function is elicited through the midpoint chaining certainty equivalent method using the probability elicited at the first stage. Finally, the probability weighting function is elicited through the probability equivalent method.     

Choices at various levels of uncertainty: An experimental test of the restated diversification theorem

Our “Restated diversification theorem” (Skogh and Wu, 2005) says that risk-averse agents may pool risks efficiently without assignment of subjective probabilities to outcomes, also at genuine uncertainty. It suffices that the agents presume that they face equal risks. Here, the theorem is tested in an experiment where the probability of loss, and the information about this probability, varies. The result supports our theorem. Moreover, it tentatively supports an evolutionary theory of the insurance industry—starting with mutual pooling at uncertainty, turning into insurance priced ex ante when actuarial information is available.     

Are people overoptimistic about the effects of heavy drinking?

We test whether heavy or binge drinkers are overly optimistic about probabilities of adverse consequences from these activities or are relatively accurate about these probabilities. Using data from a survey in eight cities, we evaluate the relationship between subjective beliefs and drinking. We assess accuracy of beliefs about several outcomes of heavy/binge drinking: reduced longevity, liver disease onset, link between alcohol consumption and Driving While Intoxicated (DWI), probability of an accident after drinking, accuracy of beliefs about encountering intoxicated drivers on the road, and legal consequences of DWI—ranging from being stopped to receiving fines and jail terms. Overall, there is no empirical support for the optimism bias hypothesis. We do find that persons consuming a lot of alcohol tend to be more overconfident about their driving abilities and ability to handle alcohol. However, such overconfidence does not translate into over-optimism about consequences of high levels of alcohol consumption.     

Measures of Mortality Risks

Different risks of death are not equivalent because of differences in timing. This paper develops measures of mortality risks that recognize the probability of death, the duration of life lost, and the role of discounting. These adjustments lead to a substantial reordering of the major causes of death. Recognition of duration-related issues explains much of the public's misperception of mortality risk probabilities, which may reflect duration-related concerns rather than biases in risk beliefs. Our estimates suggest that in forming their risk beliefs the public discounts years of life lost at a rate from 3.3–12.4 percent. Standardization of lifetimes at risk also alters the relative efficacy of regulatory policies for which we provide a variety of cost-effectiveness measures.     

Sampling experience reverses preferences for ambiguity

People often need to choose between alternatives with known probabilities (risk) and alternatives with unknown probabilities (ambiguity). Such decisions are characterized by attitudes towards ambiguity, which are distinct from risk attitudes. Most studies of ambiguity attitudes have focused on the static case of single choice, where decision makers typically prefer risky over ambiguous prospects. However, in many situations, decision makers may be able to sample outcomes of an ambiguous alternative, allowing for inferences about its probabilities. The current paper finds that such sampling experience reverses the pattern of ambiguity attitude observed in the static case. This effect can only partly be explained by the updating of probabilistic beliefs, suggesting a direct effect of sampling on attitudes toward ambiguity.     

How to reveal people’s preferences: Comparing time consistency and predictive power of multiple price list risk elicitation methods

The question of how to measure and classify people’s risk preferences is of substantial importance in the field of economics. Inspired by the multitude of ways used to elicit risk preferences, we conduct a holistic investigation of the most prevalent method, the multiple price list (MPL) and its derivations. In our experiment, we find that revealed preferences differ under various versions of MPLs as well as yield unstable results within a 30-minute time frame. We determine the most stable elicitation method with the highest forecast accuracy by using multiple measures of within-method consistency and by using behavior in two economically relevant games as benchmarks. A derivation of the well-known method by Holt and Laury (American Economic Review 92(5):1644–1655, 2002), where the highest payoff is varied instead of probabilities, emerges as the best MPL method in both dimensions. As we pinpoint each MPL characteristic’s effect on the revealed preference and its consistency, our results have implications for preference elicitation procedures in general.     

Avoiding the curves: Direct elicitation of time preferences

We propose and test a new method for eliciting curvature-controlled discount rates that are invariant to the form of the utility function. Our method uses a single elicitation task and obtains individual discount rates without knowledge of risk attitude or parametric assumptions about the form of the utility function. We compare our method to a double elicitation technique in which the utility function and discount rate are jointly estimated. Our experiment shows that these methods yield consistent estimates of the discount rate, which is reassuring given the wide range of estimates in the literature. We find little evidence of probability weighting, but in a second experiment, we observe that discount rates are sensitive to the length of the front-end delay, suggesting present bias. When the front-end delay is at least two weeks, we estimate average discount rates to be 11.3 and 12.2% in the two experiments.     

Combining expert probabilities using the product of odds

We resolve a useful formulation of the question how a statistician can coherently incorporate the information in a consulted expert??s probability assessment for an event into a personal posterior probability assertion. Using a framework that recognises the total information available as composed of units available only to each of them along with units available to both, we show: that a sufficient statistic for all the information available to both the expert and the statistician is the product of their odds ratios in favour of the event; that the geometric mean of their two probabilities specifies a contour of pairs of assertions in the unit-square that yield the same posterior probability; that the information-combining function is parameterised by an unknown probability for the event conditioned only on the unspecified information common to both the statistician and the expert; and that an assessable mixing distribution over this unspecified probability allows an integrable mixture distribution to represent a computable posterior probability. The exact results allow the identification of the subclass of coherent probabilities that are externally Bayesian operators. This subclass is equivalent to the class of combining functions that honour the principles of uniformity and compromise.     

Regret theory and risk attitudes

We examine risk attitudes under regret theory and derive analytical expressions for two components—the resolution and regret premiums—of the risk premium under regret theory. We posit that regret-averse decision makers are risk seeking (resp., risk averse) for low (resp., high) probabilities of gains and that feedback concerning the foregone option reinforces risk attitudes. We test these hypotheses experimentally and estimate empirically both the resolution premium and the regret premium. Our results confirm the predominance of regret aversion but not the risk attitudes predicted by regret theory; they also clarify how feedback affects attitudes toward both risk and regret.