Our aim in this paper is to clarify the notion of independence for imprecise probabilities. Suppose that two marginal experiments are each described by an imprecise probability model, i.e., by a convex set of probability distributions or an equivalent model such as upper and lower probabilities or previsions. Then there are several ways to define independence of the two experiments and to construct an imprecise probability model for the joint experiment. We survey and compare six definitions of independence. To clarify the meaning of the definitions and the relationships between them, we give simple examples which involve drawing balls from urns. For each concept of independence, we give a mathematical definition, an intuitive or behavioural interpretation, assumptions under which the definition is justified, and an example of an urn model to which the definition is applicable. Each of the independence concepts we study appears to be useful in some kinds of application. The concepts of strong independence and epistemic independence appear to be the most frequently applicable.

In 1999 a research program was started with the goal of finding the main reasons that influence people's decisions to insure themselves against losses caused by flood. In a field study a questionnaire that measured perception of the flood risk and motives for insuring or not insuring against the consequences of this disaster was used. The data were collected on a group of 66 households that were hit by flood in 1997. As expected, the results showed that the insurance decisions were related to a few basic psychological factors of flood-risk perception. The most important finding was that people who reported greater fear while thinking about flood bought insurance more often after the flood had occurred than people who did not report fear. Another factor of flood-risk perception, knowledge about flood, was not related to any aspect of insurance decisions. The practical implications of these findings are discussed.

This study explored whether comparisons of one's perceived absolute risk of being overweight to that of other people are predictive of concern about one's risk and intentions to reduce it. One hundred and six North American college students rated their absolute risk, concern, and intentions, and compared their risk with that of several comparison referents including average targets (e.g., ‘the average same-sex student in the US’), single targets (e.g., ‘best friend’), temporal targets (e.g., ‘yourself a few years ago’), and exemplar targets (e.g., ‘prototypical person with weight problem’). Participants showed high concern and intentions when comparing unfavorably with many of the average targets, even when controlling for absolute risk. Comparisons with single targets were generally related to concern, though not to intentions. Temporal and exemplar comparisons were unrelated to concern and intentions. Comparative risk (particularly with average targets) may elicit worry about one's risk and encourage protective measures, irrespective of one's absolute risk.

This paper examines the effect of characteristics of the decision situation and of the decision maker on decision processes and outcomes in the context of risky choice. Male and female undergraduate students were presented with decisions from different domains of life. For each decision they indicated the likelihood with which they would use each of five decision modes (i.e., ways of making the decision): by following someone's advice, by weighing pros and cons, by following their intuition, etc. They also chose between two courses of action described for each decision and rated the perceived riskiness of both alternatives. We found that the content domain of the decision and/or the gender (or the interaction of both) of the decision maker influenced decision mode usage, and risk perception, behavior, and preference (derived within a risk-return model of risky choice). These results have implications for educational interventions and decision aids that attempt to influence or change the risky decisions of target populations.

In a nationwide study, we explored how Israelis, currently stricken by an intense wave of terrorism, perceive the risk of being the victim of a terrorist attack. We studied both absolute and comparative (i.e., vis-a-vis other people at the area of residence) perceived controllability and vulnerability. The picture that emerges is one of realism. We found no evidence of the comparative optimistic illusions, which characterizes the risk-perception literature. Most participants report some level of behavior change and precautions against the threat of terrorism, but most of them were doubtful about the effectiveness of these precautionary attempts. Perceived absolute vulnerability was the only risk perception variable related to precautionary behaviors. We discuss the disappearance of comparative optimistic biases when the threat is clearly realistic.

Climatological and agricultural research has shown that El Nino cycles in the Pacific Ocean are a good predictor of maize yields in southern Africa, particularly Zimbabwe. However, forecasters can only offer probabilistic predictions, rather than saying with certainty whether Zimbabwe will experience wet or dry conditions. In an effort to avoid confusing farmers, extension service officers translate the forecast into deterministic terms. This approach conflicts with the literature in risk communication, which suggests that participatory discussions of the full forecast content is necessary to maintain credibility over time. But most of the research on which this literature is based has taken place in industrialized countries, and it is unclear whether the lessons apply as well in places like rural Zimbabwe. To test for this, an experiment was conducted with farmers in villages throughout Zimbabwe, in which they revealed their ability to make decisions under situations of uncertainty. The results are qualitatively similar to those of similar experiments conducted in industrialized countries. This suggests that improvements could be made to current forecast communication practices in Zimbabwe.

The current study examined the hypothesis that adolescents' absolute and comparative perceptions of vulnerability to potential negative consequences of substance use mediate the relation between traditional predictors of use, and actual substance use. The data support the hypothesis in that absolute risk perceptions mediate the relations between parental communication about substances, peer substance use and risk-taking tendency, and subsequent adolescent use. Comparative risk perceptions, however, are only weakly related to these three predictors, and do not predict subsequent use. The data also replicate earlier findings that behavioral willingness mediates the relation between risk perceptions and behavior. Methodological and applied implications are discussed.

‘Ambiguous beliefs’ are beliefs which are inconsistent with a unique, additive prior. The problem of their update in face of new information has been dealt with in the theoretical literature, and received several contradictory answers. In particular, the ‘maximum likelihood update’ and the ‘full Bayesian update’ have been axiomatized. This experimental study attempts to test the descriptive validity of these two theories by using the Ellsberg experiment framework.

This study examined the applicability of Huber's (1997) model of risk management to a real-world consumer insurance decision, namely whether to insure a recently purchased item against possible mechanical breakdown in the future. Huber argued that decision makers manage the risks of negative outcomes by applying one or more defusing operators. Respondents in this study asked for whatever information they felt necessary to decide whether to take out an extended warranty on two consumer products of differing values. We found support for most aspects of the model, particularly in relation to risk defusing operators, but also identified some respondents who could not easily be accommodated within it, i.e. those who perceived risk, but did not seem prepared to take any action. We also found evidence for recognition primed insurance decisions. The results are interpreted from a bounded rationality perspective.

This study examined the relations between risk cognitions and risk behavior. Adolescents’ perceptions of the risks associated with driving after drinking (DAD) and their perceptions of the prevalence of this behavior among their peers were assessed, and these perceptions were used to predict their DAD behavior. Results provided evidence of a type of cognitive social influence: the more common adolescents thought the behavior was, the less risk, both personal and general, they attributed to it. As expected, however, this relation was significant only for those who were high in a tendency to engage in social comparison. Perceptions of risk, in turn, were prospectively related to risk behavior for all participants. Specifically, low perceived risk, especially personal risk, was associated with an increase in DAD behavior. Implications of the results for DAD interventions are discussed.

The influence of the probabilistic set-up (i.e., formal aspects of the presented probability information) and of the task domain on the active search for probability information in quasi-natural risky decision tasks was investigated. In each of four tasks (domains: business, medicine, social, epidemic control) 72 subjects chose between a risky alternative and one without risk. There were three conditions in relation to the probabilistic set-up: (a) In the single condition the decision concerned a single case (e.g., one person). In the two multiple conditions (b and c) the decision was for many cases (e.g., 100 persons). In (b) the decision maker had to make an either-or decision (same alternative for all cases). In (c) the subject could assign a proportion of cases to one alternative and the rest to the other one. While the probabilistic set-up had no effect on the search for probabilistic information, the task domain had a strong impact.

This paper surveys results of a research program investigating human judgments of imprecise probabilities under sample-space ignorance (i.e., ignorance of what the possible outcomes are in a decision). The framework used for comparisons with human judgments is primarily due to Walley (1991, 1996). Five studies are reported which test four of Walley's prescriptions for judgment under sample-space ignorance, as well as assessing the impact of the number of observations and types of events on subjective lower and upper probability estimates. The paper concludes with a synopsis of future directions for empirical research on subjective imprecise probability judgments.

Two views of game theory are discussed: (1) game theory as a description of the behavior of rational individuals who recognize each other's reationality and reasoning abilities, and (2) game theory as an internally consistent recommendation to individuals on how to act in interactive situations. It is shown that the same mathematical tool, namely modal logic, can be used to explicitly model both views.

The aim of this research is to determine whether people in the UK associate genetically modified food (GM food) with the ‘mad cow’ disease (or BSE) issue, and explain any such association in terms of analogical reasoning. Further, to test whether a lack of perceived ‘control over exposure’ to GM food accounts for the heightened vulnerability it is typically associated with. Two consecutive questionnaires were administered in various regions of the UK. The first asked 200 participants to indicate whether they regarded GM food as a health risk, and to spontaneously identify issues similar to GM food. The second asked a further 200 to rate the similarity between GM food and BSE, Salmonella, and food irradiation, and to give personal and general risk ratings for consuming GM food, and tolerance ratings for its availability and availability if ‘always clearly labelled’. The results indicate GM food is overwhelmingly identified as a health risk, and that BSE is widely held to be analogous with GM food. However, BSE was rated as less similar to GM food than food irradiation. The widespread identification of BSE as an analogue is explained in terms of it representing a ‘worst-case scenario’. The risk ratings are consistent with the heightened sense of vulnerability associated with GM food; the tolerance ratings are greater for the availability of GM food if ‘always clearly labelled’, indicating a perceived lack of control over exposure to GM food accounts, in part, for GM food risk perceptions. However, the results also indicate that, in terms of practical risk management, labelling may only be effective with respect to a younger age group.

Using an experimental manipulation embedded in a national survey, this study investigated the effect on smokers' risk judgments of receiving accurate information about the cigarette consumption of the average smoker. It was hypothesized that this information would reduce smokers' estimates of the risk of lung cancer faced by the average smoker (‘other's risk’) and hence influence their comparative risk judgments. As predicted, the information made lighter smokers more optimistic and heavier smokers less optimistic. However, the experimental manipulation had no effect on intention to give up smoking. The difference score (other's risk minus own risk) correlated 0.52 with the single-item comparative risk measure. The former measure showed a small but significant optimistic bias whereas the latter measure showed a small but significant pessimistic bias. The findings are discussed in terms of measurement issues and the implications for interventions designed to influence risk perceptions.

Everyone likes clean seawater to bathe in and standards for acceptable seawater quality are set by the European Commission (CEC, 1976). In 1994, proposals to revise these standards were announced. These proposals were the subject of a House of Lords Select Committee Inquiry (HMSO, 1994, 1995), which deplored the fact that a soundly based cost–benefit analysis of the proposed revision had not been produced. This paper considers the question of developing means to assess the adequacy of the proposed revision from a social/public perception standpoint, using a mixed methodology of quantitative survey and qualitative focus groups. The aim of using such an approach is to provide a more in-depth and informative input into the decision-making process for policy makers. Our results show that mean willingness-to-pay amounts, representing the economic benefits of the revision to the 1976 EC bathing water standard, are roughly of the same order of magnitude as the estimated potential cost increases in average annual household water bills necessary to implement the revision. This result is qualified by analysis of how preferences are constructed in terms of socioeconomic variables, perceptions and attitudes towards risk, and issues such as trust, blame and accountability of institutions and regulatory processes involved in setting standards for bathing water quality.

In processing expert judgements, it is common to assign experts’ opinions different degrees of plausibility or probability. This results in the necessity of constructing some kind of hierarchical model. Hierarchical models are widely used in Bayesian inference. Several models have been proposed by the adherents of imprecise statistical reasoning. This paper describes a new hierarchical uncertainty model of a more general form: first- and second-order uncertainty models are both imprecise. We demonstrate that depending on what kind of evidence the modeller has at hand and the reliability quantifying the quality of expert judgement one comes to different rules of combination. It is shown that the rules of combination known in imprecise statistical reasoning can be inferred from the approach proposed in the paper. We describe how the model can be used to generate first-order coherent interval-valued previsions. The paper is concluded by addressing the issue of computing the probability of an event ‘one imprecise value is greater than another’. This case is prominent in light of decision making based on imprecise previsions.

Klein (1997) found that participants were more influenced by information about their comparative risk standing than information about their absolute risk standing. If reliable, these findings have important implications for understanding and improving risk communication. In this paper we report the findings of several unsuccessful attempts by us to replicate Klein's findings in the UK, using one of his experimental paradigms, and discuss the findings of other recent attempts to replicate his work. Findings are inconsistent from study to study but, overall, provide some evidence that people respond to comparative and not just to absolute risk information. Issues that need to be addressed systematically in future research include: the ambiguity of absolute information, proportional differences in risk magnitude, cross-cultural and individual differences in preferences for social comparison information, and the systematic exploration of responses to absolute and comparative risk information in real choice situations.

The standard account of the ‘reflection effect’ (Kahneman and Tversky, 1979) is that attitude toward risk changes across gain or loss framings of outcomes. Weber and Bottom (1989) proposed an alternative account in which decision makers have stable risk attitudes, but changing risk perceptions. Undergraduates were randomly assigned to read one of three hypothetical informed consent documents from a trial of a cholesterol-lowering drug. Documents used gain, loss or both framings to describe expected benefits. Respondents rated riskiness of participation and non-participation in the trial and made a choice about whether they would participate in the trial.

The reflection effect was replicated. In addition, as predicted by the Weber and Bottom account, respondents in the gain condition were more likely to rate participation as riskier than non-participation compared to respondents in the loss condition, and in each condition more than 70 per cent of respondents chose to avoid the option they judged as riskier. Implications for informed consent are discussed.

This paper deals with a model of pollution accumulation in which a catastrophic environmental event occurs once the pollution stock exceeds some uncertain critical level. This problem is studied in a context of ‘hard uncertainty’ since we consider that the available knowledge concerning the value taken by the critical pollution threshold contains both randomness and imprecision. Such a general form of knowledge is modelled as a (closed) random interval. This approach is mathematically tractable and amenable to numerical simulations. In this framework we investigate the effect of hard uncertainty on the optimal pollution/consumption trade-off and we compare the results with those obtained both in the certainty case and in the case of ‘soft uncertainty’ (where only randomness prevails).