The last 20 years or so has seen an intense search carried out within Dempster–Shafer theory, with the aim of finding a generalization of the Shannon entropy for belief functions. In that time, there has also been much progress made in credal set theory—another generalization of the traditional Bayesian epistemic representation—albeit not in this particular area. In credal set theory, sets of probability functions are utilized to represent the epistemic state of rational agents instead of the single probability function of (...) traditional Bayesian theory. The Shannon entropy has been shown to uniquely capture certain highly intuitive properties of uncertainty, and can thus be considered a measure of that quantity. This article presents two measures developed with the purpose of generalizing the Shannon entropy for (1) unordered convex credal sets and (2) possibly non-convex credal sets ordered by second order probability, thereby providing uncertainty measures for such epistemic representations. There is also a comparison with the results of the measure AU developed within Dempster–Shafer theory in a few instances where unordered convex credal set theory and Dempster–Shafer theory overlap. (shrink)

This paper analyzes concepts of independence and assumptions of convexity in the theory of sets of probability distributions. The starting point is Kyburg and Pittarelli’s discussion of “convex Bayesianism” (in particular their proposals concerning E-admissibility, independence, and convexity). The paper offers an organized review of the literature on independence for sets of probability distributions; new results on graphoid properties and on the justification of “strong independence” (using exchangeability) are presented. Finally, the connection between Kyburg and Pittarelli’s results and recent developments (...) on the axiomatization of non-binary preferences, and its impact on “complete” independence, are described. (shrink)

Uncertainty plays an important role in The General Theory, particularly in the theory of interest rates. Keynes did not provide a theory of uncertainty, but he did make some enlightening remarks about the direction he thought such a theory should take. I argue that some modern innovations in the theory of probability allow us to build a theory which captures these Keynesian insights. If this is the right theory, however, uncertainty cannot carry its weight in Keynes’s arguments. This does not (...) mean that the conclusions of these arguments are necessarily mistaken; in their best formulation they may succeed with merely an appeal to risk. (shrink)

The degrees of belief of rational agents should be guided by the evidence available to them. This paper takes as a starting point the view—argued elsewhere—that the formal model best able to capture this idea is one that represents degrees of belief using Dempster–Shafer belief functions. However degrees of belief should not only respect evidence: they also guide decision and action. Whatever formal model of degrees of belief we adopt, we need a decision theory that works with it: that takes (...) as input degrees of belief so represented. The task of this paper is to develop such a decision theory for the belief function model of degrees of belief. This is not the first paper to attempt that task, but compared to the existing literature it takes a more abstract route to its destination, via a consideration of the very idea of rational decision making in light of one’s beliefs and desires. After presenting the new decision theory and comparing it to existing views, the paper goes on to consider diachronic decision situations. (shrink)

Unlike other classical arguments for the existence of God, Pascal’s Wager provides a pragmatic rationale for theistic belief. Its most popular version says that it is rationally mandatory to choose a way of life that seeks to cultivate belief in God because this is the option of maximum expected utility. Despite its initial attractiveness, this long-standing argument has been subject to various criticisms by many philosophers. What is less discussed, however, is the rationality of this choice in situations where the (...) decision-makers are confronted with greater uncertainty. In this paper, I examine the imprecise version of Pascal’s Wager: those scenarios where an agent’s credence that God exists is imprecise or vague rather than precise. After introducing some technical background on imprecise probabilities, I apply five different principles for decision-making to two cases of state uncertainty. In the final part of the paper, I argue that it is not rationally permitted to include zero as the lower probability of God’s existence. Although the conditions for what makes an act uniquely optimal vary significantly across those principles, I also show how the option of wagering for God can defeat any mixed strategy under two distinct interpretations of salvation. (shrink)

This paper aims to draw attention to an important but underappreciated aspect of the instrumental value of freedom: its diagnostic value. This is the value freedom has insofar as it makes it possible for us to discover ourselves and improve ourselves in our capacity to make value judgements. Diagnostic value, I argue, has an important role to play in explaining the value we attach to freedom. Accordingly, this paper is aimed at elucidating this concept, examining its relevance to our lives, (...) and defending an account of how to measure the diagnostic value of opportunities and sets of opportunities. (shrink)

– We offer a new motivation for imprecise probabilities. We argue that there are propositions to which precise probability cannot be assigned, but to which imprecise probability can be assigned. In such cases the alternative to imprecise probability is not precise probability, but no probability at all. And an imprecise probability is substantially better than no probability at all. Our argument is based on the mathematical phenomenon of non-measurable sets. Non-measurable propositions cannot receive precise probabilities, but there is a natural (...) way for them to receive imprecise probabilities. The mathematics of non-measurable sets is arcane, but its epistemological import is far-reaching; even apparently mundane propositions are liable to be affected by non-measurability. The phenomenon of non-measurability dramatically reshapes the dialectic between critics and proponents of imprecise credence. Non-measurability offers natural rejoinders to prominent critics of imprecise credence. Non-measurability even reverses some of the critics’ arguments—by the very lights that have been used to argue against imprecise credences, imprecise credences are better than precise credences. (shrink)

I formulate and tentatively defend the view that we cannot be rationally required to have one type of doxastic attitude (e.g., beliefs, credences, imprecise credences, etc.) because we have another type; in other words, we can only be required to have, say, given credences because we have some other credences already. I explore an argument that appeals to the idea that there is no good reasoning from one type to the other type. I consider some important possible responses, and conclude (...) by sketching a different argument for the same conclusion, appealing to the nature of inquiry and belief-formation. (shrink)

This paper examines preferences among uncertain prospects when the decision maker is uneasy about his assignment of subjective probabilities. It proposes a two-stage lottery framework for the analysis of such prospects, where the first stage represents an assessment of the vagueness (ambiguity) in defining the problem's randomness and the second stage represents an assessment of the problem for each hypothesized randomness condition. Standard axioms of rationality are prescribed for each stage, including weak ordering, continuity, and strong independence. The ‘Reduction of (...) Compound Lotteries' axiom is weakened, however, so that the two lottery stages have consistent, but not collapsible, preference structures. The paper derives a representation theorem from the primitive preference axioms, and the theorem asserts that preference-consistent decisions are made as if the decision maker is maximizing a modified expected utility functional. This representation and its implications are compared to alternative decision models. Criteria for assigning the relative empirical power of the alternative models are suggested. (shrink)

The notion of risk plays a central role in economics, finance, health, psychology, law and elsewhere, and is prevalent in managing challenges and resources in day-to-day life. In recent work, Duncan Pritchard (2015, 2016) has argued against the orthodox probabilistic conception of risk on which the risk of a hypothetical scenario is determined by how probable it is, and in favour of a modal conception on which the risk of a hypothetical scenario is determined by how modally close it is. (...) In this article, we use Pritchard’s discussion as a springboard for a more wide-ranging discussion of the notion of risk. We introduce three different conceptions of risk: the standard probabilistic conception, Pritchard’s modal conception, and a normalcy conception that is new (though it has some precursors in the psychological literature on risk perception). Ultimately, we argue that the modal conception is ill-suited to the roles that a notion of risk is required to play and explore the prospects for a form of pluralism about risk, embracing both the probabilistic and the normalcy conceptions. (shrink)

According to the traditional Bayesian view of credence, its structure is that of precise probability, its objects are descriptive propositions about the empirical world, and its dynamics are given by conditionalization. Each of the three essays that make up this thesis deals with a different variation on this traditional picture. The first variation replaces precise probability with sets of probabilities. The resulting imprecise Bayesianism is sometimes motivated on the grounds that our beliefs should not be more precise than the evidence (...) calls for. One known problem for this evidentially motivated imprecise view is that in certain cases, our imprecise credence in a particular proposition will remain the same no matter how much evidence we receive. In the first essay I argue that the problem is much more general than has been appreciated so far, and that it’s difficult to avoid without compromising the initial evidentialist motivation. The second variation replaces descriptive claims with moral claims as the objects of credence. I consider three standard arguments for probabilism with respect to descriptive uncertainty—representation theorem arguments, Dutch book arguments, and accuracy arguments—in order to examine whether such arguments can also be used to establish probabilism with respect to moral uncertainty. In the second essay, I argue that by and large they can, with some caveats. First, I don’t examine whether these arguments can be given sound non-cognitivist readings, and any conclusions therefore only hold conditional on cognitivism. Second, decision-theoretic representation theorems are found to be less convincing in the moral case, because there they implausibly commit us to thinking that intertheoretic comparisons of value are always possible. Third and finally, certain considerations may lead one to think that imprecise probabilism provides a more plausible model of moral epistemology. The third variation considers whether, in addition to conditionalization, agents may also change their minds by becoming aware of propositions they had not previously entertained, and therefore not previously assigned any probability. More specifically, I argue that if we wish to make room for reflective equilibrium in a probabilistic moral epistemology, we must allow for awareness growth. In the third essay, I sketch the outline of such a Bayesian account of reflective equilibrium. Given that this account gives a central place to awareness growth, and that the rationality constraints on belief change by awareness growth are much weaker than those on belief change by conditionalization, it follows that the rationality constraints on the credences of agents who are seeking reflective equilibrium are correspondingly weaker. (shrink)

There is currently much discussion about how decision making should proceed when an agent's degrees of belief are imprecise; represented by a set of probability functions. I show that decision rules recently discussed by Sarah Moss, Susanna Rinard and Rohan Sud all suffer from the same defect: they all struggle to rationalize diachronic ambiguity aversion. Since ambiguity aversion is among the motivations for imprecise credence, this suggests that the search for an adequate imprecise decision rule is not yet over.

In this paper I offer an alternative - the ‘dispositional account’ - to the standard account of imprecise probabilism. Whereas for the imprecise probabilist, an agent’s credal state is modelled by a set of credence functions, on the dispositional account an agent’s credal state is modelled by a set of sets of credence functions. On the face of it, the dispositional account looks less elegant than the standard account – so why should we be interested? I argue that the dispositional (...) account is actually simpler, because the dispositional choice behaviour that fixes an agent’s credal state is faithfully depicted in the model of that agent’s credal state. I explore some of the implications of the account, including a surprising implication for the debate over dilation. (shrink)

Innovative research on decision making under ‘deep uncertainty’ is underway in applied fields such as engineering and operational research, largely outside the view of normative theorists grounded in decision theory. Applied methods and tools for decision support under deep uncertainty go beyond standard decision theory in the attention that they give to the structuring of decisions. Decision structuring is an important part of a broader philosophy of managing uncertainty in decision making, and normative decision theorists can both learn from, and (...) contribute to, the growing deep uncertainty decision support literature. (shrink)

This paper proposes a new model of graded modal judgment. It begins by problematizing the phenomenon: given plausible constraints on the logic of epistemic modality, it is impossible to model graded attitudes toward modal claims as judgments of probability targeting epistemically modal propositions. This paper considers two alternative models, on which modal operators are non-proposition-forming: (1) Moss (2015), in which graded attitudes toward modal claims are represented as judgments of probability targeting a “proxy” proposition, belief in which would underwrite belief (...) in the modal claim. (2) A model on which graded attitudes toward modal claims are represented as judgments of credence taking as their objects (non-propositional) modal representations (rather than proxy propositions). The second model, like Moss’ model, is shown to be semantically and mathematically tractable. The second model, however, can be straightforwardly integrated into a plausible model of the role of graded attitudes toward modal claims in cognition and normative epistemology. (shrink)

The standard Bayesian recipe for selecting the rational choice is presented. A familiar example in which the recipe fails to produce any definite result is introduced. It is argued that a generalization of Gärdenfors’ and Sahlin’s theory of unreliable probabilities — which itself does not guarantee a solution to the problem — offers the best available approach. But a number of challenges to this approach are also presented and discussed.

Many have argued that a rational agent's attitude towards a proposition may be better represented by a probability range than by a single number. I show that in such cases an agent will have unstable betting behaviour, and so will behave in an unpredictable way. I use this point to argue against a range of responses to the ‘two bets’ argument for sharp probabilities.

Traditional Bayesianism requires that an agent’s degrees of belief be represented by a real-valued, probabilistic credence function. However, in many cases it seems that our evidence is not rich enough to warrant such precision. In light of this, some have proposed that we instead represent an agent’s degrees of belief as a set of credence functions. This way, we can respect the evidence by requiring that the set, often called the agent’s credal state, includes all credence functions that are in (...) some sense compatible with the evidence. One known problem for this evidentially motivated imprecise view is that in certain cases, our imprecise credence in a particular proposition will remain the same no matter how much evidence we receive. In this article I argue that the problem is much more general than has been appreciated so far, and that it’s difficult to avoid it without compromising the initial evidentialist motivation. _1_ Introduction _2_ Precision and Its Problems _3_ Imprecise Bayesianism and Respecting Ambiguous Evidence _4_ Local Belief Inertia _5_ From Local to Global Belief Inertia _6_ Responding to Global Belief Inertia _7_ Conclusion. (shrink)

The question of how the probabilistic opinions of different individuals should be aggregated to form a group opinion is controversial. But one assumption seems to be pretty much common ground: for a group of Bayesians, the representation of group opinion should itself be a unique probability distribution, 410–414, [45]; Bordley Management Science, 28, 1137–1148, [5]; Genest et al. The Annals of Statistics, 487–501, [21]; Genest and Zidek Statistical Science, 114–135, [23]; Mongin Journal of Economic Theory, 66, 313–351, [46]; Clemen and (...) Winkler Risk Analysis, 19, 187–203, [7]; Dietrich and List [14]; Herzberg Theory and Decision, 1–19, [28]). We argue that this assumption is not always in order. We show how to extend the canonical mathematical framework for pooling to cover pooling with imprecise probabilities by employing set-valued pooling functions and generalizing common pooling axioms accordingly. As a proof of concept, we then show that one IP construction satisfies a number of central pooling axioms that are not jointly satisfied by any of the standard pooling recipes on pain of triviality. Following Levi, 3–11, [39]), we also argue that IP models admit of a much better philosophical motivation as a model of rational consensus. (shrink)

This paper considers a puzzling conflict between two positions that are each compelling: it is irrational for an agent to pay to avoid `free' evidence before making a decision, and rational agents may have imprecise beliefs and/or desires. Indeed, we show that Good's theorem concerning the invariable choice-worthiness of free evidence does not generalise to the imprecise realm, given the plausible existing decision theories for handling imprecision. A key ingredient in the analysis, and a potential source of controversy, is the (...) general approach taken for resolving sequential decision problems { we make explicit what the key alternatives are and defend our own approach. Furthermore, we endorse a resolution of the aforementioned puzzle { we privilege decision theories that merely permit avoiding free evidence over decision theories for which avoiding free evidence is uniquely admissible. Finally, we situate this particular result about free evidence within the broader `dynamic-coherence' literature. (shrink)

The Intergovernmental Panel on Climate Change has developed a novel framework for assessing and communicating uncertainty in the findings published in their periodic assessment reports. But how should these uncertainty assessments inform decisions? We take a formal decision-making perspective to investigate how scientific input formulated in the IPCC’s novel framework might inform decisions in a principled way through a normative decision model.

The menu-dependent nature of regret-minimization creates subtleties when it is applied to dynamic decision problems. It is not clear whether forgone opportunities should be included in the menu. We explain commonly observed behavioral patterns as minimizing regret when forgone opportunities are present. If forgone opportunities are included, we can characterize when a form of dynamic consistency is guaranteed.

It is natural to think of precise probabilities as being special cases of imprecise probabilities, the special case being when one’s lower and upper probabilities are equal. I argue, however, that it is better to think of the two models as representing two different aspects of our credences, which are often vague to some degree. I show that by combining the two models into one model, and understanding that model as a model of vague credence, a natural interpretation arises that (...) suggests a hypothesis concerning how we can improve the accuracy of aggregate credences. I present empirical results in support of this hypothesis. I also discuss how this modeling interpretation of imprecise probabilities bears upon a philosophical objection that has been raised against them, the so-called inductive learning problem. (shrink)

Many fine-grained decisions concerning climate change involve significant, even severe, uncertainty. Here, we focus on modelling the decisions of single agents, whether individual persons or groups perceived as corporate entities. We offer a taxonomy of the sources and kinds of uncertainty that arise in framing these decision problems, as well as strategies for making a choice in spite of uncertainty. The aim is to facilitate a more transparent and structured treatment of uncertainty in climate decision making.

In this paper we present two distinctly epistemological puzzles that arise for one who aspires to defend some plausible version of the precautionary principle. The first puzzle involves an application of contextualism in epistemology; and the second puzzle concerns the task of defending a plausible version of the precautionary principle that would not be invalidated by de minimis.

Probabilistic belief contraction has been a much neglected topic in the field of probabilistic reasoning. This is due to the difficulty in establishing a reasonable reversal of the effect of Bayesian conditionalization on a probabilistic distribution. We show that indifferent contraction, a solution proposed by Ramer to this problem through a judicious use of the principle of maximum entropy, is a probabilistic version of a full meet contraction. We then propose variations of indifferent contraction, using both the Shannon entropy measure (...) as well as the Hartley entropy measure, with an aim to avoid excessive loss of beliefs that full meet contraction entails. (shrink)

In Chapter 12 of the General Theory, on “The State of Long-Term Expectation,” Keynes writes: “It would be foolish, in forming our expectations, to attach great weight to matters which are very uncertain”. In a footnote to this sentence, Keynes points out that by “very uncertain” he does not mean the same as “very improbable” and refers to the chapter on “The Weight of Arguments” in his earlier Treatise on Probability. The purpose of this article, in the first place, is (...) to provide an account of, and to sort out the relations between, Keynes's views on probability, uncertainty, and the weight of arguments. (shrink)

Ellsberg (The Quarterly Journal of Economics 75, 643–669 (1961); Risk, Ambiguity and Decision, Garland Publishing (2001)) argued that uncertainty is not reducible to risk. At the center of Ellsberg’s argument lies a thought experiment that has come to be known as the three-color example. It has been observed that a significant number of sophisticated decision makers violate the requirements of subjective expected utility theory when they are confronted with Ellsberg’s three-color example. More generally, such decision makers are in conflict with (...) either the ordering assumption or the independence assumption of subjective expected utility theory. While a clear majority of the theoretical responses to these violations have advocated maintaining ordering while relaxing independence, a persistent minority has advocated abandoning the ordering assumption. The purpose of this paper is to consider a similar dilemma that exists within the context of multiattribute models, where it arises by considering indeterminacy in the weighting of attributes rather than indeterminacy in the determination of probabilities as in Ellsberg’s example. (shrink)

According to a standard account of evidence, one piece of information is stronger evidence for an hypothesis than is another iff the probability of the hypothesis on the one is greater than it is on the other. This condition, I argue, is neither necessary nor sufficient because various factors can strengthen the evidence for an hypothesis without increasing (and even decreasing) its probability. Contrary to what probabilists claim, I show that this obtains even if a probability function can take these (...) evidential factors into account in ways they suggest and yield a unique probability value. Nor will the problem be solved by appealing to second-order probabilities. (shrink)

The aim of this paper is to provide a detailed characterisation of some ways in which our preferences reflect our reasons. I will argue that practical reasons can be characterised along two dimensions that influence our preferences: their balance and their weight. This is analogous to a similar characterisation of the way in which probabilities reflect the balance and weight of evidence in epistemology. In this paper, I will illustrate the distinction between the balance and weight of reasons, and show (...) how this is crucial for an adequate account of preference and choice. The upshot is a more complete picture of a particular kind of decision, labelled by Isaac Levi and, more recently, Ruth Chang as ‘hard choices’. These are choices in which one option is better than another in some ways, the other is better than the first in some ways, but neither seems better overall. The distinction between the balance and weight of reasons presents a new way of understanding how hard choices vary by degree and what it is that makes them so hard. (shrink)

Abstract:The standard, Bayesian account of rational belief and decision is often argued to be unable to cope properly with severe uncertainty, of the sort ubiquitous in some areas of policy making. This paper tackles the question of what should replace it as a guide for rational decision making. It defends a recent proposal, which reserves a role for the decision maker’s confidence in beliefs. Beyond being able to cope with severe uncertainty, the account has strong normative credentials on the main (...) fronts typically evoked as relevant for rational belief and decision. It fares particularly well, we argue, in comparison to other prominent non-Bayesian models in the literature. (shrink)

Standard applications of evolutionary game theory look at a single game and focus on the evolution of behavior for that game alone. Instead, this article uses tools from evolutionary game theory to study the competition between choice mechanisms in a rich and variable multigame environment. A choice mechanism is a way of subjectively representing a decision situation, paired with a method for choosing an act based on this subjective representation. We demonstrate the usefulness of this approach by a case study (...) that shows how subjective representations in terms of regret that differ from the actual fitness can be evolutionarily advantageous. (shrink)

Most investigations into how people make risky choices have employed a simple drosophila: monetary gambles involving stated outcomes and probabilities. People are asked to make decisions from description . When people decide whether to back up their computer hard drive, cross a busy street, or go out on a date, however, they do not enjoy the convenience of stated outcomes and probabilities. People make such decisions either in the void of ignorance or in the twilight of their own often limited (...) experience of such real-world options. In the latter case, they make decisions from experience . Recent research has consistently documented that decisions from description and decisions from experience can lead to substantially different choices. Key in this description–experience gap is people’s treatment of rare events. In this paper, I briefly review studies that have documented the description–experience gap, offer several explanations for this gap, and discuss to what extent people’s decisions from experience are in conflict with benchmarks of rationality. (shrink)

Safety is an important topic for a wide range of disciplines, such as engineering, economics, sociology, psychology, political science and philosophy, and plays a central role in risk analysis and risk management. The aim of this thesis is to develop a concept of safety that is relevant for decision-making, and to elucidate its consequences for risk and safety research and practices. Essay I provides a conceptual analysis of safety in the context of societal decision-making, focusing on some fundamental distinctions and (...) aspects, and argues for a more complex notion than what is commonly given. This concept of safety explicitly includes epistemic uncertainty, the degree to which we are uncertain of our knowledge of the situation at hand. It is discussed the extent to which such a concept may be considered an objective concept, and concluded that it is better seen as an intersubjective concept. Some formal versions of a comparative safety concept are also proposed. Essay II explores some consequences of epistemic uncertainty. It is commonly claimed that the public is irrational in its acceptance of risks. An underlying presumption in such a claim is that the public should follow the experts’ advice in recommending an activity whenever the experts have better knowledge of the risk involved. This position is criticised based on considerations from epistemic uncertainty and the goal of safety. Furthermore, it is shown that the scope of the objection covers the entire field of risk research, risk assessment as well as risk management. Essay III analyses the role of epistemic uncertainty for principles of achieving safety in an engineering context. The aim is to show that to account for common engineering principles we need the understanding of safety that has been argued for in Essays I-II. Several important principles in engineering safety are analysed, and it is argued that we cannot fully account for them on a narrow interpretation of safety as the reduction of risk. An adequate concept of safety must include not only the reduction of risk but also the reduction of uncertainty. (shrink)

Imprecise probabilities are an increasingly popular way of reasoning about rational credence. However they are subject to an apparent failure to display convincing inductive learning. This paper demonstrates that a small modification to the update rule for IP allows us to overcome this problem, albeit at the cost of satisfying only a weaker concept of coherence.

According to Jon Elster, mechanisms are frequently occurring and easily recognizable causal patterns that are triggered under generally unknown conditions or with indeterminate consequences. In the absence of laws, moreover, mechanisms provide explanations. In this paper I argue that Elster’s view has difficulties with progressing knowledge. Normally, filling in the causal picture without revising it should not threaten one’s explanation. But this seems to be Elster’s case. The critique is constructive in the sense that it is built up from a (...) discussion of a mechanism that might explain ‘unwarranted’ risk taking in connection with swimming—a mechanism that is mirrored in the proverb: The best swimmers drown. (shrink)

This paper studies a generalization of rational choice theory. I briefly review the motivations that Helzner gives for his conditional choice construction . Then, I focus on the important class of conditional choice functions with vacuous second tiers. This class is interesting for both formal and philosophical reasons. I argue that this class makes explicit one of conditional choice’s normative motivations in terms of an account of neutrality advocated within a certain tradition in decision theory. The observations recorded—several of which (...) are generalizations of central results in the standard theory of rational choice—are intended to provide further insight into how conditional choice generalizes the standard account and are offered as additional evidence of the fruitfulness of the conditional choice framework. (shrink)

This article deals with the impact of governmental assistance on insurance demand under ambiguity, i.e., in situations where probabilities are uncertain. First, using a model of insurance demand under ambiguity, we derive theoretical predictions about the impact of several governmental assistance programmes on optimal insurance demand. For example, governmental assistance through a fixed public support scheme implies that partial insurance is always optimal under fair insurance with ambiguity. Second, we present the results of an experiment designed to test these predictions. (...) We find support for several of our theoretical predictions. For example, the presence of governmental assistance through a fixed public support scheme decreases individuals’ willingness to pay to be fully insured. Finally, we compare these results with those obtained for a risk situation. We find that, regardless of the form of governmental assistance, participants in the ambiguity context are consistently willing to pay more to be fully insured than participants in the risk situation. (shrink)