Showing papers on "Ambiguity aversion published in 2010"
TL;DR: A common system, consisting of at least the striatum and the medial prefrontal cortex, was found to represent subjective value under both conditions, and this work concluded that risk aversion and ambiguity aversion are distinct phenomena.
Abstract: Risk and ambiguity are two conditions in which the consequences of possible outcomes are not certain. Under risk, the probabilities of different outcomes can be estimated, whereas under ambiguity, even these probabilities are not known. Although most people exhibit at least some aversion to both risk and ambiguity, the degree of these aversions is largely uncorrelated across subjects, suggesting that risk aversion and ambiguity aversion are distinct phenomena. Previous studies have shown differences in brain activations for risky and ambiguous choices and have identified neural mechanisms that may mediate transitions from conditions of ambiguity to conditions of risk. Unknown, however, is whether the value of risky and ambiguous options is necessarily represented by two distinct systems or whether a common mechanism can be identified. To answer this question, we compared the neural representation of subjective value under risk and ambiguity. fMRI was used to track brain activation while subjects made choices regarding options that varied systematically in the amount of money offered and in either the probability of obtaining that amount or the level of ambiguity around that probability. A common system, consisting of at least the striatum and the medial prefrontal cortex, was found to represent subjective value under both conditions.
405 citations
TL;DR: This article studied the impact of ambiguity and ambiguity aversion on equilibrium asset prices and portfolio holdings in competitive financial markets, and found that attitudes toward ambiguity are heterogeneous across the population, just as attitudes toward risk are heterogeneities across the populations, but that heterogeneity of attitudes towards ambiguity has different implications than heterogeneity of attitude toward risk, and that investors who are sufficiently ambiguity averse find open sets of prices for which they refuse to hold an ambiguous portfolio.
Abstract: This paper studies the impact of ambiguity and ambiguity aversion on equilibrium asset prices and portfolio holdings in competitive financial markets. It argues that attitudes toward ambiguity are heterogeneous across the population, just as attitudes toward risk are heterogeneous across the population, but that heterogeneity of attitudes toward ambiguity has different implications than heterogeneity of attitudes toward risk. In particular, when some state probabilities are not known, agents who are sufficiently ambiguity averse find open sets of prices for which they refuse to hold an ambiguous portfolio. This suggests a different cross section of portfolio choices, a wider range of state price/probability ratios, and different rankings of state price/probability ratios than would be predicted if state probabilities were known. Experiments confirm all of these suggestions. Our findings contradict the claim that investors who have cognitive biases do not affect prices because they are inframarginal: ambiguity-averse investors have an indirect effect on prices because they change the per capita amount of risk that is to be shared among the marginal investors. Our experimental data also suggest a positive correlation between risk aversion and ambiguity aversion that might explain the "value effect" in historical data. (JEL G11, G12, C92, D53)
335 citations
TL;DR: In this article, the authors review models of ambiguity aversion and show that such models have implications for portfolio choice and asset pricing that are very different from those of SEU and that help to explain otherwise puzzling features of the data.
Abstract: The Ellsberg paradox suggests that people behave differently in risky situations -- when they are given objective probabilities -- than in ambiguous situations when they are not told the odds (as is typical in financial markets). Such behavior is inconsistent with subjective expected utility theory (SEU), the standard model of choice under uncertainty in financial economics. This article reviews models of ambiguity aversion. It shows that such models -- in particular, the multiple-priors model of Gilboa and Schmeidler -- have implications for portfolio choice and asset pricing that are very different from those of SEU and that help to explain otherwise puzzling features of the data.
330 citations
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TL;DR: In this article, a generalized recursive smooth ambiguity model is proposed to allow a three-way separation among risk aversion, ambiguity aversion, and intertemporal substitution in a consumption-based asset-pricing model.
Abstract: We propose a novel generalized recursive smooth ambiguity model which permits a three-way separation among risk aversion, ambiguity aversion, and intertemporal substitution. We apply this utility model to a consumption-based asset-pricing model in which consumption and dividends follow hidden Markov regime-switching processes. Our calibrated model can match the mean equity premium, the mean risk-free rate, and the volatility of the equity premium observed in the data. In addition, our model can generate a variety of dynamic asset-pricing phenomena, including the procyclical variation of price–dividend ratios, the countercyclical variation of equity premia and equity volatility, the leverage effect, and the mean reversion of excess returns. The key intuition is that an ambiguity-averse agent behaves pessimistically by attaching more weight to the pricing kernel in bad times when his continuation values are low.
323 citations
TL;DR: This article found that when faced with a decision how to split their investment between a risky lottery and an asset with a fixed return, people increase the proportion invested in the risky option the more they like the lottery.
Abstract: We study the following basic intuition: when faced with a decision how to split their investment between a risky lottery and an asset with a fixed return, people increase the proportion invested in the risky option the more they like the lottery. We find counter-examples to this, and in fact we find no simple relation between preferences between lotteries and the fraction invested in them. We use three well-documented biases (ambiguity aversion, the illusion of control and myopic loss aversion) to show this. First we replicate the previous results in a laboratory experiment with financial incentives, and then test whether participants are willing to explicitly pay a small sum of money in line with the bias (pay for less ambiguity, more perceived control, or more frequent information about portfolio performance). We then study how portfolio choice depends on these biases. With the parameters chosen, the illusion of control was eliminated when participants were asked to pay to gain more control, and the bias did not affect investment behavior (i.e., participants invested in a risky option the same fraction when faced with more or less control). In the ambiguity treatment, people were willing to pay for less ambiguity, but again the level of ambiguity did not influence investment. Finally, in the myopic loss aversion treatment participants were willing to pay money to have more freedom to choose, even though (in line with the documented bias) they invested less when having more freedom to change their investment.
210 citations
TL;DR: A dual system model (DSM) of decision making under risk and uncertainty according to which the value of a gamble is a combination of the values assigned to it independently by the affective and deliberative systems is presented.
Abstract: This article presents a dual system model (DSM) of decision making under risk and uncertainty according to which the value of a gamble is a combination of the values assigned to it independently by the affective and deliberative systems. On the basis of research on dual process theories and empirical research in Hsee and Rottenstreich (2004) and Rottenstreich and Hsee (2001) among others, the DSM incorporates (a) individual differences in disposition to rational versus emotional decision making, (b) the affective nature of outcomes, and (c) different task construals within its framework. The model has good descriptive validity and accounts for (a) violation of nontransparent stochastic dominance, (b) fourfold pattern of risk attitudes, (c) ambiguity aversion, (d) common consequence effect, (e) common ratio effect, (f) isolation effect, and (g) coalescing and event-splitting effects. The DSM is also used to make several novel predictions of conditions under which specific behavior patterns may or may not occur.
175 citations
TL;DR: It is demonstrated that people can adopt a favorable view of ambiguous risks relative to ones with known probabilities, contrary to the usual attitude of ambiguity aversion, when doing so permits justification for unfair behavior.
165 citations
TL;DR: Barberis et al. as mentioned in this paper characterized generalized disappointment aversion (GDA) risk preferences that can overweight lower-tail outcomes relative to expected utility, and showed in an endowment economy that recursive utility with GDA risk preferences generates effective risk aversion that is countercyclical.
Abstract: We characterize generalized disappointment aversion (GDA) risk preferences that can overweight lower-tail outcomes relative to expected utility. We show in an endowment economy that recursive utility with GDA risk preferences generates effective risk aversion that is countercyclical. This feature comes from endogenous variation in the probability of disappointment in the representative agent's intertemporal consumption-saving problem that underlies the asset pricing model. The variation in effective risk aversion produces a large equity premium and a risk-free rate that is procyclical and has low volatility in an economy with a simple autoregressive endowment-growth process. The observed facts of aggregate risks and asset prices in the post-war U.S. economy have led researchers to explore models of intertemporal preferences that generalize the well-developed time-additive expected utility specification used in the asset pricing economy of Lucas (1978). In particular, models that allow for countercyclical risk aversion have been shown to perform much better than the standard model (see, e.g., Campbell and Cochrane (1999), Gordon and St-Amour (2000), and Barberis, Huang, and Santos (2001)).1 This leads us to explore the preference foundations that may produce this behavior endogenously. Our preference specification is a one-parameter extension of the Gul (1991) disappointment aversion utility function. These new preferences
163 citations
TL;DR: The authors derived the Arrow-Pratt approximation of the certainty equivalent under model uncertainty as defined by the smooth model of decision making under ambiguity of Klibanoff, Marinacci and Mukerji.
Abstract: We derive the analogue of the classic Arrow-Pratt approximation of the certainty equivalent under model uncertainty as defined by the smooth model of decision making under ambiguity of Klibanoff, Marinacci and Mukerji (2005). We study its scope via a portfolio allocation exercise that delivers a tractable mean-variance model adjusted for model uncertainty. In a problem with a risk-free asset, a risky asset, and an ambiguous asset, we find that portfolio rebalancing in response to higher model uncertainty only depends on the ambiguous asset's alpha, setting the performance of the risky asset as benchmark. In addition, the portfolios recommended by our model are not systematically conservative on the share held in the ambiguous asset: indeed, in general, it is not true that greater ambiguity reduces the optimal demand for the ambiguous asset. The analytical tractability of the enhanced Arrow-Pratt approximation renders our model especially well suited for calibration exercises aimed at exploring the consequences of ambiguity aversion on equilibrium asset prices.
132 citations
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TL;DR: The concepts of ambiguity and ambiguity aversion are used to formalize the idea of an investor's “familiarity” toward assets and show that for any given level of expected returns, the optimal portfolio depends on two quantities.
Abstract: We develop a model of portfolio choice to nest the views of Keynes - who advocates concentration in a few familiar assets - and Markowitz - who advocates diversification across assets. We rely on the concepts of ambiguity and ambiguity aversion to formalize the idea of an investors "familiarity" toward assets. The model shows that when an investor is equally ambiguous about all assets, then the optimal portfolio corresponds to Markowitzs fully diversified portfolio. In contrast, when an investor exhibits different degrees of familiarity across assets, the optimal portfolio depends on (i) the relative degree of ambiguity across assets, and (ii) the standard deviation of the estimate of expected return on each asset. If the standard deviation of the expected return estimate and the difference between the ambiguity about familiar and unfamiliar assets are low, then the optimal portfolio is composed of a mix of both familiar and unfamiliar assets; moreover, an increase in correlation between assets causes an investor to increase concentration in the assets with which they are familiar (flight to familiarity). Alternatively, if the standard deviation of the expected return estimate and the difference between the ambiguity of familiar and unfamiliar assets are high, then the optimal portfolio contains only the familiar asset(s) as Keynes would have advocated. In the extreme case in which the ambiguity about all assets and the standard deviation of the estimated mean are high, then no risky asset is held (non-participation). The model also has empirically testable implications for trading behavior: in response to a change in idiosyncratic volatility, the Keynesian portfolio always exhibits more trading than the Markowitz portfolio, while the opposite is true for a change in systematic volatility. In the equilibrium version of the model with heterogeneous investors who are familiar with different assets, we find that the risk premium of stocks depends on both systematic and idiosyncratic volatility, and that the equity risk premium is significantly higher than in the standard model without ambiguity.
127 citations
TL;DR: Robust control theory as discussed by the authors is a tool for assessing decision rules when a decision maker distrusts either the specification of transition laws or the distribution of hidden state variables or both, and it can be applied to asset pricing uncertainty premia and design of robust macroeconomic policies.
Abstract: Robust control theory is a tool for assessing decision rules when a decision maker distrusts either the specification of transition laws or the distribution of hidden state variables or both. Specification doubts inspire the decision maker to want a decision rule to work well for a ∅ of models surrounding his approximating stochastic model. We relate robust control theory to the so-called multiplier and constraint preferences that have been used to express ambiguity aversion. Detection error probabilities can be used to discipline empirically plausible amounts of robustness. We describe applications to asset pricing uncertainty premia and design of robust macroeconomic policies.
TL;DR: In this article, the authors take the Anscombe-Aumann framework with horse and roulette lotteries, and apply the Savage axioms to the horse and the von Neumann-Morgenstern axiomas to the roulette lotsteries.
Abstract: This paper takes the Anscombe-Aumann framework with horse and roulette lotteries, and applies the Savage axioms to the horse lotteries and the von Neumann-Morgenstern axioms to the roulette lotteries. The resulting representation of preferences yields a subjective probability measure over states and two utility functions, one governing risk attitudes and one governing ambiguity attitudes. The model is able to accommodate the Ellsberg paradox and preferences for reductions in ambiguity.
TL;DR: In this article, the value of information in a non-expected utility model of ambiguity with second-order probabilities is studied, and it is shown that information that reduces ambiguity has a positive value for ambiguity-averse decision makers.
Abstract: 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.
TL;DR: The authors survey the literature that has explored the implications of decision-making under ambiguity for financial market outcomes, such as portfolio choice and equilibrium asset prices, leading to a number of significant advances in our ability to rationalize empirical features of asset returns and portfolio decisions.
Abstract: We survey the literature that has explored the implications of decision-making under ambiguity for financial market outcomes, such as portfolio choice and equilibrium asset prices. This ambiguity literature has led to a number of significant advances in our ability to rationalize empirical features of asset returns and portfolio decisions, such as the failure of the two-fund separation theorem in portfolio decisions, the modest exposure to risky securities observed for a majority of investors, the home equity preference in international portfolio diversification, the excess volatility of asset returns, the equity premium and the risk-free rate puzzles, and the occurrence of trading break-downs.
TL;DR: The authors showed that ambiguity aversion increases the value of a statistical life if the marginal utility of an increase in wealth is larger if one is alive rather than dead, and showed that the total effect of ambiguity aversion on self-protection is unclear.
TL;DR: In this article, two Ellsberg-style thought experiments are described that reflect on the smooth ambiguity decision model developed by Klibanoff, Marinacci, and Mukerji (2005).
Abstract: Two Ellsberg-style thought experiments are described that reflect on the smooth ambiguity decision model developed by Klibanoff, Marinacci, and Mukerji (2005). The first experiment poses difficulties for the model's axiomatic foundations and, as a result, also for its interpretation, particularly for the claim that the model achieves a separation between ambiguity and the attitude toward ambiguity. Given the problematic nature of its foundations, the behavioral content of the model and how it differs from multiple priors, for example, are not clear. The second thought experiment casts some light on these questions.
TL;DR: In this article, a decision-theoretic framework for preferences under uncertainty is developed, where the agent considers a full-dimensional set of possible priors and abandons her status quo option only if she finds an alternative that returns a higher expected utility for each of these priors.
TL;DR: It is demonstrated that a preference for unambiguous options is shared with rhesus macaques, and data indicate that ambiguity aversion reflects fundamental cognitive biases shared with other animals rather than uniquely human factors guiding decisions.
Abstract: People generally prefer risky options, which have fully specified outcome probabilities, to ambiguous options, which have unspecified probabilities. This preference, formalized in economics, is strong enough that people will reliably prefer a risky option to an ambiguous option with a greater expected value. Explanations for ambiguity aversion often invoke uniquely human faculties like language, self-justification, or a desire to avoid public embarrassment. Challenging these ideas, here we demonstrate that a preference for unambiguous options is shared with rhesus macaques. We trained four monkeys to choose between pairs of options that both offered explicitly cued probabilities of large and small juice outcomes. We then introduced occasional trials where one of the options was obscured and examined their resulting preferences; we ran humans in a parallel experiment on a nearly identical task. We found that monkeys reliably preferred risky options to ambiguous ones, even when this bias was costly, closely matching the behavior of humans in the analogous task. Notably, ambiguity aversion varied parametrically with the extent of ambiguity. As expected, ambiguity aversion gradually declined as monkeys learned the underlying probability distribution of rewards. These data indicate that ambiguity aversion reflects fundamental cognitive biases shared with other animals rather than uniquely human factors guiding decisions.
TL;DR: The authors reviewed models of ambiguity aversion and showed that such models have implications for portfolio choice and asset pricing that are very different from those of SEU and that help to explain otherwise puzzling features of the data.
TL;DR: In a recent article, Machina (Am Econ Rev forthcoming, 2008) suggested choice problems in the spirit of Ellsberg (Q J Econ 75:643-669, 1961), which challenge tail-separability, an implication of Choquet expected utility as discussed by the authors.
Abstract: In a recent article, Machina (Am Econ Rev forthcoming, 2008) suggested choice problems in the spirit of Ellsberg (Q J Econ 75:643-669, 1961), which challenge tail-separability, an implication of Choquet expected utility (CEU), to a similar extent as the Ellsberg paradox challenged the sure-thing principle implied by subjective expected utility (SEU). We have tested choice behavior for bets on one of Machina's choice problems, the reflection example. Our results indicate that tail-separability is violated by a large majority of subjects (over 70% of the sample). These empirical findings complement the theoretical analysis of Machina (Am Econ Rev forthcoming, 2008) and, together, they confirm the need for new approaches in the analysis of ambiguity for decision making.
TL;DR: In this paper, the problem of the timing of an investment decision under partial information is analyzed in a framework where the firm is ambiguity averse, and the analysis yields the description of a robust decision rule for an investment in a finite life project in presence of a stochastic instantaneous return.
Abstract: The problem of the timing of an investment decision under partial information is analyzed in a framework where the firm is ambiguity averse. The analysis yields the description of a robust decision rule for an investment in a finite life project in presence of a stochastic instantaneous return. It is demonstrated that ambiguity aversion may accelerate investment in the short run. Ex post validation of the determined investment policy treats the impact of ambiguity aversion on the proper way of discounting of the profit flow resulting from the project and the fair price of risk associated with ambiguity aversion.
TL;DR: The authors apply static and dynamic versions of a smooth ambiguity model to climate mitigation policy and obtain a general result on the comparative statics of optimal abatement and ambiguity aversion and illustrate this sufficient condition in some simple examples.
Abstract: Economic evaluation of climate policy traditionally treats uncertainty by appealing to expected utility theory. Yet our knowledge of the impacts of climate policy may not be of
sufficient quality to justify probabilistic beliefs. In such circumstances, it has been argued that the axioms of expected utility theory may not be the correct standard of rationality. By contrast, several axiomatic frameworks have recently been proposed that account for ambiguous beliefs. In this paper, we apply static and dynamic versions of a smooth ambiguity model to climate mitigation policy. We obtain a general result on the comparative statics of optimal abatement and ambiguity aversion and illustrate this sufficient condition in some simple examples. We then extend our analysis to a more realistic, dynamic setting, and adapt a well-known empirical model of the climate-economy system to show that the value of emissions abatement increases as ambiguity aversion increases, and that this ‘ambiguity premium’ can in some plausible cases be very large.
TL;DR: Meyer et al. as discussed by the authors showed that it is generally reasonable to make such an assumption, especially when the form of the utility function and the bounds on the degree of risk aversion are carefully chosen.
Abstract: A recent contribution by Meyer et al. (2009, p. 521) corrected an error of fact by Hardaker et al. (2004b, p. 253) about the comparison between stochastic dominance with respect to a function (SDRF) and stochastic efficiency with respect to a function (SERF). While both methods compare risky prospects for a bounded range of degrees of risk aversion, SERF, unlike SDRF, also demands an assumption that a chosen measure of risk aversion is constant over all levels of outcomes being evaluated. It is argued that it is generally reasonable to make such an assumption, especially when the form of the utility function and the bounds on the degree of risk aversion are carefully chosen. Then SERF has the advantage that it can lead to a smaller efficient set than that identified by SDRF. SERF also has advantages of ease and transparency in use.
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TL;DR: In this paper, it was shown that when individuals have (a special form of) maximin expected utility (MEU) preferences, then any efficient allocation is incentive compatible, whereas only MEU preferences have this property.
Abstract: The conflict between Pareto optimality and incentive compatibility, that is, the fact that some Pareto optimal (efficient) allocations are not incentive compatible is a fundamental fact in information economics, mechanism design and general equilibrium with asymmetric information. This important result was obtained assuming that the individuals are expected utility maximizers. Although this assumption is central to Harsanyi’s approach to games with incomplete information, it is not the only one reasonable. In fact, a huge literature criticizes EU’s shortcomings and propose alternative preferences. Thus, a natural question arises: does the mentioned conflict extend to other preferences? We show that when individuals have (a special form of) maximin expected utility (MEU) preferences, then any efficient allocation is incentive compatible. Conversely, only MEU preferences have this property. We also provide applications of our results to mechanism design and show that Myerson-Satterthwaite’s negative result ceases to hold in our MEU framework.
TL;DR: In this article, a game-theoretic framework that allows for explicitly randomized strategies is used to study the effect of ambiguity aversion on equilibrium outcomes, and the notions of independent strategies as well as of "common priors" are amended to render them applicable to games in which players lack probabilistic sophistication.
Abstract: A game-theoretic framework that allows for explicitly randomized strategies is used to study the effect of ambiguity aversion on equilibrium outcomes. The notions of "independent strategies" as well as of "common priors" are amended to render them applicable to games in which players lack probabilistic sophistication. Within this framework the equilibrium predictions of two player games with ambiguity averse and with ambiguity neutral players are observationally equivalent. This equivalence result does not extend to the case of games with more than two players. A translation of the concept of equilibrium in beliefs to the context of ambiguity aversion yields substantially different predictions - even for the case with just two players.
TL;DR: In this paper, the similarities between Ellsberg's and Shackle's frameworks for discussing the limits of the probabilistic approach to decision theory are more important than usually admitted.
Abstract: This paper argues that the similarities between Ellsberg's and Shackle's frameworks for discussing the limits of the probabilistic approach to decision theory are more important than usually admitted. The paper discusses the grounds on which the ambiguity surrounding the decision-maker in Ellsberg's urn experiments can be deemed analogous to the uncertainty faced by Shackle's entrepreneur taking 'crucial decisions'. The two authors' insights are assessed, and special attention is paid to the criteria for decision under uncertainty they put forward. The paper establishes a link between Ellsberg's and Shackle's perspectives and the non-additive probability approach of Gilboa and Schmeidler, an approach that offers an alternative to standard probability calculus, which can be of use to analyse both ambiguity and uncertainty. The comparison between Ellsberg and Shackle draws on an interpretation of Keynes's Treatise on Probability emphasising Keynes's rejection of both well-defined probability functions and maximisation as a guide to human conduct. It is shown that Keynes's viewpoint implies a reconsideration of the boundaries of probability theory that is in the same vein of Ellsberg's and Shackle's concern in the years of the consolidation of Savage's new probabilistic mainstream. Copyright The Author 2009. Published by Oxford University Press on behalf of the Cambridge Political Economy Society. All rights reserved., Oxford University Press.
TL;DR: It is demonstrated that a Uniform increase in prudence accompanied by a uniform increase (decrease) in risk aversion is sufficient to indicate greater downside risk aversion, provided prudence is greater than three times the degree of risk aversion.
TL;DR: In this paper, it is argued that loss aversion is a property of observable choice behavior and two new definitions of loss averse behavior are advocated under prospect theory, the new properties hold if the commonly used utility-based measures of loss aversion are corrected by a probability-based measure of risk aversion and their product exceeds 1.
Abstract: This paper reviews the most common approaches that have been adopted to analyze and describe loss aversion under prospect theory. Subsequently, it is argued that loss aversion is a property of observable choice behavior and two new definitions of loss averse behavior are advocated. Under prospect theory, the new properties hold if the commonly used utility based measures of loss aversion are corrected by a probability based measure of loss aversion and their product exceeds 1. It is shown that prominent parametric families of weighting functions, while successful in accommodating empirical findings on probabilistic risk attitudes, may not fit well with the theoretical implications of the new loss averse behavior conditions.
TL;DR: The authors show that risk sharing relying on wealth as a proxy for risk aversion is identified only insofar as the econometrician is willing to assume that the principal is risk neutral or her preferences exhibit constant absolute risk aversion (CARA).
Abstract: Tests of risk sharing in the contracting literature often rely on wealth as a proxy for risk aversion. The intuition behind these tests is that since contract choice is monotonic in the coefficients of risk aversion, which are themselves assumed monotonic in wealth, the effect of a change in wealth on contract choice is clearly identified. We show that tests of risk sharing relying on wealth as a proxy for risk aversion are identified only insofar as the econometrician is willing to assume that (a) the principal is risk neutral or her preferences exhibit constant absolute risk aversion (CARA); and (b) the agent is risk neutral.
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TL;DR: In this article, the authors provide a multiple-priors representation of ambiguous beliefs for any preference that admits an affine utility representation when restricted to constant acts, and suitably continuous.
Abstract: This paper provides a multiple-priors representation of ambiguous beliefs a la Ghirardato, Maccheroni, and Marinacci (2004) and Nehring (2002) for any preference that is (i) monotonic, (ii) Bernoullian, i.e. admits an affine utility representation when restricted to constant acts, and (iii) suitably continuous. Monotonicity is the main substantive assumption: we do not require either Certainty Independence or Uncertainty Aversion. We characterize the set of ambiguous beliefs in terms of Clarke-Rockafellar differentials. This allows us to provide an explicit calculation of the set of priors for several recent decision models: multiplier preferences, the smooth ambiguity model, the vector expected utility model, as well as confidence function, variational, general "uncertainty-averse" preferences, and mean-dispersion preferences.