Two-parameter decision models and rank-dependent expected utility

This paper extends the existing literature concerning the relationship between two parameter decision models and those based on expected utility in two main directions. The first relaxes Meyer's location and scale (or Sinn's linear class) condition and shows that a two-parameter representation of preferences over uncertain prospects and the expected utility representation yield consistent rankings of random variables when the decision maker's choice set is restricted to random variables differing by mean shifts and monotone meanpreserving spreads. The second shows that the rank-dependent expected utility model is also consistent with two-parameter ranking methods if the probability transform satisfies certain dominance conditions. The main implication of these results is that the simple two-parameter model can be used to analyze the comparative statics properties of a wide variety of economic models, including those with multiple sources of uncertainty when the random variables are comonotonic. To illustrate this point, we apply our results to the problem of optimal portfolio investment with random initial wealth. We find that it is relatively easy to obtain strong global comparative statics results even if preferences do not satisfy the independence axiom.     

Ambiguity Aversion in the Field of Insurance: Insurers’ Attitude to Imprecise and Conflicting Probability Estimates

This article presents the results of a survey designed to test, with economically sophisticated participants, Ellsberg’s ambiguity aversion hypothesis, and Smithson’s conflict aversion hypothesis. Based on an original sample of 78 professional actuaries (all members of the French Institute of Actuaries), this article provides empirical evidence that ambiguity (i.e. uncertainty about the probability) affect insurers’ decision on pricing insurance. It first reveals that premiums are significantly higher for risks when there is ambiguity regarding the probability of the loss. Second, it shows that insurers are sensitive to sources of ambiguity. The participants indeed, charged a higher premium when ambiguity came from conflict and disagreement regarding the probability of the loss than when ambiguity came from imprecision (imprecise forecast about the probability of the loss). This research thus documents the presence of both ambiguity aversion and conflict aversion in the field of insurance, and discuses economic and psychological rationales for the observed behaviours.     

Ambiguity aversion in the small and in the large for weighted linear utility

The widely observed preference for lotteries involving precise rather than vague of ambiguous probabilities is called ambiguity aversion. Ambiguity aversion cannot be predicted or explained by conventional expected utility models. For the subjectively weighted linear utility (SWLU) model, we define both probability and payoff premiums for ambiguity, and introduce alocal ambiguity aversion function a(u) that is proportional to these ambiguity premiums for small uncertainties. We show that one individual's ambiguity premiums areglobally larger than another's if and only if hisa(u) function is everywhere larger. Ambiguity aversion has been observed to increase 1) when the mean probability of gain increases and 2) when the mean probability of loss decreases. We show that such behavior is equivalent toa(u) increasing in both the gain and loss domains. Increasing ambiguity aversion also explains the observed excess of sellers' over buyers' prices for insurance against an ambiguous probability of loss.     

Experimental evidence on valuation with multiple priors

Popular models for decision making under ambiguity assume that people use not one but multiple priors. This paper is a first attempt to experimentally elicit the min and the max of multiple priors directly. In an ambiguous scenario we measure a participant’s single prior, her min and max of multiple priors, and the valuation of an ambiguous asset with the same underlying states as the ambiguous scenario. We use the min and the max of multiple priors to directly test two popular multiple priors models: the maxmin model and the α maxmin model. We find more support for the α maxmin model: although people put about twice the weight on the minimum of multiple priors, they also consider the maximum. Furthermore, we indirectly elicit confidence weights over the whole set of multiple priors and test two additional models: variational preferences and the smooth model of ambiguity. Two particular versions of the variational preferences model explain less than the α maxmin but more than the maxmin model. Overall, the smooth model of ambiguity performs best among all models tested.     

Evaluating second-order probability judgments with strictly proper scoring rules

Empirical studies have demonstrated that uncertainty about event probabilities, also known as ambiguity or second-order uncertainty, can affect decision makers choice preferences. Despite the importance of second-order uncertainty in decision making, almost no effort has been directed towards the development of methods that evaluate the accuracy of second-order probabilities. In this paper, we describe conditions under which strictly proper scoring rules can be used to assess the accuracy of second-order probability judgments. We investigate the effectiveness of using a particular strictly proper scoring rule the ranked probability score - to discourage biased assessments of second-order uncertainty.     

Observing different orders of risk aversion

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

Ellsberg paradox: Ambiguity and complexity aversions compared

We present a simple model where preferences with complexity aversion, rather than ambiguity aversion, resolve the Ellsberg paradox. We test our theory using laboratory experiments where subjects choose among lotteries that “range” from a simple risky lottery, through risky but more complex lotteries, to one similar to Ellsberg’s ambiguity urn. Our model ranks lotteries according to their complexity and makes different—at times contrasting—predictions than most models of ambiguity in response to manipulations of prizes. The results support that complexity aversion preferences play an important and separate role from beliefs with ambiguity aversion in explaining behavior under uncertainty.     

The price for information about probabilities and its relation with risk and ambiguity

In this article, ambiguity attitude is measured through the maximum price a decision maker is willing to pay to know the probability of an event. Two problems are examined in which the decision maker faces an act: in one case, buying information implies playing a lottery, while, in the other case, buying information gives also the option to avoid playing the lottery. In both decision settings, relying on the Choquet expected utility model, we study how the decision maker??s risk and ambiguity attitudes affect the reservation price for ambiguity resolution. These effects are analyzed for different levels of ambiguity of the act. Operating instructions for the elicitation of the reservation price for ambiguity resolution in an experimental setting are provided at the end of the article.     

Measuring ambiguity attitude: (Extended) multiplier preferences for the American and the Dutch population

Empirical studies of ambiguity aversion often use measures that are not grounded in theory. This paper shows how a theoretically-founded measure of ambiguity aversion can be derived from Hansen and Sargent’s theory of multiplier preferences. Multiplier preferences are used in macroeconomics to capture model uncertainty. At the micro level, they have not been applied yet, because they do not permit ambiguity seeking, which is usually observed for a substantial proportion of subjects. We give a preference foundation for (extended) multiplier preferences accommodating both ambiguity aversion and ambiguity seeking and we propose a simple method to measure them using matching probabilities. We illustrate our method in two large representative samples (Dutch and American) and obtain the first micro estimates of multiplier preferences.     

Is imprecise knowledge better than conflicting expertise? Evidence from insurers’ decisions in the United States

This paper reports the results of the first experiment in the United States designed to distinguish between two sources of ambiguity: imprecise ambiguity (expert groups agree on a range of probability, but not on any point estimate) versus conflict ambiguity (each expert group provides a precise probability estimate which differs from one group to another). The specific context is whether risk professionals (here, insurers) behave differently under risk (when probability is well-specified) and different types of ambiguity in pricing catastrophic risks (floods and hurricanes) and non-catastrophic risks (house fires). The data show that insurers charge higher premiums when faced with ambiguity than when the probability of a loss is well specified (risk). Furthermore, they tend to charge more for conflict ambiguity than imprecise ambiguity for flood and hurricane hazards, but less in the case of fire. The source of ambiguity also impacts causal inferences insurers make to reduce their uncertainty.     

A simplified axiomatic approach to ambiguity aversion

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.     

Ambiguity aversion and the propensities for self-insurance and self-protection

Two models of ambiguity preferences that permit comparative statics analysis of greater ambiguity aversion yield definite predictions concerning propensities for self-insurance and self-protection: The levels of both activities that are optimal for an ambiguity-averse decision maker are higher in the presence of ambiguity than in its absence, and demands for both activities increase with greater ambiguity aversion. The reason is that, at levels optimal for one decision maker, an increase in either activity results in a mean-preserving contraction in the distribution of expected utility in the presence of ambiguity, which is valuable to anyone with the same risk preferences who is more ambiguity averse.     

Insurer ambiguity and market failure

A series of studies investigate the decision processes of actuaries, underwriters, and reinsurers in setting premiums for ambiguous and uncertain risks. Survey data on prices reveal that all three types of these insurance decision makers are risk averse and ambiguity averse. In addition, groups appear to be influenced in their premium-setting decisions by specific reference points such as expected loss and the concern with insolvency. This behavior is consistent with a growing analytical and empirical literature in economics and decision processes that investigates the role that uncertainty plays on managerial choices. Improved risk-assessment procedures and government involvement in providing protection against catastrophic losses may induce insurers to reduce premiums and broaden available coverage.This article is part of a larger effort supported by the National Science Foundation on The Role of Insurance, Compensation, Regulation, and Protective Behavior in Decision Making about Risk and Misfortune. We greatly appreciate the many helpful comments and suggestions by our colleagues on the project: Jon Baron, Colin Camerer, Neil Doherty, Jack Hershey, Eric Johnson, and Paul Kleindorfer. Support from NSF Grant #SES8809299 is gratefully acknowledged.     

The strength of sensitivity to ambiguity

We report an experiment where each subject’s ambiguity sensitivity is measured by an ambiguity premium, a concept analogous to and comparable with a risk premium. In our design, some tasks feature known objective risks and others uncertainty about which subjects have imperfect, heterogeneous, information (“ambiguous tasks”). We show how the smooth ambiguity model can be used to calculate ambiguity premia. A distinctive feature of our approach is estimation of each subject’s subjective beliefs about the uncertainty in ambiguous tasks. We find considerable heterogeneity among subjects in beliefs and ambiguity premia; and that, on average, ambiguity sensitivity is about as strong as risk sensitivity.     

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.     

Feddersen and Pesendorfer meet Ellsberg

The Condorcet Jury Theorem formalises the “wisdom of crowds”: binary decisions made by majority vote are asymptotically correct as the number of voters tends to infinity. This classical result assumes like-minded, expected utility maximising voters who all share a common prior belief about the right decision. Ellis (Theor Econo 11(3): 865–895, 2016) shows that when voters have ambiguous prior beliefs—a (closed, convex) set of priors—and follow maxmin expected utility (MEU), such wisdom requires that voters’ beliefs satisfy a “disjoint posteriors” condition: different private signals lead to posterior sets with disjoint interiors. Both the original theorem and Ellis’s generalisation assume symmetric penalties for wrong decisions. If, as in the jury context, errors attract asymmetric penalties then it is natural to consider voting rules that raise the hurdle for the decision carrying the heavier penalty for error (such as conviction in jury trials). In a classical model, Feddersen and Pesendorfer (Am Politi Sci Rev 92(1):23–35, 1998) have shown that, paradoxically, raising this hurdle may actually increase the likelihood of the more serious error. In particular, crowds are not wise under the unanimity rule: the probability of the more serious error does not vanish as the crowd size tends to infinity. We show that this “Jury Paradox” persists in the presence of ambiguity, whether or not juror beliefs satisfy Ellis’s “disjoint posteriors” condition. We also characterise the strictly mixed equilibria of this model and study their properties. Such equilibria cannot exist in the absence of ambiguity but may exist for arbitrarily large jury size when ambiguity is present. In addition to uninformative strictly mixed equilibria, analogous to those exhibited by Ellis (Theor Econo 11(3): 865–895, 2016), there may also exist strictly mixed equilibria which are informative about voter signals.

     

Indecisiveness aversion and preference for commitment

We present an axiomatic model of preferences over menus that is motivated by three assumptions. First, the decision maker is uncertain ex ante (i.e., at the time of choosing a menu) about her ex post (i.e., at the time of choosing an option within her chosen menu) preferences over options, and she anticipates that this subjective uncertainty will not resolve before the ex post stage. Second, she is averse to ex post indecisiveness (i.e., to having to choose between options that she cannot rank with certainty). Third, when evaluating a menu she discards options that are dominated (i.e., inferior to another option whatever her ex post preferences may be) and restricts attention to the undominated ones. Under these assumptions, the decision maker has a preference for commitment in the sense of preferring menus with fewer undominated alternatives. We derive a representation in which the decision maker’s uncertainty about her ex post preferences is captured by means of a subjective state space, which in turn determines which options are undominated in a given menu, and in which the decision maker fears, whenever indecisive, to choose an option that will turn out to be the worst (undominated) one according to the realization of her ex post preferences.     

Living without state-independence of utilities

This article is concerned with the representation of preferences which do not satisfy the ordinary axioms for state-independent utilities. After suggesting reasons for not being satisfied with solutions involving state-dependent utilities, an alternative representation shall be proposed involving state-independent utilities and a situation-dependent factor. The latter captures the interdependencies between states and consequences. Two sets of axioms are proposed, each permitting the derivation of subjective probabilities, state-independent utilities, and a situation-dependent factor, and each operating in a different framework. The first framework involves the concept of a decision situation—consisting of a set of states, a set of consequences and a preference relation on acts; the probabilities, utilities and situation-dependent factor are elicited by referring to other, appropriate decision situations. The second framework, which is technically related, operates in a fixed decision situation; particular “subsituations” are employed in the derivation of the representation. Possible interpretations of the situation-dependent factor and the notion of situation are discussed.     

Preference and belief: Ambiguity and competence in choice under uncertainty

We investigate the relation between judgments of probability and preferences between bets. A series of experiments provides support for the competence hypothesis that people prefer betting on their own judgment over an equiprobable chance event when they consider themselves knowledgeable, but not otherwise. They even pay a significant premium to bet on their judgments. These data connot be explained by aversion to ambiguity, because judgmental probabilities are more ambiguous than chance events. We interpret the results in terms of the attribution of credit and blame. The possibility of inferring beliefs from preferences is questioned.¹