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We provide two extensions of Gilboa and Schmeidler (J Math Econ 18:141-153, 1989)'s maxmin expected utility decision rule to accommodate a decision maker's changing ambiguity attitudes. The two rules are, respectively, a weighted maxmin rule and a variant constraint rule. The former evaluates an act by a weighted average of its worst and best possible expected utilities over a set of priors, with the weights depending on the act. The latter evaluates an act by its worst expected utility over a neighborhood of a set of approximating priors, with the size of the neighborhood depending on the act. Canonical representations of the two rules are provided for classes of preference relations that exhibit, respectively, ambiguity aversion à la Schmeidler (Econometrica 57:571-587, 1989) and ambiguity aversion à la Ghirardato and Marinacci (J Econ Theory 102:251-289, 2002). In the second part of this paper, we study wealth effect under ambiguity. We propose axioms on absolute and relative ambiguity aversion and derive three representations for the ambiguity averse preference relations exhibiting decreasing (increasing) absolute ambiguity aversion. In particular, decreasing absolute ambiguity aversion implies that as the baseline utility of an act increases, a weighted maxmin decision maker puts less weight on the worst case, and a variant constraint decision maker considers a smaller neighborhood of approximating priors.

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.

Coherent-ambiguity aversion is defined within the (Klibanoff et al., Econometrica 73:1849–1892, 2005 ) smooth-ambiguity model (henceforth KMM ) as the combination of choice-ambiguity and value-ambiguity aversion. Five ambiguous decision tasks are analyzed theoretically, where an individual faces two-stage lotteries with binomial, uniform, or unknown second-order probabilities. Theoretical predictions are then tested through a 10-task experiment. In (unambiguous) tasks 1–5, risk aversion is elicited through both a portfolio choice method and a BDM mechanism. In (ambiguous) tasks 6–10, choice-ambiguity aversion is elicited through the portfolio choice method, while value-ambiguity aversion comes about through the BDM mechanism. The behavior of over 75 % of classified subjects is in line with the KMM model in all tasks 6–10, independent of their degree of risk aversion. Furthermore, the percentage of coherent-ambiguity-averse subjects is lower in the binomial than in the uniform and in the unknown treatments, with only the latter difference being significant. The most part of coherent-ambiguity-loving subjects show a high risk aversion.

Theory predicts that entrepreneurs have distinct attitudes toward risk and uncertainty, but empirical evidence is mixed. To better understand the unique behavioral characteristics of entrepreneurs and the causes of these mixed results, we perform a large “lab-in-the-field” experiment comparing entrepreneurs to managers (a suitable comparison group) and employees ( n = 2,288). The results indicate that entrepreneurs perceive themselves as less risk averse than managers and employees, in line with common wisdom. However, when using experimental incentivized measures, the differences are subtler. Entrepreneurs are only found to be unique in their lower degree of loss aversion, and not in their risk or ambiguity aversion. This combination of results might be explained by our finding that perceived risk attitude is not only correlated to risk aversion but also to loss aversion. Overall, we therefore suggest using a broader definition of risk that captures this unique feature of entrepreneurs: their willingness to risk losses. This paper was accepted by Uri Gneezy, behavioral economics .

Using a theorem showing that matching probabilities of ambiguous events can capture ambiguity attitudes, we introduce a tractable method for measuring ambiguity attitudes and apply it in a large representative sample. In addition to ambiguity aversion, we confirm an ambiguity component recently found in laboratory studies: a-insensitivity, the tendency to treat subjective likelihoods as 50-50, thus overweighting extreme events. Our ambiguity measurements are associated with real economic decisions; specifically, a-insensitivity is negatively related to stock market participation. Ambiguity aversion is also negatively related to stock market participation, but only for subjects who perceive stock returns as highly ambiguous. This paper was accepted by James Smith, decision analysis .

We propose and characterize a model of preferences over acts such that the decision maker prefers act f to act g if and only if${\\Bbb E}_{\\mu}\\phi ({\\Bbb E}_{\\pi }u\\circ f)\\geq {\\Bbb E}_{\\mu}\\phi ({\\Bbb E}_{\\pi }u\\circ g)$, where E is the expectation operator, u is a von Neumann-Morgenstern utility function, φ is an increasing transformation, and μ is a subjective probability over the set Π of probability measures π that the decision maker thinks are relevant given his subjective information. A key feature of our model is that it achieves a separation between ambiguity, identified as a characteristic of the decision maker's subjective beliefs, and ambiguity attitude, a characteristic of the decision maker's tastes. We show that attitudes toward pure risk are characterized by the shape of u, as usual, while attitudes toward ambiguity are characterized by the shape of φ. Ambiguity itself is defined behaviorally and is shown to be characterized by properties of the subjective set of measures Π. One advantage of this model is that the well-developed machinery for dealing with risk attitudes can be applied as well to ambiguity attitudes. The model is also distinct from many in the literature on ambiguity in that it allows smooth, rather than kinked, indifference curves. This leads to different behavior and improved tractability, while still sharing the main features (e.g., Ellsberg's paradox). The maxmin expected utility model (e.g., Gilboa and Schmeidler (1989)) with a given set of measures may be seen as a limiting case of our model with infinite ambiguity aversion. Two illustrative portfolio choice examples are offered.

Perturbed utility functions—the sum of expected utility and a nonlinear perturbation function—provide a simple and tractable way to model various sorts of stochastic choice. We provide two easily understood conditions each of which characterizes this representation: One condition generalizes the acyclicity condition used in revealed preference theory, and the other generalizes Luce's IIA condition. We relate the discrimination or selectivity of choice rules to properties of their associated perturbations, both across different agents and across decision problems. We also show that these representations correspond to a form of ambiguity-averse preferences for an agent who is uncertain about her true utility.

Measurements of ambiguity attitudes have so far focused on artificial events, where (subjective) beliefs can be derived from symmetry of events and can be then controlled for. For natural events as relevant in applications, such a symmetry and corresponding control are usually absent, precluding traditional measurement methods. This paper introduces two indexes of ambiguity attitudes, one for aversion and the other for insensitivity/perception, for which we can control for likelihood beliefs even if these are unknown. Hence, we can now measure ambiguity attitudes for natural events. Our indexes are valid under many ambiguity theories, do not require expected utility for risk, and are easy to elicit in practice. We use our indexes to investigate time pressure under ambiguity. People do not become more ambiguity averse under time pressure but become more insensitive (perceive more ambiguity). These findings are plausible and, hence, support the validity of our indexes.

This paper investigates when an increase in ambiguity raises demand for coinsurance under uncertainty of a net-wealth (loss) distribution. Unlike previous papers that have studied the determining conditions limited to a specific class of ambiguity-averse individuals for ambiguity increases, this paper studies determining conditions characterized by changes in the possible net-wealth distributions resulting from an increase in ambiguity, for all risk-and weakly ambiguity-averse (risk-averse and weakly ambiguity-loving) individuals. When the preferences are characterized by an α-maxmin model, we find that, for risk- and weakly ambiguity-averse (risk-averse and weakly ambiguity-loving) individuals, the determining conditions involve a greater location-weighted probability mass function under both the worst (best) and possible net-wealth distributions on average for a positive scalar less than one, which are analogous to smaller central riskiness. We further discuss an effect of the increase in ambiguity raising the demand for coinsurance on expected utility under ambiguity.

Ellsberg and others suggested that decision under ambiguity is a rich empirical domain with many phenomena to be investigated beyond the Ellsberg urns. We provide a systematic empirical investigation of this richness by varying the uncertain events, the outcomes, and combinations of both. Although ambiguity aversion is prevailing, we also find systematic ambiguity seeking, confirming insensitivity. We find that ambiguity attitudes depend on the kind of uncertainty (the source) but not on the kind of outcomes. Ambiguity attitudes are closer to rationality (ambiguity neutrality) for natural uncertainties than for artificial Ellsberg urn uncertainties. This also appears from the reductions of monotonicity violations and of insensitivity. Ambiguity attitudes have predictive power across different outcomes and sources of uncertainty, with individual-specific components. Our rich domain serves well to test families of weighting functions for fitting ambiguity attitudes. We find that two-parameter families, capturing not only aversion but also insensitivity, are desirable for ambiguity even more than for risk. The Goldstein–Einhorn family performed best for ambiguity. Data and the online appendix are available at https://doi.org/10.1287/mnsc.2017.2777 . This paper was accepted by Manel Baucells, decision analysis.