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We extend Ellsberg's two-urn paradox and propose three symmetric forms of partial ambiguity by limiting the possible compositions in a deck of 100 red and black cards in three ways. Interval ambiguity involves a symmetric range of 50 - to 50 + n red cards. Complementarity, disjoint ambiguity arises from two nonintersecting intervals of 0 to and 100 - n to 100 red cards. Two-point ambiguity involves n or 100 - n red cards. We investigate experimentally attitudes towards partial ambiguity and the corresponding compound lotteries in which the possible compositions are drawn with equal objective probabilities. This yields three key findings: distinct attitudes towards the three forms of partial ambiguity, significant association across attitudes towards partial ambiguity and compound risk, and source preference between two-point ambiguity and two-point compound risk. Our findings help discriminate among models of ambiguity in the literature.

Foundations are provided for rank-dependent preferences within the popular two-stage framework of Anscombe–Aumann, in which risk and ambiguity feature as distinct sources of uncertainty. We advance the study of attitudes towards ambiguity without imposing expected utility for risk. As a result, in our general model, ambiguity attitude can be captured by non-additive subjective probabilities as under Choquet expected utility or by a specific utility for ambiguity as in recursive expected utility or, if required, by both. The key property for preferences builds on (discrete) rates of substitution which are standardly applied in economics. By demanding consistency for these rates of substitution across events and within or across sources of uncertainty, we obtain a model that nests popular theories for risk and ambiguity. This way, new possibilities for theoretical and empirical analyses of these theories emerge.

We report a portfolio-choice experiment that enables us to estimate parametric models of ambiguity aversion at the level of the individual subject. The assets are Arrow securities that correspond to three states of nature, where one state is risky with known probability and two states are ambiguous with unknown probabilities. We estimate two specifications of ambiguity aversion, one kinked and one smooth, that encompass many of the theoretical models in the literature. Each specification includes two parameters: one for ambiguity attitudes and another for risk attitudes. We also estimate a three-parameter specification that includes an additional parameter for pessimism/optimism (underweighting/overweighting the probabilities of different payoffs). The parameter estimates for individual subjects exhibit considerable heterogeneity. We cannot reject the null hypothesis of subjective expected utility for a majority of subjects. Most of the remaining subjects exhibit statistically significant ambiguity aversion or seeking and/or pessimism or optimism.

We use the multiple price list method and a recursive expected utility theory of smooth ambiguity to separate out attitude towards risk from that towards ambiguity. Based on this separation, we investigate if there are differences in agent behaviour under uncertainty over gain amounts vis-a-vis uncertainty over loss amounts. On an aggregate level, we find that (i) subjects are risk averse over gains and risk seeking over losses, displaying a “reflection effect” and (ii) they are ambiguity neutral over gains and are mildly ambiguity seeking over losses. Further analysis shows that on an individual level, and with respect to both risky and ambiguous prospects, there is limited incidence of a reflection effect where subjects are risk/ambiguity averse (seeking) in gains and seeking (averse) in losses, though this incidence is higher for ambiguous prospects. A very high proportion of such cases of reflection exhibit risk (ambiguity) aversion in gains and risk (ambiguity) seeking in losses, with the reverse effect being significantly present in the case of risk but almost absent in case of ambiguity. Our results suggest that reflection across gains and losses is not a stable individual characteristic, but depends upon whether the form of uncertainty is precise or ambiguous, since we rarely find an individual who exhibits reflection in both risky and ambiguous prospects. We also find that correlations between attitudes towards risk and ambiguity were domain dependent.

An extension to Ellsberg's experiment demonstrates that attitudes to ambiguity and compound objective lotteries are tightly associated. The sample is decomposed into three main groups: subjective expected utility subjects, who reduce compound objective lotteries and are ambiguity neutral, and two groups that exhibit different forms of association between preferences over compound lotteries and ambiguity, corresponding to alternative theoretical models that account for ambiguity averse or seeking behavior.

This paper axiomatizes static and dynamic quantile preferences. Static quantile preferences specify that a prospect should be preferred if it has a higher τ-quantile, for some τ ϵ (0, 1), while its dynamic counterpart extends this to take into account a sequence of decisions and information disclosure. An important motivation for the axiomatization that leads to this preference is the separation of tastes and beliefs. We first axiomatize quantile preferences for the static case with finite state space and then extend the axioms to the dynamic context. The dynamic preferences induce an additively separable quantile model with standard discounting, that is, the recursive equation is characterized by the sum of the current period utility function and the discounted value of the certainty equivalent, which is a quantile function. These preferences are time consistent and have a simple quantile recursive representation, which gives the model the analytical tractability needed in several fields in financial and economic applications. Finally, we study the notion of risk attitude in both the static and recursive quantile models. In quantile models, the risk attitude is completely captured by the quantile τ, a single-dimensional parameter. This is simpler than in expected utility models, where in general the risk attitude is determined by a function.

Models of utility in stochastic continuous-time settings typically assume that beliefs are represented by a probability measure, hence ruling out a priori any concern with ambiguity. This paper formulates a continuous-time intertemporal version of multiple-priors utility, where aversion to ambiguity is admissible. In a representative agent asset market setting, the model delivers restrictions on excess returns that admit interpretations reflecting a premium for risk and a separate premium for ambiguity.

The paper analyses optimal spending of an endowment fund. The purpose is to find a spending rule which is optimal for the owners and which secures that the fund will last “forever”. This we do by finding closed form solutions of the optimal consumption to wealth ratio. We solve this problem using the life cycle model, where the agent can have preferences represented by expected utility or recursive utility. We apply our results to a sovereign wealth fund, and demonstrate that the optimal spending rate is significantly lower than the fund’s expected real rate of return, a rule which is in common use. Employing the latter as the spending rate, implies that the fund’s value deteriorates both in probability and in expectation, as time goes. For both kinds of long term convergence we find closed form threshold values. Spending below these values secures a sustainable fund.

We commonly think of information as an instrument for better decisions, yet evidence suggests that people often decline free information in non-strategic scenarios. This paper provides a theory for how a dynamically-consistent decision maker can be averse to partial information as a consequence of ambiguity aversion. It introduces a class of recursive preferences on an extended choice domain, which allows the preferences to depend on how information is dynamically revealed and to depart from the standard expected-utility theory. A new notion of ambiguity aversion, called Event Complementarity, exactly characterizes aversion to partial information. Familiar static ambiguity-averse preferences are embedded into the general recursive model, in which conditions for partial information aversion are identified. The findings suggest that Event Complementarity overlaps with yet still differs from the conventional notion of ambiguity aversion.

Experimental evidence suggests that individuals are more risk averse when they perceive risk that is gradually resolved over time. We address these findings by studying a decision maker who has recursive, nonexpected utility preferences over compound lotteries. The decision maker has preferences for one-shot resolution of uncertainty if he always prefers any compound lottery to be resolved in a single stage. We establish an equivalence between dynamic preferences for one-shot resolution of uncertainty and static preferences that are identified with commonly observed behavior in Allais-type experiments. The implications of this equivalence on preferences over information systems are examined. We define the gradual resolution premium and demonstrate its magnifying effect when combined with the usual risk premium.