Recursive multiple-priors

This paper axiomatizes an intertemporal version of multiple-priors utility. A central axiom is dynamic consistency, which leads to a recursive structure for utility, to ‘rectangular’ sets of priors and to prior-by-prior Bayesian updating as the updating rule for such sets of priors. It is argued that dynamic consistency is intuitive in a wide range of situations and that the model is consistent with a rich set of possibilities for dynamic behavior under ambiguity.     

Choice under uncertainty with the best and worst in mind: Neo-additive capacities

We develop the simplest generalization of subjective expected utility that can accommodate both optimistic and pessimistic attitudes towards uncertainty—Choquet expected utility with non-extreme-outcome-additive (neo-additive) capacities. A neo-additive capacity can be expressed as the convex combination of a probability and a special capacity, we refer to as a Hurwicz capacity, that only distinguishes between whether an event is impossible, possible or certain. We show that neo-additive capacities can be readily applied in economic problems, and we provide an axiomatization in a framework of purely subjective uncertainty.     

Higher-order generalizations of Arrow-Pratt and Ross risk aversion: A comparative statics approach

We analyze comparative risk aversion in a new way, through a comparative statics problem in which, for a cost, agents can shift from an initial probability distribution toward a preferred distribution. The Ross characterization arises when the original distribution is riskier than the preferred distribution and the cost is monetary, and the Arrow-Pratt characterization arises when the original distribution differs from the preferred distribution by a simple mean-preserving spread and the cost is a utility cost. Higher-order increases in risk lead to higher-order generalizations, and the comparative statics method yields a unified approach to the problem of comparative risk attitudes.     

Substituting one risk increase for another: A method for measuring risk aversion

This paper defines the rate of substitution of one stochastic change to a random variable for another. It then focuses on the case where one of these changes is an nth degree risk increase, and the other is an m  th degree risk increase, where n>m?1$n>m?1$. The paper shows that the rate of substitution for these two risk increases can be used to provide a broader definition and two additional characterizations of the nth degree Ross more risk averse partial order. The implications for local intensity measures of nth degree risk aversion are also examined. The analysis organizes the existing results as well as generates new ones.     

Unequal uncertainties and uncertain inequalities: an axiomatic approach

In this paper, we provide an axiomatic characterization of social welfare functions for uncertain incomes. Our most general result is that a small number of reasonable assumptions regarding welfare orderings under uncertainty rule out pure ex ante as well as pure ex post evaluations. Any social welfare function that satisfies these axioms should lie strictly between the ex ante and the ex post evaluations of income distributions. We also provide an axiomatic characterization of the weighted average of the minimum and the maximum of ex post and ex ante evaluations.     

Co-monotonicity: Toward a utility function capturing envy

Envy affects economic decisions, and can lead to monotonicity violation. We introduce co-monotonicity—a generalization of monotonicity, expected to hold even in the presence of envy. Experimental results and implications for the form of possible utility functions are discussed.     

Small worlds: Modeling attitudes toward sources of uncertainty

We introduce the concept of a conditional small world event domain—an extension of Savage's [The Foundations of Statistics, Wiley, New York, 1954] notion of a ‘small world’—as a self-contained collection of comparable events. Under weak behavioral conditions we demonstrate probabilistic sophistication in any small world event domain without relying on monotonicity or continuity. Probabilistic sophistication within, though not necessarily across, small worlds provides a foundation for modeling a decision maker that has source-dependent risk attitudes. This also helps formalize the idea of source preference and suggests an interpretation of ambiguity aversion, often associated with Ellsberg-type behavior, in terms of comparative risk aversion across small worlds.     

The rationality and efficacy of decisions under uncertainty and the value of an experiment

Summary A decision maker faces a known prior distribution over payoff relevant states. We compare the expected utility of this individual under two scenarios. In the first, the decision maker makes a choice without further information. In the second, the decision maker has access to an experiment before choosing an action. However, the decision maker does not know the true joint distribution over states and messages. The value of the experiment as measured by the difference in the two utility levels can be negative as well as positive. We give a condition which is necessary and sufficient for the experiment to be valuable in our sense, for any decision problem.An earlier version of this paper was circulated under the title Noisy Bayes Updating and the Value of Information. We have gained from the comments of Stephen Coate, John Geanakoplos, Larry Samuelson, Timothy Van Zandt and seminar participants at Harvard Business School, Princeton, Boston University, the international conference on game theory at Stony Brook 1992 and the Winter meeting of the Econometric Society at Anaheim 1993. The first author received support for this project from NSF grant #SES-9308515 and a University of Pennsylvania Research Foundation Grant.     

The monetary utility premium and interpersonal comparisons

The utility premium is generally defined as the pain or reduction in expected utility caused by an n$n$th-degree risk increase, where n≥2$n\ge 2$. While it is a very useful concept in understanding a decision maker’s choice in uncertain situations, the utility premium is not interpersonally comparable. This note shows that the monetary utility premium–the utility premium divided by the expected marginal utility at the random starting wealth–is interpersonally comparable, and the comparison is characterized by Ross more risk aversion of the corresponding degree.     

Greater prudence and greater downside risk aversion

Variation in the degree of downside risk aversion across decision makers has implications for efficient risk sharing. However, except for small differences in risk preferences, there is no index, analogous to the Arrow-Pratt index of risk aversion, that depends only on local properties of the utility function and indicates the degree of aversion to downside risk. A measure that does depend only on local properties of the utility function u, the index of prudence p=−u^?/u^″, is related to downside risk aversion, which is indicated by a positive value for u^?. Although we show that the degree of prudence is not an accurate indicator of the degree of downside risk aversion, we nonetheless demonstrate 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 (less) than three times the degree of risk aversion.     

Ambiguous events and maxmin expected utility

We study the properties associated to various definitions of ambiguity [L.G. Epstein, J. Zhang, Subjective probabilities on subjectively unambiguous events, Econometrica 69 (2001) 265-306; P. Ghirardato et al., Differentiating ambiguity and ambiguity attitude, J. Econ. Theory 118 (2004) 133-173; K. Nehring, Capacities and probabilistic beliefs: a precarious coexistence, Math. Soc. Sci. 38 (1999) 197-213; J. Zhang, Subjective, ambiguity, expected utility and Choquet expected utility, Econ. Theory 20 (2002) 159-181] in the context of Maximin Expected Utility (MEU). We show that each definition of unambiguous events produces certain restrictions on the set of priors, and completely characterize each definition in terms of the properties it imposes on the MEU functional. We apply our results to two open problems. First, in the context of MEU, we show the existence of a fundamental incompatibility between the axiom of “Small unambiguous event continuity” (Epstein and Zhang, 2001) and the notions of unambiguous event due to Zhang (2002) and Epstein and Zhang (2001). Second, we show that, in the context of MEU, the classes of unambiguous events according to either Zhang (2002) or Epstein and Zhang (2001) are always λ-systems. Finally, we reconsider the various definitions in light of our findings, and identify some new objects (Z-filters and EZ-filters) corresponding to properties which, while neglected in the current literature, seem relevant to us.     

Regret aversion and opportunity dependence

This paper provides an axiomatic model of decision making under uncertainty in which the decision maker is driven by anticipated ex post regrets. Our model allows both regret aversion and likelihood judgement over states to coexist. Also, we characterize two special cases, minimax regret with multiple priors that generalizes Savage's minimax regret, and a smooth model of regret aversion.     

Mean-dispersion preferences and constant absolute uncertainty aversion

We axiomatize, in an Anscombe–Aumann framework, the class of preferences that admit a representation of the form V(f)=μ−ρ(d)$V(f)=\mu -\rho (d)$, where μ is the mean utility of the act f with respect to a given probability, d   is the vector of state-by-state utility deviations from the mean, and ρ(d)$\rho (d)$ is a measure of (aversion to) dispersion that corresponds to an uncertainty premium. The key feature of these mean-dispersion   preferences is that they exhibit constant absolute uncertainty aversion. This class includes many well-known models of preferences from the literature on ambiguity. We show what properties of the dispersion function ρ(⋅)$\rho (\cdot )$ correspond to known models, to probabilistic sophistication, and to some new notions of uncertainty aversion.     

An operational interpretation and existence of the Aumann-Serrano index of riskiness

In this note we provide an operational interpretation of the economic index of riskiness of Aumann and Serrano (2008) and discuss its existence in the case of non-finite gambles.     

Convex and decreasing absolute risk aversion is proper

Proper risk aversion, a pivotal concept in the study of behavioral conditions on utility functions, states that an undesirable risk can never be made desirable by the presence of an independent risk. It is well known that standard risk aversion is sufficient for this concept. We show in this short article that convex and decreasing absolute risk aversion is an alternative sufficient condition.     

Certainty Independence and the Separation of Utility and Beliefs

Economists often operate under an implicit assumption that the tastes of a decision maker are quite stable, while his beliefs change with the availability of new information. We show that for a general class of preferences, a separation of a key component of tastes, the utility function, from the other components of the representation is possible only if the decision maker's preferences satisfy a mild but not completely innocuous condition, called ‘certainty independence’. We also outline the axiomatic characterization of the preferences that obtain such separation, which are a subset of the biseparable preferences.     

Risk taking by agents: The role of ex-ante and ex-post accountability

We study the role of accountability in situations where an agent makes risky decisions for a principal. We observe that in the absence of accountability, agents choose less risk averse investments for the principal than investors who invest for their own account. Accountability mitigates the observed decrease in risk aversion. Differences are observed between situations where agents are accountable for their decision (“ex-ante”) and where they are accountable for the outcome (“ex-post”).     

Anonymity and individual risk

Adverse selection economies with private information are generally studied under the assumption that contracts are exclusive. That is, retrading is prohibited. An alternative market mechanism, the anonymous mechanism, is studied here. Risk averse agents trade contingent claims directly and side markets are in equilibrium. The result is the anonymous equilibrium. The anonymous equilibrium results in a set of endogenous transfers and subsidies.     

Subjective mortality expectations and consumption and saving behaviours among the elderly

Abstract Life expectancy is an important factor that individuals have to take into account for saving and consumption choices. The life‐cycle model of consumption and saving behaviour predicts that consumption growth should decrease with higher mortality rates. The aim of this study is to test this hypothesis based on data about subjective longevity expectations from the Health and Retirement Study merged with detailed consumption data from two waves of the Consumption and Activities Mail Survey. This study finds that an increase in subjective mortality by 1% corresponds to an annual decrease in consumption of non‐durable goods of around 1.8%.     

Differentiating ambiguity and ambiguity attitude

The objective of this paper is to show how ambiguity, and a decision maker (DM)'s response to it, can be modelled formally in the context of a general decision model. We introduce a relation derived from the DM's preferences, called “unambiguous preference”, and show that it can be represented by a set of probabilities. We provide such set with a simple differential characterization, and argue that it is a behavioral representation of the “ambiguity” that the DM may perceive. Given such revealed ambiguity, we provide a representation of ambiguity attitudes. We also characterize axiomatically a special case of our decision model, the “α-maxmin” expected utility model.