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.     

Unintended hedging in ambiguity experiments

We describe an ambiguity hedging problem in Ellsberg experiments, where combinations of individually ambiguous bets eliminate aggregate ambiguity, and which may yield incorrect classifications of ambiguity averse subjects. We propose a new classification consistent with this hedging possibility.     

A class of incomplete and ambiguity averse preferences

This paper characterizes models of ambiguous beliefs in the absence of the completeness axiom. We axiomatize multiple-selves versions of some of the most important examples of complete and ambiguity averse preferences, and characterize when those incomplete preferences are ambiguity averse.     

Robustness and ambiguity in continuous time

We use statistical detection theory in a continuous-time environment to provide a new perspective on calibrating a concern about robustness or an aversion to ambiguity. A decision maker repeatedly confronts uncertainty about state transition dynamics and a prior distribution over unobserved states or parameters. Two continuous-time formulations are counterparts of two discrete-time recursive specifications of Hansen and Sargent (2007) [16]. One formulation shares features of the smooth ambiguity model of Klibanoff et al. (2005) and (2009) [24] and [25]. Here our statistical detection calculations guide how to adjust contributions to entropy coming from hidden states as we take a continuous-time limit.     

Hierarchies of ambiguous beliefs

We present a theory of interactive beliefs analogous to Mertens and Zamir [Formulation of Bayesian analysis for games with incomplete information, Int. J. Game Theory 14 (1985) 1-29] and Brandenburger and Dekel [Hierarchies of beliefs and common knowledge, J. Econ. Theory 59 (1993) 189-198] that allows for hierarchies of ambiguity. Each agent is allowed a compact set of beliefs at each level, rather than just a single belief as in the standard model. We propose appropriate definitions of coherency and common knowledge for our types. Common knowledge of coherency closes the model, in the sense that each type homeomorphically encodes a compact set of beliefs over the others’ types. This space universally embeds every implicit type space of ambiguous beliefs in a beliefs-preserving manner. An extension to ambiguous conditional probability systems [P. Battigalli, M. Siniscalchi, Hierarchies of conditional beliefs and interactive epistemology in dynamic games, J. Econ. Theory 88 (1999) 188-230] is presented. The standard universal type space and the universal space of compact continuous possibility structures are epistemically identified as subsets.     

Risk-taking for others under accountability

We let subjects take risky decisions that affect themselves and a passive recipient. Adding a requirement to justify their choices significantly reduces loss aversion. This indicates that such an accountability mechanism may be effective at debiasing loss aversion in agency relations.     

A note on Kuhn’s Theorem with ambiguity averse players

Kuhn’s Theorem shows that extensive games with perfect recall can equivalently be analyzed using mixed or behavioral strategies, as long as players are expected utility maximizers. This note constructs an example that illustrates the limits of Kuhn’s Theorem in an environment with ambiguity averse players who use a maxmin decision rule and full Bayesian updating.     

Ecological-economic viability as a criterion of strong sustainability under uncertainty

Strong sustainability, according to the common definition, requires that different natural and economic capital stocks be maintained as physical quantities separately. Yet, in a world of uncertainty this cannot be guaranteed. To therefore define strong sustainability under uncertainty in an operational manner we propose to use the concept of viability. Viability means that the different components and functions of a dynamic, stochastic system at any time remain in a domain where the future existence of these components and functions is guaranteed with sufficiently high probability. We develop a unifying and general ecological-economic concept of viability that encompasses the traditional ecological and economic notions of viability as special cases. It provides an operational criterion of strong sustainability under a mild form of uncertainty and for medium spatial and temporal scales. We illustrate this concept and demonstrate its usefulness by applying it to livestock grazing management in semi-arid rangelands.     

The effect of ambiguity aversion on reward scheme choice

We test the implications of ambiguity aversion in a principal–agent problem with multiple agents. Models of ambiguity aversion suggest that, under ambiguity, comparative compensation schemes may become more attractive than independent wage contracts. We test this by presenting agents with a choice between comparative reward schemes and independent contracts, which are designed such that under uncertainty about output distributions (that is, under ambiguity), ambiguity averse agents should typically prefer comparative reward schemes, independent of their degree of risk aversion. We indeed find that the share of agents who choose the comparative scheme is higher under ambiguity.     

Efficient allocations under ambiguity

Important implications of the expected utility hypothesis and risk aversion are that if agents have the same probability belief, then consumption plans in every efficient allocation of resources under uncertainty are comonotone with the aggregate endowment, and if their beliefs are concordant, then the consumption plans are measurable with respect to the aggregate endowment. We study these two properties of efficient allocations for models of preferences that exhibit ambiguity aversion using the concept of conditional beliefs, which we introduce in this paper. We provide characterizations of such conditional beliefs for the standard models of preferences used in applications.     

Sequentially consistent rules of choice under complete uncertainty

This work analyzes the problem of individual choice under complete uncertainty. In this context, each alternative action consists of a set of different possible outcomes with no associated probability distribution. The work examines and defines a class of rules such that: (a) the evaluation of sets (actions) follows a certain procedural pattern; and (b) an assumption of sequential contraction consistency, which is an adaptation of Sen's α condition, is satisfied. In this framework, some results of characterization show that several well-known rules for comparing sets of outcomes can be reinterpreted in procedural terms.     

The demand for a risky asset in the presence of a background risk

We examine the demand for a risky asset in the presence of two risks: a financial risk and a background risk which need not be financial. First, we compute the necessary and sufficient condition for a positive demand for a risky asset, showing that it depends on two terms capturing respectively the direct effect of risk premium and the dependence between the two risks. Second, we develop higher order expectation dependence concept and show that the more information about the sign of higher cross derivatives of the utility function we have, the weaker dependence conditions on distribution we achieve.     

Representation and aggregation of preferences under uncertainty

We axiomatize in the Anscombe-Aumann setting a wide class of preferences called rank-dependent additive preferences that includes most known models of decision under uncertainty as well as state dependent versions of these models. We prove that aggregation is possible and necessarily linear if and only if (society's) preferences are uncertainty neutral. The latter means that society cannot have a non-neutral attitude toward uncertainty on a subclass of acts. A corollary to our theorem is that it is not possible to aggregate multiple prior agents, even when they all have the same set of priors. A number of ways to restore the possibility of aggregation are then discussed.     

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.     

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.     

Interim efficient allocations under uncertainty

This paper considers an exchange economy under uncertainty with asymmetric information. Uncertainty is represented by multiple priors and posteriors of agents who have either Bewley's incomplete preferences or Gilboa-Schmeidler's maximin expected utility preferences. The main results characterize interim efficient allocations under uncertainty; that is, they provide conditions on the sets of posteriors, thus implicitly on the way how agents update the sets of priors, for non-existence of a trade which makes all agents better off at any realization of private information. For agents with the incomplete preferences, the condition is necessary and sufficient, but for agents with the maximin expected utility preferences, the condition is sufficient only. A couple of necessary conditions for the latter case are provided.     

Strategies as states

We define rationality and equilibrium when states specify agents’ actions and agents have arbitrary partitions over these states. Although some suggest that this natural modeling step leads to paradox, we show that Bayesian equilibrium is well defined and puzzles can be circumvented. The main problem arises when player j's partition informs j of i's move and i knows j's strategy. Then i's inference about j's move will vary with i's own move, and i may consequently play a dominated action. Plausible conditions on partitions rule out these scenarios. Equilibria exist under the same conditions, and more generally ε equilibria usually exist.     

Risk taking under heterogenous revenue sharing

We examine the impact of informal risk sharing on risk taking incentives when transfers are organized through a social network. A bilateral partial sharing rule satisfies that neighbors share equally a part of their revenue. In such a society, correlated technologies generate interdependent risk levels. We obtain three findings. First, there is a unique and interior Nash-equilibrium risk profile, and it is in general differentiated and related to the Bonacich measure of the risk sharing network. Second, more revenue sharing enhances risk taking on average, although some agents may lower their risk level. Last, we find that under investment might often be observed.     

Subjective probability over a subjective decision tree

A state space has been assumed as a primitive for modeling uncertainty, which presumes that the analyst knows all the uncertainties a decision maker (DM) perceives. This is problematic because states are private information of the DM, and hence are not directly observable to the analyst. Dekel et al. [Representing preferences with a unique subjective state space, Econometrica 69 (2001) 891-934] derive, rather than assume, the subjective state space from preference over suitable choice objects.In a dynamic setting, a decision tree, that is, a pair consisting of a state space and a filtration, has been taken as a primitive. This assumption is also problematic—a decision tree should be derived rather than assumed as a primitive. We formulate a three-stage extension of the above literature in order to model a DM who anticipates subjective uncertainty to be resolved gradually over time. We identify also subjective beliefs on the subjective state space.     

A mechanism for eliciting a probability distribution

This work extends Karni’s direct revelation mechanism for eliciting agents’ subjective beliefs over the distribution of a random variable.