A challenge to the compound lottery axiom: A two-stage normative structure and comparison to other theories

This paper examines preferences among uncertain prospects when the decision maker is uneasy about his assignment of subjective probabilities. It proposes a two-stage lottery framework for the analysis of such prospects, where the first stage represents an assessment of the vagueness (ambiguity) in defining the problem's randomness and the second stage represents an assessment of the problem for each hypothesized randomness condition. Standard axioms of rationality are prescribed for each stage, including weak ordering, continuity, and strong independence. The Reduction of Compound Lotteries' axiom is weakened, however, so that the two lottery stages have consistent, but not collapsible, preference structures. The paper derives a representation theorem from the primitive preference axioms, and the theorem asserts that preference-consistent decisions are made as if the decision maker is maximizing a modified expected utility functional. This representation and its implications are compared to alternative decision models. Criteria for assigning the relative empirical power of the alternative models are suggested.     

Non-expected utility risk premiums: The cases of probability ambiguity and outcome uncertainty

This paper discusses two problems. (a) What happens to the conditional risk premium that a decision maker is willing to pay out of the middle prize in a lottery to avoid uncertainty concerning the middle prize outcome, when the probabilities of other prizes change? (b) What happens to the increase that a decision maker is willing to accept in the probability of an unpleasant outcome in order to avoid ambiguity concerning this probability, when this probability increases? We discuss both problems by using anticipated utility theory, and show that the same conditions on this functional predict behavioral patterns that are consistent both with a natural extension of the concept of diminishing risk aversion and with some experimental findings.     

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.     

A possibilistic hierarchical model for behaviour under uncertainty

Hierarchical models are commonly used for modelling uncertainty. They arise whenever there is a `correct' or `ideal' uncertainty model but the modeller is uncertain about what it is. Hierarchical models which involve probability distributions are widely used in Bayesian inference. Alternative models which involve possibility distributions have been proposed by several authors, but these models do not have a clear operational meaning. This paper describes a new hierarchical model which is mathematically equivalent to some of the earlier, possibilistic models and also has a simple behavioural interpretation, in terms of betting rates concerning whether or not a decision maker will agree to buy or sell a risky investment for a specified price. We give a representation theorem which shows that any consistent model of this kind can be interpreted as a model for uncertainty about the behaviour of a Bayesian decision maker. We describe how the model can be used to generate buying and selling prices and to make decisions.     

On the nonexistence of Blackwell's theorem-type results with general preference relations

A well-known theorem of Blackwell states that, when quantity of information is properly defined, every expected utility decision maker prefers more information to less; for more general preferences, however, the theorem is no longer true. In this article, we investigate the extent to which Blackwell's Theorem does not hold and describe conditions, and situations, under which information is still valuable. We also show that, for many types of additions of information, there exists a decision maker who will reject this information.We thank Niv Ahituv, Larry Epstein, Uzi Segal, and an anonymous referee for their helpful comments. This article was partially financed by the Israel Institute of Business Research.     

Maximizing expected utility is a survival criterion

The conventional expected utility of a lottery establishes a tight myopic lower bound on the probability of indefinitely avoiding economic ruin, the probability of survival. The analysis presented assumes that the decision maker is able to decline lotteries that are unacceptable in the expected utility sense. Application of the result to the assessment of utility functions is discussed.     

Process fairness,outcome fairness,and dynamic consistency: Experimental evidence for risk and ambiguity

Literature on fairness preferences distinguishes between outcome fairness, concerning the final allocation of payoffs, and process fairness, concerning the expected allocation of payoffs. It is not obvious, however, whether process fairness can consistently be implemented. Once uncertainty is resolved and outcomes are determined, the ex-ante procedurally fair decision maker may become consequentialist ex-post, and reconsider her choice on the basis of the observed outcomes. We present experimental evidence on dynamic consistency of social preferences under both known risk and ambiguity. A significant share of people subscribe to process fairness both before and after the resolution of uncertainty.     

Ambiguity,pessimism, and rational religious choice

Using a subclass of the α-maximin expected-utility preference model, in which the decision maker’s degree of ambiguity and degree of pessimism are each parameterized, we present a theory of religious choice in the Pascalian decision theory tradition, one that can resolve dilemmas, address the “many Gods objection,” and address the ambiguity inherent in religious choice. Parameterizing both the degree of ambiguity and the degree of pessimism allows one to examine how the two interact to impact choice, which is useful regardless of the application. Applying this model to religious choice is a move beyond subjective expected-utility theory, allowing us to show that a change in either the degree of ambiguity or the degree of pessimism can lead a decision maker to “convert” from one religion to another.     

Strategic communication with reporting costs

A decision maker relies on information of parties affected by her decision. These parties try to influence her decision by selective disclosure of facts. As is well known from the literature, competition between the informed parties constrains their ability to manipulate information. We depart from this literature by introducing a cost to communicate. Our parties trade off their reporting cost against the effect on the decision. Some information is never revealed. In contrast to setups without communication costs, our decision maker can benefit by ex ante committing to an ex post suboptimal decision rule. Moreover, committing ex ante not to listen to one of the parties may also be beneficial for the decision maker.

     

Ambiguity and risk taking in organizations

Kunreuther, Meszaros, and Hogarth (1993) argue that insurers are risk averse and ambiguity averse, and that they use cognitive reference points and constraints in making pricing decisions. They further claim that insurer ambiguity may be a factor that has a role in market failure at the industry level. Arguably, ambiguity may be an important aspect of decision behavior. In this article, research on managerial risk taking is reviewed with a focus on the relationship between ambiguity and risk taking. In particular, the effects of the organizational and institutional context are highlighted. It is argued that the political aspects of insurer decision behavior should be considered as well. Implications for further study and understanding of decision making are discussed.     

Partial information on decisional competences and the desirability of the expert rule in uncertain dichotomous choice situations

The paper analyzes the expert resolution problem by employing extended versions of the uncertain dichotomous choice model. The main purpose of this study is to illustrate how the analysis of optimal decision rules can be carried out while dispensing with the common restrictive assumption of full information regarding individual decisional competences. In contrast to most previous studies in this field we here evaluate the expert rule under alternative assumptions regarding the available partial information on judgmental competences rather than compare it in an ad hoc manner to some common alternative rules, such as simple majority rule. A fuller optimality analysis allowing the evaluation of all relevant decision rules, and not merely the expert rule, is attempted for a five-member panel of experts assuming a uniform distribution of individual competences. For three-member groups the optimality issue is resolved by assuming no information on individual competences and interpreting the expert rule as an even-chance lottery on skills.     

Are beliefs a matter of taste? A case for objective imprecise information

We argue, in the spirit of some of Jean-Yves Jaffray’s work, that explicitly incorporating the information, however imprecise, available to the decision maker is relevant, feasible, and fruitful. In particular, we show that it can lead us to know whether the decision maker has wrong beliefs and whether it matters or not, that it makes it possible to better model and analyze how the decision maker takes into account new information, even when this information is not an event and finally that it is crucial when attempting to identify and measure the decision maker’s attitude toward imprecise information.     

Uncertainty Aversion Vs. Competence: An Experimental Market Study

Heath and Tversky (1991, Journal of Risk and Uncertainty 4:5–28) posed that reaction to ambiguity is driven by perceived competence. Competence effects may be inconsistent with ambiguity aversion if betting on own judgement is preferred to betting on a chance event, because judgemental probabilities are more ambiguous than chance events. This laboratory experiment analyses whether ambiguity affects prices and volumes in a double auction market, and contrasts ambiguity aversion to competence effects. In order to test for the presence of competence effects, in the experiment uncertainty is tied to the realisation of events about which the decision maker is more or less knowledgeable. Two experiments are presented: in the first, knowledge is exogenous, whereas in the second the knowledge judgement is endogenous. Market prices provide evidence in favour of the competence hypothesis only when competence is self-assessed. Comparable volumes are observed in both experiments.      

Ambiguity and decision modeling: A preference-based approach

In this article, we develop a model that permits a decision maker's preferences to depend on the decision maker's ambiguity about the probability of an event that is relevant for decision-making purposes. We deal with ambiguity through preference modeling, with ambiguity leading to modifications in the utilities of outcomes. The behavior of ambiguity premiums and probability premiums as the payoffs are varied depends on the nature of the modifications in utilities. Particular forms of the model that arise under different sets of assumptions about preferences include additive, bilinear, and ratio forms. We conclude with a brief example and some thoughts about potential generalizations and implications of the model.     

Invariant multiattribute utility functions

We present a method to characterize the preferences of a decision maker in decisions with multiple attributes. The approach modifies the outcomes of a multivariate lottery with a multivariate transformation and observes the change in the decision maker’s certain equivalent. If the certain equivalent follows this multivariate transformation, we refer to this situation as multiattribute transformation invariance, and we derive the functional form of the utility function. We then show that any additive or multiplicative utility function that is formed of continuous and strictly monotonic utility functions of the individual attributes must satisfy transformation invariance with a multivariate transformation. This result provides a new interpretation for multiattribute utility functions with mutual utility independence as well as a necessary and sufficient condition that must be satisfied when assuming these widely used functional forms. We work through several examples to illustrate the approach.     

Choices Under Ambiguity With Familiar And Unfamiliar Outcomes

This paper considers a decision-making process under ambiguity in which the decision-maker is supposed to split outcomes between familiar and unfamiliar ones. She is assumed to behave differently with respect to unfamiliar gains, unfamiliar losses and customary (familiar) outcomes. In particular, she is supposed to be pessimistic on gains, optimistic on losses and ambiguity neutral on the familiar outcomes. A generalization of the usual Choquet Integral is formalized when the decision maker holds capacities and probabilities. A characterization of the decision-maker’s behavior is provided for a specific subset of capacities, in which it is shown that the decision-maker underestimates the unfamiliar outcomes while is linear in probabilities on customary ones.     

The role of ambiguity in manipulating voter behavior

In the standard account of ambiguous political rhetoric, ambiguity is equated with voter uncertainty: a politician takes an ambiguous stance on an issue by announcing to the voters a lottery over the positions he might take on that issue. The present account equates ambiguity with multiple meanings: a politician takes an ambiguous stance on an issue if his rhetoric admits of different meanings in different contexts. The linguistic account of ambiguity is based upon the formal logical concept of a piecewise definition. Piecewise defined terms provide the politician with the means of uttering statements each of which has seperate meanings for different voters.This paper formalizes the multiple-meaning account of ambiguity and traces the implications of the formalization for political strategy. The findings are: ambiguous rhetoric is often optimal; the conditions under which it is optimal are simple; and the selection of optimal ambiguous rhetoric is rarely simple.