Applications of belief functions in business decisions: A review <s> Judgment under Ambiguity <s> I. Are there uncertainties that are not risks? 643. — II. Uncertainties that are not risks, 647. — III. Why are some uncertainties not risks? — 656. <s> BIB001 </s> Applications of belief functions in business decisions: A review <s> Judgment under Ambiguity <s> In uncertainty situations where knowledge is described by a Dempster-Shafer belief function (which is more general than a probability measure), von Neumann-Morgenstern linear utility theory applies and leads to a generalized expected utility representation of preference which allows for risk-attitude and ambiguity-attitude (pessimism/optimism). <s> BIB002 </s> Applications of belief functions in business decisions: A review <s> Judgment under Ambiguity <s> Abstract A primary motivation for reasoning under uncertainty is to derive decisions in the face of inconclusive evidence. Shafer's theory of belief functions, which explicity represents the underconstrained nature of many reasoning problems, lacks a formal procedure for making decisions. Clearly, when sufficient information is not available, no theory can prescribe actions without making additional assumptions. Faced with this situation, some assumption must be made if a clearly superior choice is to emerge. This paper offers a probabilistic interpretation of a simple assumption that disambiguates decision problems represented with belief functions. It is proved that it yields expected values identical to those obtained by a probabilistic analysis that makes the same assumption. A strict separation is maintained between evidence that carries information about a situation and assumptions that may be made for disambiguation of choices. In addition, it is shown how the decision analysis methodology frequently employed in probabilistic reasoning can be extended for use with belief functions. This generalization of decision analysis allows the use of belief functions within the familiar framework of decision trees. <s> BIB003 </s> Applications of belief functions in business decisions: A review <s> Judgment under Ambiguity <s> This article presents a descriptive theory for complex choice problems. In line with the bounded rationality assumption, we hypothesize that decision makers modify a complex choice into some coarse approximations, each of which is a binary lottery. We define the value of a best coarse approximation to be the utility of the choice. Using this paradigm, we axiomatize and justify a new utility function called the "coarse utility function." We show that the coarse utility function approximates the rank- and sign-dependent utility function. It satisfies dominance but admits violations of independence. It reduces judgmental load and allows flexible judgmental information. It accommodates phenomena associated with probability distortions and provides a better resolution to the St. Petersburg paradox than the expected and rank-dependent theories. Copyright 1995 by Kluwer Academic Publishers <s> BIB004 </s> Applications of belief functions in business decisions: A review <s> Judgment under Ambiguity <s> Behavioral accounting research deals with a complex set of phenomenon including the broad domain of human decision making under uncertainty. Two aspects of decision making of particular relevance to accounting and auditing research are two constructs that are inexorably interrelated: uncertainty and information (evidence). This paper introduces a theoretical perspective that enriches the knowledge-set that may be used in behavioral accounting research when confronting decision contexts that involve uncertainly. ::: ::: The main body of the paper is an introduction to belief functions. The introduction includes a discussion of the fundamental constructs and then illustrates the use of belief functions in two audit settings: traditional financial statement audit planning and the evaluation of evidence in a cascaded-inference setting involving the ebaluation of internal accounting control. The paper concludes with a brief exploration of some of the research issues and opportunities that are related to the potential use of belief functions in behavioral accounting research. <s> BIB005 </s> Applications of belief functions in business decisions: A review <s> Judgment under Ambiguity <s> Auditors are required to perform assessments of risk during the planning stages of an audit. Consequently, the efficiency and effectiveness of audits should be affected by risk assessments. Various models that could assist auditors in the risk assessment task have been discussed in the auditing literature. <s> BIB006
Ellsberg ( According to BIB001 , many people prefer Act 1 to Act 2 and Act 4 to Act 3. If we represent the ambiguity using probabilities, according to the subjective expected utility theory of  , the choice of majority implies the following inequalities: where ρ(r), ρ(b), and ρ(y) are the respective the probabilities of drawing red, black, and yellow colors, and V(x) is the utility of payoff x with V(0) = 0. Note that, the above two inequalities are conflicting with each other because they can be respectively reduced to ρ(r) > ρ(b) and ρ(r) < ρ(b). This conflict is now often referred to as the Ellsberg paradox. In the Ellsberg's example, the choice of majority is reasonable in the sense that people are ambiguity averse. In the first comparison, a third of balls are red while the proportion of black ones is unspecified. In the second comparison, two-thirds of the balls are black and yellow while the proportion of red and yellow balls is unknown. In other words, there is ambiguity for event "black" and so for the outcome of Act 2 in the first comparison. Similarly, there is ambiguity for event "red or yellow" and so for the outcome of Act 3 when comparing Act 3 and Act 4. The obvious force of ambiguity aversion drives the choice of majority to be the one we observed. There have been many resolutions proposed to explain the Ellsberg paradox. For example,  argues that the expectation formulism lacks plausible semantics when it applies to a one-shot gamble. An expected utility was initially interpreted by Bernoulli as a measure of one's long-run average in multiple plays of a gamble. These semantics are adequate when uncertainty is represented by empirical frequencies, and the same gamble is repeated. However, applying the expected utility principle to a one-shot play of a gamble is counterintuitive (see also Einhorn 1990, Liu and BIB004 . Churchill Eisenhart illustrates this predicament nicely (see  . A Pacific naval base has a machine selling Coca-Cola at 5 cents a bottle. The price of Coke goes up to 6 cents. The machine takes only nickels. If at random one in every six bottles in the machine is empty, then the machine is undoubtedly fair. Such a mechanical vendor may be fine for the regular patrons but a casual visitor who has to pay a dime before he or she gets a drink is not likely to think the game fair even if he or she is warned beforehand. The Ellsberg paradox raised a fundamental issue on whether uncertainties can be represented as probabilities. The foundation of the Bayesian theory  ) builds on the affirmative answer to the question. However, the proponents of DST think otherwise. For example, many have suggested representing ambiguity using belief functions. Accordingly, BIB002  , BIB003 BIB003 , and  have proposed to extend the subjective expected utility theory to embed the belief function representation. In a study related to audit decisionmaking,  applies belief functions to understand how auditors make value judgments under ambiguity. Later, based on the model developed by BIB003 BIB003 , Srivastava (1997a) further explores decision-making behavior under ambiguity. He argues that the Ellsberg paradox is due to the difficulty in modeling ambiguity as probabilities. More interestingly, he notes that the belief-function approach models human decision behavior and value judgment more naturally than the traditional probabilistic approach. BIB005 discuss, in general, the relevance of belief functions in behavioral research in auditing. Along the same line of inquiries as in BIB002  , BIB003 BIB003 , and Yager (1989),  suggests that, besides risk aversion, additional parameter(s) such as ambiguity aversion might be needed to explain one's attitudes to ambiguity. This is evidenced in some existing resolutions to the Ellsberg paradox. For example, BIB003 and  introduce a single parameter, interpreted as a pessimism index, to characterize one's attitude to ambiguity.  Turner (1994) has examined, in a quasi-experimental setting, three models that are descriptive of the audit judgment process: the Audit risk model, the Dempster-Shafer Belief Function Model, and the Hogarth and Einhorn Belief Adjustment Model (1992).  have performed empirical studies to compare the audit risk model of SAS 47 with two other models: the firm-based model, and a belief-based model.  have used belief functions to analyze auditors' behavior in aggregating items of evidence to determine the likelihood of material misstatements in the valuation of inventory. Extending the research on audit risk, BIB006 conduct an experiment where auditors are asked to make judgment about audit risk using their intuition and compare this with the value generated by various models. Interestingly, their findings indicate that there is no statistical difference between the auditor's intuitive assessments and the values generated by the belief function model.  conducted an experiment where he asked the auditor subjects to make judgment about whether the evidence is positive or negative pertaining to a specific audit objective and the level of support provided to various audit objectives without contaminating their thought process whether they express their judgment in terms of probabilities or belief functions. It is interesting to note that 80% of the responses were modeled only through belief functions.  