# A general framework for accounting and audit decision making under ambiguity.

INTRODUCTIONThis study shows that ambiguity and vagueness in accounting and auditing (hereafter accounting) involves different sources (ambiguous probability judgments, vague states of nature, imprecise payoffs, and varying degrees of precision) and that decision theory (e.g., von Neumann & Morgenstern, 1944; Savage, 1954) fails to deal with these sources. The failure to deal with the ambiguity in accounting leads to incomplete and unrealistic representation of accounting problems. In addition, ambiguity should not be ignored for it affects decisions.

The problem of ambiguity and the inability of decision theory to deal with ambiguity were noted by fuzzy set theorists, behavioralists, and expected utility theorists. Savage (1954, 57) himself noted, "neither the theory of personal probability ... nor any other device known" to him at that time would accommodate vagueness. Savage (1954, 169) also noted that "the aura of vagueness ... attached to ... judgments" may lead to violations of his axioms.

The effect of ambiguity (in particular, vague probabilities) on decision making was supported by the empirical research on Ellsberg's paradox (Bernasconi & Graham, 1992; Curley & Yates, 1985; Dolan & Jones, 2004; Du & Budescu, 2005; Einhorn & Hogarth, 1985, 1986; Ellsberg, 1961; Hayashi & Wada, 2010; MacCrimmon & Larsson, 1979; Mukerji, 2000; Yates & Zukowski, 1976). The paradox suggests when given a choice between options with vague probabilities and options with known probabilities, the majority of people display ambiguity avoidance (prefer options with known probabilities) and a small number of people display ambiguity seeking (prefer options with vague probabilities). Both ambiguity avoidance and seeking violate the axioms of decision theory including Savage's sure thing axiom and the additivity of probabilities principle.

Accountants also noted that ambiguity in accounting may affect the lobbying activities of the firm (Stephens, Dillard, & Dennis, 1985), decision to comply with government regulations (Maher, 1981), demand for auditing (Wallin, 1992), auditors' loss contingency reporting decision (Nelson & Kinney, 1997), the planning of audit engagements (Guess, Louwers, & Strawser, 2000), quality control decisions (Ghosh & Ray, 1997), and audit risk assessments (Wright & Davidson, 2000). Research also showed that ambiguity in the tax area affects tax decisions including the decision to comply with tax laws (Spicer & Thomas, 1982).

Different models to address the problem of ambiguity have been provided by behavioralists, expected utility theorists, and fuzzy set theorists and have been used by accountants to address the ambiguity in some accounting problems. As explained later, these models allow for only some of the different aspects of the ambiguity in accounting and/or do not address (or fully explain) the empirical evidence on the effect of ambiguity on decisions.

Therefore, the objective of this study is to provide a general framework for decision making under ambiguity. Using the calculus of fuzzy sets theory (Zadeh, 1965), the proposed framework generalizes decision theory and expands the previous ambiguity models to allow for the different sources of the ambiguity in accounting (including, imprecise payoffs, vague probabilities, fuzzy states of nature, and varying degrees of precision). Therefore, it provides for more complete and realistic representation of accounting problems.

In addition, the model is flexible in that it incorporates decision theory and decision making under risk as a special case. The study proves that, when there is no ambiguity in the decision situation (i.e., for decision making under risk), the framework results in decisions identical to those obtained by decision theory. Thus, it satisfies the consistency with decision theory test which is used (e.g., Ashton, 1982) in evaluating models that allow for risk. The framework also explains the empirical evidence on Ellsberg's paradox. Thus, it satisfies Einhorn & Hogarth's (1985, 436) suggestion that, any theory for decision making under ambiguity should "explain the pattern of choices elicited by Ellsberg's problems."

The remainder of the paper is organized as follows. The next section examines the problem of ambiguity in accounting. Section III discusses some of the earlier ambiguity models. The proposed framework is presented in sections IV and V and illustrated in section VI. Section VII shows that the framework incorporates decision theory and decision making under risk as a special case. In section VIII, the framework is used to explain the empirical evidence on Ellsberg's paradox. Section IX provides a summary and suggestions for future research. Finally, Appendix A provides a brief overview of fuzzy set theory calculus.

DECISION THEORY AND THE AMBIGUITY IN ACCOUNTING

Ambiguity and vagueness are different from randomness. Randomness deals with the uncertainty (in terms of probability) regarding the occurrence of events. Ambiguity, however, has to do with the uncertainty due to the imprecision of words, stimuli, events, estimates, and judgments (such as probability judgments). For example, the term Material Account Error involves vagueness because of the inexact meaning of the word Material. But, the question about the probability of an error of $1000 involves randomness. The event is well defined (an error is either $1000 or not); the uncertainty lies with the occurrence/nonoccurrence of the event.

Randomness, as measured by probability, can be related to vague events as when one asks about the probability of having a Material Account Error or a Large Cost Variance. On the other hand, probability judgments can be vague and ambiguous as when one expresses his probability judgments by using values such as About 40% and Pretty Likely.

Similar definitions of ambiguity were provided in other fields. These definitions emphasize imprecision but differ from field to field to reflect the subject matter of the fields. For example, following Zadeh (1965), fuzzy set theorists have been concerned with set theory and suggested that ambiguity (fuzziness) has to do with fuzzy classes (e.g., Tall Men, Young Women); classes of objects with no exact/sharp boundaries between what is and what is not.

Some philosophers distinguish between ambiguity and vagueness. Black (1963), for example, noted that ambiguity exists when a word or concept has multiple meanings and is used to describe distinguishable subconcepts, while vagueness exists when the word or concept lacks precise shape and boundaries. However, in this paper the two terms are used interchangeably for vague concepts are often ambiguous. For example, the term Immaterial Error has no sharp boundaries (vague) and can be used to describe an error of $10 and an error of $20 (ambiguous).

In behavioral and expected utility literature, researchers (e.g., Einhorn & Hogarth, 1985, 1986; Ellsberg, 1961; Savage, 1954) distinguished between decisions made under risk (with known probabilities) and decisions made under ambiguity (with inexact probabilities). Thus, ambiguity is used to imply inexact probabilities or uncertainty about probabilities. However, the use of inexact probabilities to define ambiguity may not cover other sources of ambiguity including, inexact payoffs, fuzzy states of nature, and varying levels of precision.

To allow for the other sources of ambiguity, the definition of decision making under ambiguity is modified to include decisions with inexact probabilities, imprecise payoffs, fuzzy states of nature, and/or varying degrees of precision. Moreover, decision making under risk is redefined to include decisions with not only known probabilities but also with precise payoffs, well-defined states of nature, and constant level of precision. The remainder of this section examines the sources of ambiguity and the inability of decision theory to deal with these sources.

First, decision theory requires precise probability judgments. However, probability judgments in accounting are not always precise and accountants, like other professionals, often use verbal (vague) expressions rather than precise numerical values to express their probability judgments. For example, auditors use terms such as High, Moderate, and Low in estimating audit risk (Boritz, Gaber, & Lemon, 1987). Similarly, accounting pronouncements use the words Probable, Reasonably Possible, Remote, Likely, and Unlikely in characterizing the probabilities of contingencies (Schultz & Reckers, 1981) and the words Too Uncertain and Reasonably Certain to express the likelihood of benefits from R&D projects (Chesley, 1985).

Second, decision theory assumes the states of nature to be well-defined (simple) events. Decision theory and probability calculus are based on ordinary set theory which treats presence or absence (membership) in a binary manner. In ordinary set theory, an object [x.sub.i] can be either a member or nonmember of set A. That is, ordinary set theory and, thus, decision theory are based on what is known by philosophers (e.g., Black, 1963) as the Law of the Excluded Middle or the two-valued logic of either/or, yes/no, true/false, black/white, and all-or-nothing.

However, in many accounting problems the states of nature (Material Errors, Significant Variances, Efficient Sampling Procedure, Strong Internal Control System) are vague (ill-defined) and represent fuzzy classes (classes with no sharp transitions between what is and what is not). Stated differently, the states of nature in accounting are inherently vague in the sense that they involve concepts that defy binary classification and violate the Law of the Excluded Middle and the two-valued logic. For example, Ro (1982, 404) noted, "Materiality is not ... a dichotomous concept." Similarly, Ijiri and Jaedicke (1966, 477) noted, "objectivity is not a black-or-white issue." Finally, Kaplan (1975, 323) noted, in cost variance investigation "the forced dichotomy between in-control and out-of-control may be an unrealistic aggregation of reality."

Third, decision theory requires precise outcomes. However, the costs and benefits do not lend themselves to precise measurements in many accounting problems and areas including capital budgeting (Anthony, Dearden, & Bedford, 1984, 403), the choice of accounting systems (Horngren, Datar, Foster, Rajan, & Ittner 2009, 12), evaluation of internal control (Cooley & Hicks, 1983), evaluation of discretionary expense centers (Kaplan & Atkinson, 1998, 295), cost variance investigation (Zebda, 1984), social accounting (Jensen, 1976, 60), and nonprofit organizations (Anthony, Dearden, & Bedford, 1984, 745).

Fourth, decision theory assumes a fixed level of accuracy. However, in accounting the accuracy of the information, including both probability judgments and payoff estimates, are not always fixed. For example, Larsson and Chesley (1986) noted that probability judgments in auditing are subject to varying levels of accuracy. Similarly, the accuracy of payoffs is not fixed in cost variance investigation (Bierman & Dyckman, 1971, 504), capital budgeting (Anthony, Dearden, & Bedford, 1984, 403), and social and financial accounting (Jensen, 1976, 96).

In summary, ambiguity is an important characteristic of many accounting problems and, thus, the failure to deal with ambiguity may lead to incomplete and unrealistic representation of decisions problems. More importantly, ambiguity should not be ignored for it affects decisions. Such effect, as mentioned earlier, has been noted by many accountants, expected utility theorists, and behavioralists, and widely supported by the empirical research on Ellsberg's paradox.

Some researchers provided new theories as alternatives to expected utility theory (e.g., Chew, Karni, & Safra, 1987; Kahneman & Tversky, 1979; Machina, 1982; Shafer, 1976). These theories, however, are not designed to accommodate the ambiguity in decision making. The next section discusses some of the previous attempts to address the problem of ambiguity.

PREVIOUS AMBIGUITY MODELS

Over the years, researchers have provided models that extend decision theory to allow for some aspects of the problem of ambiguity in decision making. These models can be divided into two groups. The first group of ambiguity decision models were proposed by behavioralists and expected utility theorists (e.g., Becker & Brownson, 1964; Einhorn & Hogarth, 1985, 1986; Ekenberg & Thorbiornson, 2001; Ekenberg, Thorbiornson, & Baidya, 2005; Ellsberg, 1961; Fine, 1988; Gajdos, Hayashi, Tallon, & Vergnaud, 2008; Gardenfors & Sahlin, 1982, 1983; Kyburg, 1988; Larsson, 1976; Marschak, 1975; Utkin, 2003; Weichselberger, 2000). The second group of ambiguity decision models appeared in fuzzy set theory (and the related possibility theory) literature (e.g., de Campos & Huete, 2001; Freeling, 1980, 1984; Jain, 1976; Georgescu, 2009; Pan & Yuan, 1997; Tanaka, Okuda, & Asai, 1976; Toth, 1992; Walley & de Cooman, 2001; Watson, Weiss, & Donnell, 1979; Yager, 1979, 1987; Zadeh, 2002).

Some of these models have been used in addressing the ambiguity in accounting. For example, Larsson & Chesley (1986) extended the model by Larsson (1976) to address vagueness in audit inference. Other researchers (e.g., Ghosh & Ray, 1992; Guess, Louwers, & Strawser, 2000; Nelson & Kinney, 1997) used the model by Einhorn & Hogarth (1985) to examine the effect of ambiguity on accounting decisions. Moreover, researchers used some of the models provided by fuzzy set theorists (and fuzzy set theory) to address the ambiguity in internal control evaluation (Cooley & Hicks, 1983), cost variance investigation (Zebda, 1984), cost-volume-profit analysis (Lilian & Yuan, 1990), materiality judgment (McEacharn, Zebda, & Calloway, 1995), peer review (Omer, Leavins, & Chandra, 1998), tax planning (Hagan, de Korvin, & Siegel, 1996), corporate acquisition (McIvor, McCloskey, Humphreys, & Maguire, 2004), going-concern assessment (Lenard, Alam, & Booth, 2000), human resource allocation in a CPA firm (Kwak, Shi, & Jung, 2003), and detecting fraud (Pathak, Vidyarthi, & Summers, 2005).

Models Proposed by Behavioralists and Expected Utility Theorists

These models used different methods to deal with ambiguous probabilities. For example, some researchers (e.g., Ekenberg & Thorbiornson, 2001; Marschak, 1975; Utkin, 2003) suggested the use of second order probabilities to address the problem of ambiguity. However, the second order probability approach was rejected by many researchers (e.g., Curley & Yates, 1985; Einhorn & Hogarth, 1985; Yates & Zukowski, 1976) for failing to explain the behavioral evidence on Ellsberg's paradox. Moreover, the second order probability approach allows for only ambiguous probabilities and, therefore, it does not fully capture the different sources of the ambiguity in decision making. One can also criticize the second order probability approach for treating ambiguity as if it was randomness that can be handled by probability theory.

The majority of the other models used the idea of presenting a vague probability by a class of possible probability distributions. For example, Einhorn and Hogarth (1985, 435) noted that vague probabilities involve situations where a number of distributions cannot be ruled out, while precise probabilities involve situations where "all but one distribution is ruled out."

Two of these models (Ellsberg, 1961; Gardenfors & Sahlin, 1982, 1983), like the model proposed in this study, suggested the use of a reliability measure to allow for the idea that the distributions in the set of possible distributions may have different levels of reliability. The two models allow but do not require the reliability measure to be a second-order probability measure.

More specifically, Ellsberg's model (1961) assumes that decision makers react to vague probabilities in two stages. In the first stage, the decision makers identify the set of all possible distributions and (implicitly) their reliability. This set of distributions (denoted Y) is reduced to a set of distributions (denoted [Y.sup.[omicron]]) that are reasonable and cannot be ruled out. The distributions in set [Y.sup.[omicron]] are then combined to calculate a composite (estimated) distribution (denoted by [y.sup.[omicron]]).

In the second stage, the decision makers calculate for each action x two values: (1) the expected value given the estimated composite distribution (denoted [est.sub.x]), and (2) the minimum expected value over all reasonable distributions in the set [Y.sup.[omicron]] (denoted [min.sub.x]). The decision makers are assumed to maximize an index representing a linear combination of these two values. For each action x, the linear combination of the above two values is calculated by p*[est.sub.x] + (1-p)*[min.sub.x], where p represents the decision maker's confidence in the composite distribution [y.sup.[omicron]].

The model by Gardenfors and Sahlin (1982, 1983) is explicit in allowing the probability distributions in the set of possible distributions to have different levels of reliability for it adds a new measure to each of the possible distributions indicating their (epistemic) reliability. The class of probability measures (denoted by P) is assumed to include all epistemically possible probability measures over the states of nature. The reliability measure is denoted by p.

Their decision procedures consist of two steps. First, the set P is restricted to a set of probability measures with a "satisfactory" degree of reliability. The restricted set is denoted by P/[p.sub.o], where [p.sub.o] is the satisfactory level of reliability. The second step involves the calculation of the expected utility for each alternative and each probability distribution in the restricted set of probability measures. Then, the minimum expected utility of each alternative is determined and the alternative with the largest minimum expected utility is selected as the optimal alternative.

Some researchers provided inference/judgment models for assessing vague probabilities. The model by Einhorn and Hogarth (1985) is of special interest for it has been used by accountants to study the effect of ambiguity on accounting decisions. The model assumes that people use an "anchoring-and-adjustment" process when they assess vague probabilities. The probability judgment resulting from this strategy is defined by S([p.sub.A]) = [p.sub.A] + k, where the anchor ([p.sub.A]) is an initial probability judgment from which adjustment is made. The adjustment (k) is assumed to be affected by 3 factors: (1) level of the anchor ([p.sub.A]), (2) amount of ambiguity perceived in the situation, and (3) the individual's attitude toward ambiguity that allows one to differentially weight values in the set of possible probability values.

In their 1986 paper, Einhorn and Hogarth extended their model of inference (judgment) under ambiguity to provide a decision model for choice under ambiguity. Like Kahneman and Tversky (1979), they defined the subjective worth of outcomes over gains and losses instead of the final asset position. The decision makers are assumed to be maximizers of expected worth under ambiguity (denoted EWA) which is defined for a two-outcome gamble as EWA = [W.sub.G]*S([p.sub.G]) + [W.sub.L]*S([p.sub.L]), where [W.sub.G] and [W.sub.L] are the subjective worths of outcome gained and lost in the gamble, and S(pG) and S(pL) are the ambiguous probabilities of gaining and losing.

The above models addressed the empirical evidence on Ellsberg's paradox. However, they allow for only vague probabilities. Thus, like the second order probability approach, they do not fully capture the ambiguity in accounting. Einhorn and Hogarth (1985, 458) alluded to this problem by noting "our model does not explicate all aspects of ambiguous choice."

Moreover, some of these models do not fully explain the evidence on Ellsberg's paradox and the effect of ambiguity on decision making. For example, as noted by Einhorn and Hogarth (1985), the models by Ellsberg (1961) and Gardenfors and Sahlin (1982, 1983) do not explain ambiguity seeking behavior. Ellsberg's model can also be criticized for not being explicit in its treatment of varying reliabilities of probability distributions in the set of possible distributions.

Gardenfors and Sahlin's model is explicit in addressing the problem of varying reliabilities of probability judgments for it allows the possible probability distributions to be explicitly weighted by a reliability measure. However, this reliability measure is ignored by Gardenfors and Sahlin in the final selection of the optimal decision. Such neglect is inconsistent with Gardenfors and Sahlin's suggestion that the reliability of probabilities affects decisions and, thus, may explain why their model did not explain the ambiguity seeking behavior.

It should be noted that the models by Ellsberg and Gardenfors and Sahlin adapted the existing decision theory to account for the ambiguity in decision making. However, Einhorn and Hogarth's choice model differs from the other models in its treatment of vague probabilities because it requires precise estimates of S([p.sub.A]) in calculating EWA for the alternatives. In essence, rather than adapting the expected value model to the existence of ambiguity in probability judgments, the choice model provided by Einhorn and Hogarth reduces the ambiguity so that it becomes consistent with the existing expected value model.

Finally, some models (e.g., Becker & Brownson, 1964; Fine, 1988; Kyburg, 1988; Larsson, 1976; Weichselberger, 2000) allow vague probabilities to be represented by an interval probability and use the range (the difference between the maximum and minimum probabilities) as a measure of the degree of ambiguity. Thus, these models allow (indirectly) for the idea of treating probability judgments by a class of probability distributions. However, these models do not allow the distributions in the set of probability distributions to have different levels of reliability because they assume (implicitly) that the possible values in the probability interval (range) to have the same level of reliability. Moreover, like other models, these models allow for only vague probabilities and, thus, they do not fully capture the ambiguity in decision making.

Models Proposed by Fuzzy Set Theorists

These models are based on fuzzy set theory. The theory was introduced to engineering and computer sciences by Zadeh (1965) and subsequently it has been widely applied to many fields such as system theory, decision making, artificial intelligence and expert systems, pattern recognition, and medicine. Today, fuzzy set theory and the related fuzzy logic are used by many companies in the development and production of many everyday products and services such as cameras, camcorders, airconditioners, refrigerators, televisions, vacuum cleaners, washing machines, microwave ovens, cars, trains, elevators, financial analysis, and personal computers.

As defined by Zadeh (1965, 339), fuzzy sets are classes of objects (e.g., Young Men, Large Variances, Material Errors) with no sharp boundaries to separate those objects belonging to the class from those that do not. Formally, a fuzzy set A of a universe E is defined by:

A = {[u.sub.A]([x.sub.i])/[x.sub.i]}, (1)

where [u.sub.A]([x.sub.i]), called membership or compatibility function, associates with every element [x.sub.i] in the universe E its compatibility with set A (i.e., the degree to which [x.sub.i] belongs to A). The symbol "/" is used to separate an element from its compatibility and not to indicate "division."

Unlike the characteristic function for an ordinary set which takes on only the value zero or 1, the membership function takes any value in the interval [0, 1]. The closer xi to satisfying the requirements of set A, the closer its membership grade is to 1. Thus, an element xi may be a nonmember of A [[u.sub.A]([x.sub.i]) = 0], slightly a member of A [[u.sub.A]([x.sub.i]) near 0], more or less a member of A [[u.sub.A]([x.sub.i]) close to .5], strongly a member of A [[u.sub.A]([x.sub.i]) near 1], or a member of A [[u.sub.A]([x.sub.i]) = 1].

Ordinary set theory treats membership in a binary manner; an object can be either a member or a nonmember of a set. Fuzzy set theory, on the other hand, allows for gradual membership and replaces the two-valued logic by a logic where different degrees of membership and shade are allowed. Thus, technically fuzzy set theory represents a generalization of ordinary set theory; an ordinary set can be thought of as a special fuzzy set whose membership function takes on only the value of zero or 1 rather than any value in the closed interval [0, 1].

Let, for example, E = {10, 20, ..., 90, 100} be the set of possible account errors. Then, a fuzzy set of E representing Moderate Error may be subjectively defined by: A = {(0/10), (0/20), (0/30), (.5/40), (1/50), (1/60), (.5/70), (0/80), (0/90), (0/100)}. By ignoring the errors with zero membership grades, set A is simplified as A = {(.5/40), (1/50), (1/60), (.5/70)}.

The membership grades for a fuzzy set represent the degrees to which the objects are compatible with the set and, thus, should not be interpreted as probabilities. For instance, the $70 error in the above example does not have a 50% chance of being moderate. Rather, the $70 error is only .5 compatible with Moderate Error. Moreover, as the example shows, the membership grades, unlike probabilities, do not have to add to one.

Rather than being defined for a finite set of elements, the membership function can be defined analytically. For example, let the fuzzy set A = {Small Variance} be defined by: [u.sub.A](x) = [[1 + [(.04x).sup.2]].sup.-1]. Thus, the compatibilities of $25 and $50 variances with set A are .50 and .20.

The membership functions can be derived either subjectively (Zadeh 1972) or empirically (Boucher & Cogus, 2002; Narazaki & Relescu, 1994; Reventos, 1999; Saaty, 1974; Wallsten, Budescu, Rapoport, Zwick, & Forsyth, 1986; Witteman & Renooij, 2003; Zimmer, 1984). Some researchers (Watson, Weiss, & Donnell, 1979) support the use of the empirical methods in the derivation of membership functions. However, it should be noted that subjectivity is the essence of ambiguity (Ellsberg 1961). Moreover, as noted by Zadeh (quoted in Watson, Weiss, & Donnell, 1979, 2-3), "it is not in keeping with the spirit of the fuzzy-set approach to be too concerned about the precision of [the grades of membership]."

One should note that fuzzy set theory is not a decision theory but rather a calculus and a modeling language (like probability calculus) whereby imprecise phenomena in decision processes and humanistic systems are dealt with systematically. The models proposed by fuzzy set theorists used fuzzy set theory calculus the same way probability calculus is used in the development of decision theory. More specifically, these models allow decision variables (e.g., payoffs) to be treated as fuzzy sets, which can be manipulated using the calculus of fuzzy sets.

Unlike the models by behavioralists and expected utility theorists, the majority of these models did not address the empirical evidence on Ellsberg's paradox. These models also ignore some aspects of the problem of ambiguity and, thus, they do not fully capture the ambiguity in accounting. For example, the models by Jain (1976) and Yager (1979, 1987) allow for ambiguous states of nature and payoffs but do not address the problem of ambiguous probabilities. The models by Tanaka, Okuda, & Asai (1976) and Toth (1992) allow for fuzzy states of nature (events) but fail to consider inexact probabilities and payoffs. The models by Watson, Weiss, & Donnell (1979) and Freeling (1980, 1984) allow for inexact payoffs and probabilities but do not allow for fuzzy states of nature because, as noted by Freeling (1980, p. 80), they are based on the assumption of "crisp" events or states of nature. Finally, many of the models (e.g., Pan & Yuan 1997; de Campos & Huete 2001; Walley & de Cooman 2001; Zadeh 2002) have focused primarily on the problem of ambiguous probabilities.

The next section generalizes decision theory and expands previous ambiguity models to allow for the different sources of the ambiguity in accounting including, vague probabilities, imprecise payoffs, vague states of nature, and varying degrees of precision. The proposed framework is based on the calculus of fuzzy set theory. (Readers unfamiliar with fuzzy set theory are encouraged to see Appendix A before proceeding to the next section.)

THE PROPOSED EXTENSION OF DECISION THEORY

Decision making under ambiguity is described by a quadruple <X, S, D, B>, where X, S, D, and B are, respectively, set of objects (e.g., account errors, cost variances), set of states of nature (e.g., Material Errors, Significant Variances), set of permissible decisions, and set of possible payoffs (outcomes) resulting from the different combinations of states of nature and permissible decisions. The set of states of nature (S) and the set of permissible decisions (D) are represented, respectively, by S = {[s.sub.i]\ i = 1, ..., n} and D = {[d.sub.j]\ j = 1, ..., m}.

The payoffs from the different combinations of decisions and states of nature (denoted [B.sub.ij]) are assumed to be stochastic. As in much of the literature on decision theory, the probabilities of the payoff [B.sub.ij] are assumed to be independent of which alternative is selected. That is, the probabilities of the payoff [B.sub.ij] (denoted [Q.sub.ij]) = probability of state [s.sub.i] (denoted [Q.sub.i]) for all alternatives. This assumption is used only for simplification and can be easily relaxed.

Probability Judgments

As in behavioral and expected utility theory literature, the proposed framework assumes that the decision maker's probability judgment is represented by a set of possible probability distributions. The distributions are allowed to have different levels of reliability by associating with every possible distribution a reliability measure representing the decision maker's judgment, given her knowledge of the situation, that the distribution is reliable and cannot be ruled out as a member of the set of possible distributions. The reliability measure is assumed to be a real-valued measure that takes any value in the closed interval M = [0,1]. The higher is the reliability of the distribution, the closer the reliability measure is to one.

The set of possible distributions (denoted Z) is treated as a fuzzy set whose membership function is the reliability measure. Let [P.sub.k] (k = 1, ..., q) be a possible distribution and q = number of possible distributions. Moreover, let [u.sub.Z]([P.sub.k]) = the reliability of distribution [P.sub.k]. Then, the set of possible distributions and their reliabilities can be represented by the fuzzy set:

Z = {[u.sub.z]([P.sub.1])/[P.sub.1], ..., [u.sub.z]([P.sub.k])/[P.sub.k], ..., [u.sub.z]([P.sub.q])/[P.sub.q]}, (2)

Using the terminology of fuzzy set theory, [u.sub.Z]([P.sub.k])/[P.sub.k] in the above Equation represents the membership grade or compatibility of distribution [P.sub.k] with set Z and the symbol "/" is used to separate [P.sub.k] from its membership grade and not to indicate division. Moreover, as mentioned earlier, the compatibilities or membership grades should not be interpreted as probabilities and, unlike probabilities, they do not have to add to one.

To obtain the probabilities of the states of nature, the set of possible distributions (Z) is partitioned into n fuzzy subsets, where n is the number of states of nature. Let each possible probability distribution [P.sub.k] be defined by:

[P.sub.k] = {[p.sub.ik]} = {[p.sub.1k], ..., [p.sub.ik], [p.sub.nk]}, (3)

where, [p.sub.ik] = the probability of state [s.sub.i] (i = 1, ..., n) in the possible distribution [P.sub.k], 0 [less than or equal to] [p.sub.ik] [less than or equal to] 1 and [[summation].sup.n.sub.i=1] [p.sub.ik] = 1. As a result, a set of q probability values {[p.sub.1k], ..., [p.sub.ik], ..., [p.sub.iq]} is obtained from Equations 2 and 3 for each state of nature [s.sub.i]. This set of values represents the possible values of the probability of state [s.sub.i] and can be represented by the fuzzy set:

[Q.sub.i] = {[u.sub.Qi]([p.sub.ik])/[p.sub.ik]}, (4)

where, k = 1, ..., q, q = number of possible distributions (or, equivalently, number of [p.sub.ik] in [Q.sub.i]), and [u.sub.Qi]([p.sub.ik]) = [u.sub.Z]([P.sub.k]). As an example, consider two states of nature [s.sub.1] and [s.sub.2]. Let Z be given by 3 distributions [P.sub.1] = {.2, .8}, [P.sub.2] = {.5, .5}, and [P.sub.3] = {.8, .2}. Let the reliabilities of these distributions be given by [u.sub.z]([P.sub.1]) = .4, [u.sub.z]([P.sub.2]) = .7, and [u.sub.z]([P.sub.3]) = .9. Then, set Z is represented by: Z = {.4/(.2, .8), .7/(.5, .5), .9/(.8, .2)} and the probabilities of [s.sub.1] and [s.sub.2] are given by: [Q.sub.1] = {(.4/.2), (.7/.5), (.9/.8)} and [Q.sub.2] = {(.4/.8), (.7/.5), (.9/.2)}.

By relaxing the assumption of a single probability distribution and representing probability judgments by fuzzy sets as in Equation (4), the proposed framework allows the decision makers to express their probability judgments by using imprecise statements and linguistic values (e.g., About 30%, Quite Likely, Better Than Even). For example, a probability judgment of About 30% can be subjectively defined by [Q.sub.i] = {(.6/.20), (1.0/.30), (.6/.40)}.

The above treatment of probability judgments also allows for the use of precise probabilities. In the case of precise probabilities, the set of possible distributions Z includes only one distribution [P.sub.k] with a reliability of 1. For example, let [P.sub.k] = {.2, .8} in the case of two states of nature ([s.sub.1] and [s.sub.2]). Then, set Z is given by Z = {1/(.2, .8)} and, using Equation (4), the probabilities of [s.sub.1] (.2) and [s.sub.2] (.8) are given by the fuzzy sets [Q.sub.1] = {(1/.20)} and [Q.sub.2] = {(1/.80)}.

States of Nature

The proposed framework allows the states of nature to be defined imprecisely. That is, every state of nature [s.sub.i] in set S will be described by a fuzzy set,

[s.sub.i] = {[u.sub.si](x)/x}. (5)

The membership function, [u.sub.si](x), associates with each element, x, in the universe X = {0,1,2, ...} its compatibility with state [s.sub.i]. For example, if the universe of errors is given by X = {10, 20, ..., 100}, then the states Immaterial Error, Moderate Error, and Material Error may be defined by the sets: Immaterial = {(1/10), (.8/20), (.6/30), (.4/40), (.2/50)}, Moderate = {(.5/40), (1/50), (1/60), (.5/70)}, and Material = {(.2/60), (.4/70), (.6/80), (.8/90), (1/100)}. As noted above, membership functions can be derived either subjectively or empirically. Moreover, rather than being defined for a finite set of elements, the membership function can be defined by a formula (analytically).

The framework also allows for precise states of nature. Consider, for example, the case of a gamble involving a flip of a coin. In this case, the universe of objects includes {h, t}. Assume two precise states of nature: Success (if h) and Failure (if t). These states of nature can be defined, using Equation (5), by the fuzzy sets: Success = {(1/h)} and Failure = {(1/t)}.

Payoffs

The framework allows payoff estimates ([B.sub.ij]) to be imprecise. For example, estimates of payoffs can take the form of Approximately $800 and Near $1,000. Decision makers can also treat the payoff as a linguistic variable and express their payoff estimates by using linguistic values such as High, Very High, Medium, Low, and Very Low. Any of these imprecise values is described by a fuzzy set represented by:

[B.sub.ij] = {[uB.sub.ij] ([b.sub.ijt])/[b.sub.ijt]}, (6)

where, t = 1, ..., t([sub.ij]), t([sub.ij]) = the number of [b.sub.ijt]'s in [B.sub.ij], and the membership function uBij(bijt) associates with every possible outcome bijt in the universe B = {0,1,2,....} its grade of membership or compatibility with set [B.sub.ij]. For example, if the universe of payoffs is given by B = {$100, $200, ..., $1000}, then Low (L) and High (H) outcomes may be subjectively defined by: L = {(.5/100), (1.0/200), (.5/300)} and H = {(.5/700), (1.0/800), (.5/900), (.2/1000)}.

The framework also provides for the use of precise payoffs. For example, in the cases of selecting information systems and investigating cost variances, estimates of costs can be precise as when one uses estimates such as $500, while estimates of benefits can be vague as when one uses values such as High Benefits. Using Equation (6), precise payoff estimates such as $500 can be described by the fuzzy set [B.sub.ij] = {(1/500)}.

Objective Function

To allow for the ambiguity in decision making, the objective of maximizing the expected value is reformulated differently as a maximization of an index called expected value index.

This index will be denoted by evi([d.sub.j]) and calculated by evi([d.sub.j]) = [EV.sub.j] / max [EV.sub.j], where [EV.sub.j] = expected value of dj and max [EV.sub.j] = maximum possible expected value.

The reformulated objective is equivalent to the maximization of the expected value because it results in the same optimal decisions. Consider, for example, two projects ([d.sub.1] and [d.sub.2]) with [EV.sub.1] = 100 and [EV.sub.2] = 80. Then, the evi([d.sub.1]) = 1 and evi([d.sub.2]) = .8 and the optimal decision is [d.sub.1] by both the maximization of expected value and the maximization of expected value index.

The proposed framework assumes the objective of maximizing a similar index that extends the expected value index to allow for ambiguity in accounting. The proposed index (called the relative merit index) represents the relative merit of the decisions in the presence of ambiguity. The measurement scale of the index is assumed to be the closed interval M = [0,1]. The higher the relative merit, the closer the number is to one, and vice versa.

Mathematically, the set of possible decisions and their relative merits is represented by a fuzzy set (called optimal decision set). Let [u.sub.OD]([d.sub.j]) be the relative merit index of [d.sub.j]. Then, the optimal decision set is represented by OD = {[u.sub.OD]([d.sub.j])/[d.sub.j]}. The optimal decision (denoted [d.sup.*]) is then the one with uOD([d.sup.*]) = max [u.sub.OD]([d.sub.j]) over j (the action that maximizes the relative merit index or equivalently the grade of membership in set OD). A simple example may illustrate the above idea. Assume that the relative merit index in the case of two audit procedures ([d.sub.1] and [d.sub.2]) is obtained by the model to be [u.sub.OD]([d.sub.1]) = .8 and [u.sub.OD]([d.sub.2]) = .3. Consequently, the fuzzy optimal decision set is given by OD = {(.8/[d.sub.1]), (.3/[d.sub.2])} and the optimal decision is [d.sub.1].

The calculation of the relative merit index and the derivation of the optimal decision set are provided in the next section. The discussion applies to decision making with and without ambiguity. However, as shown later, in the absence of ambiguity the relative merit index is reduced to (and can be calculated by) the above-mentioned expected value index. The remainder of this section relates the model's treatment of probability judgments to behavioral literature.

Probability Judgments Revisited

The framework's treatment of probabilities allows for many of the ideas proposed by behavioralists and expected utility theorists to deal with probabilities. First, as noted above, the framework allows ambiguous probabilities to be represented by a class of probability distributions and precise probabilities to be represented by one probability distribution. Thus, the model is consistent with the conceptualization of vague and precise probabilities in behavioral and expected utility literature.

Second, like Ellsberg (1961) and Gardenfors and Sahlin (1982, 1983), the model allows the distributions in the set of possible probability distributions to be weighted by a reliability factor. The framework also allows but does not require the reliability to be a second-order probability measure. However, unlike Ellsberg's model, the framework is explicit in allowing for probabilities with varying levels of reliabilities and unlike Gardenfors and Sahlin's model, the framework does not neglect the reliability measure in the final selection of the optimal decision.

Third, the model allows for vague probabilities to be represented by a range which is used by some researchers (e.g., Becker & Brownson, 1964; Einhorn & Hogarth, 1985; Larsson, 1976) as a measure of the degree of ambiguity. Let [p.sub.max] and [p.sub.min] be the highest and lowest values of [Q.sub.i] in Equation (4). Then, R = [p.sub.max] - [P.sub.min] is the range of [Q.sub.i] and can be used as a measure of the degree of ambiguity. The greater the amount of ambiguity, the larger is the difference between [p.sub.max] and [p.sub.min]. The maximum value (R = 1) occurs when [p.sub.max] = 1 and [p.sub.min] = 0, while the lowest value (R = 0) occurs when [p.sub.max] = [p.sub.min] (or when [Q.sub.i] is precise).

Finally, the model's treatment of probability judgments allows for the three factors assumed by Einhorn and Hogarth (1985) to affect ambiguous probability judgments (the anchor or initial probability judgment, the amount of ambiguity perceived in the situation, and the individual's attitude toward ambiguity that allows one to differentially weight possible probability values). Consider Equation (4) which can be rewritten as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

where [p.sub.max] and [p.sub.min] are, as defined above, the highest and lowest values of [Q.sub.i]. In this detailed equation, [p.sub.A] represents the initial probability judgment, the difference between [p.sub.max] and [p.sub.min] represents the degree of ambiguity, and [u.sub.Qi] can be used to allow for differential weighting of different probability values.

DERIVATION OF THE OPTIMAL DECISION SET

In the presence of ambiguity, the relative merit index of each decision and the optimal decision set can be obtained by first obtaining the expected value associated with the decision. In decision theory, the expected value of an action (denoted [EV.sub.j]) is given by the summation of [Q.sub.i]*[B.sub.ij] over i, where [B.sub.ij]'s and [Q.sub.i]'s are precise. This definition of expected value can be extended to decision making under ambiguity. In the case where the payoffs and probabilities (but not the states of nature) are imprecise, the expected value under ambiguity (denoted [EVA.sub.j]) is derived by using Extension Principle II, which allows the domain of an n-ary function to be extended from points in the universe into fuzzy subsets of the universe (see Appendix A and Zadeh 1975a).

Let [Q.sub.i]'s be fuzzy probabilities described by compatibility functions [u.sub.Qi]([p.sub.ik])'s, where k = 1, ..., q. Moreover, let [B.sub.ij]'s be fuzzy outcomes described by compatibility functions [u.sub.Bij]([b.sub.ijt])'s, where t = 1, t(ij). Then, by Extension Principle II and allowing for the probabilities [Q.sub.i]'s to be related through the condition [summation].sup.n.sub.i=1] [p.sub.ik] = 1, the expected value under ambiguity of each decision [d.sub.j] (denoted [EVA.sub.j]) is given by:

[EVA.sub.j] = {[u.sub.EVAj]([b.sub.Lj])/[b.sub.L]j}, (8)

where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the case where the states of nature are also ambiguous, the [EVA.sub.j] is derived as follows. First, Extension Principle I, which allows the domain of an unary function to be extended from points in the universe into fuzzy subsets of the universe (see Appendix A; and Zadeh, 1975a), is used to allow the payoffs ([B.sub.ij]) and probabilities ([Q.sub.i]) to reflect the fuzzy knowledge about the state of nature. These payoffs and probabilities (denoted F[B.sub.ij] and F[Q.sub.i]) are given by:

[FB.sub.ij] = {[u.sub.FBij]([B.sub.ij])/[B.sub.ij]},

[FQ.sub.i] = {[u.sub.FQi]([Q.sub.i])/[Q.sub.i]}, (9)

where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Second, sets [FB.sub.ij] and [FQ.sub.i], which are fuzzy sets of fuzzy payoffs [B.sub.ij] and fuzzy probabilities [Q.sub.i], are reduced to fuzzy sets of nonfuzzy payoffs and nonfuzzy probabilities by using the operation of fuzzification which has the effect of fuzzifying a nonfuzzy set or increasing the fuzziness of a fuzzy set (see Appendix A; and Zadeh, 1972a). The reduced sets (denoted [RFB.sub.ij] and [RFQ.sub.i]) are given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

Finally, using the traditional definition of expected value and Extension Principle II and allowing for the condition [[summation].sup.n.sub.i=1] [p.sub.ik] = 1, the extended definition of the [EVA.sub.j] is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that in set [EVA.sub.j], [b.sub.Lj] represents the possible expected values, while [u.sub.EVAj]([b.sub.Lj]) represents the reliabilities or compatibilities of [b.sub.Lj] with set [EVA.sub.j] and, therefore, it reflects the ambiguous knowledge about the decision problem. The optimal decision can be selected on the basis of the maximum possible expected value, [b.sub.Lj]. Such selection, however, ignores the reliabilities or compatibilities of [b.sub.Lj] with sets [EVA.sub.j]. Alternatively, the optimal action can be selected on the basis of the highest reliability or grade of membership in sets [EVA.sub.j]. Such selection, however, ignores the possible expected values, [b.sub.Lj], in sets [EVA.sub.j].

Consider two decisions [d.sub.1] and [d.sub.2] whose [EVA.sub.1] and [EVA.sub.2] are calculated as [EVA.sub.1] = {(.2/100), (.6/300), (.1/800)} and [EVA.sub.2] = {(.8/100), (.9/300), (.5/500)}. If the decision is based on only the maximum [b.sub.Lj], decision makers would choose d1 (to obtain the $800). However, the $800 has a compatibility of only .1 with set [EVA.sub.1]. On the other hand, if the decision is based only on the highest [u.sub.EVAj]([b.sub.Lj]), decision makers would choose [d.sub.2] which has the highest membership grade (.9). However, the [b.sub.Lj] associated with the highest [u.sub.EVA2] is only $300.

Allowing for both the possible expected values, [b.sub.Lj], and their compatibilities, [u.sub.EVAj]([b.sub.Lj]), in the selection of the optimal decision can be accomplished by using the previously-mentioned idea of expected value index to obtain for each decision a new set (expected value index under ambiguity) whose membership function reflects the possible expected values, [b.sub.Lj]. This new set is then combined with set [EVA.sub.j]. The membership of the combined set will reflect both the possible expected values and their compatibilities. When the expected values are ambiguous (represented by fuzzy sets) as in the case of [EVA.sub.j], the expected value index of each decision will also be ambiguous and can be obtained by employing the concept of a maximizing set for a set (see Appendix A; Zadeh, 1972b; and Jain, 1976).

Let EVA be the set of all possible [b.sub.Lj] and let [b.sub.max] be the maximum [b.sub.Lj] in set EVA. Then, the expected value index under ambiguity for [d.sub.j] (denoted [EVI.sub.j]) is given by the fuzzy set:

[EVI.sub.j] = {[u.sub.EVIj]([b.sub.Lj])/[b.sub.Lj]},

where,

[u.sub.EVIj]([b.sub.Lj]) = [([b.sub.Lj]/[b.sub.max]).sup.c], (12)

and c is an integer that takes on the value of either one or two depending on whether or not the states of nature are ambiguous.

In the determination of [EVI.sub.j], c is selected in a way to ensure the compatibility of [EVI.sub.j] with [EVA.sub.j] because the two sets will be combined later. When the states of nature are assumed to be vague, [EVA.sub.j] is obtained by performing the operation of fuzzification and is considered a 2nd level fuzzy set. In this case, using c = 2 in the derivation of [EVI.sub.j] makes it compatible to [EVA.sub.j]. When the states of nature are assumed to be nonambiguous, [EVA.sub.j] is considered a 1st level fuzzy set. In this case, using c = 1 ensures the compatibility of [EVI.sub.j] with [EVA.sub.j].

The membership function in set [EVI.sub.j], [u.sub.EVIj]([b.sub.Lj]), represents the degree to which [b.sub.Lj] approximates the maximum possible expected value ([b.sub.max]) and, thus, it reflects the possible expected values, [b.sub.Lj]. Based on the reformulated objective of maximizing the expected value index, the optimal decision can be selected on the basis of the maximum [u.sub.EVIj]([b.sub.Lj]). Such selection, however, does not allow for the vague knowledge about the decision problem which, as noted earlier, is reflected by the compatibility of [b.sub.Lj], [u.sub.EVAj]([b.sub.Lj]), with sets [EVA.sub.j]'s. To allow for such ambiguity in the selection of the optimal decision, sets [EVI.sub.j]'s and [EVA.sub.j]'s can be combined by formulating for each decision a new fuzzy set (denoted [EVC.sub.j]) as follows:

[EVC.sub.j] = {[u.sub.EvCj]([b.sub.Lj])/[b.sub.Lj]},

where,

[u.sub.EVCj]([b.sub.Lj]) = [[u.sub.EVAj]([b.sub.Lj]) x [u.sub.EVIj]([b.sub.Lj])]. (13)

The membership function [u.sub.EVCj]([b.sub.Lj]) for set [EVC.sub.j] is defined as the product of [u.sub.EVIj]([b.sub.Lj]) and [u.sub.EVAj]([b.sub.Lj]) and, consequently, it reflects both the degree to which [b.sub.Lj] approximates the maximum possible expected value measured by [u.sub.EVIj]([b.sub.Lj]) and the fuzzy knowledge about the decision problem represented by [u.sub.EVAj]([b.sub.Lj]).

In more general terms, [u.sub.EVCj]([b.sub.Lj]) can be defined as the intersection of [u.sub.EVAj]([b.sub.Lj]) and [u.sub.EVIj]([b.sub.Lj]). According to the original definitions in Zadeh (1965), the intersection operation and the connective AND are defined by the minimum operator. But, other definitions have appeared in fuzzy sets literature (Bellman & Zadeh, 1970; Weber, 1979; Yager, 1980; Zimmermann & Zysno, 1983). The suitability of those definitions was the subject of several studies (Alsina & Trillas, 1987; Bellman & Giertz, 1973; Thole, Zimmermann, & Zysno, 1979; Zimmermann & Zysno, 1983). However, available evidence indicates that no one definition is universally preferred. In this study, the multiplicative operator is used to assure that [u.sub.EVCj]([b.sub.Lj]) reflects both [u.sub.EVAj]([b.sub.Lj]) and [u.sub.EVIj]([b.sub.Lj]). The model, however, is adaptable to other operators and definitions.

The fuzzy optimal decision set, OD = {[u.sub.oD]([d.sub.j])/[d.sub.j]}, can now be obtained. The membership grade or relative merit index, [u.sub.OD]([d.sub.j]), of each decision [d.sub.j] in D is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)

Then, as defined above, the optimal decision ([d.sup.*]) is the alternative with the highest relative merit index or equivalently the highest grade of membership, [u.sub.OD]([d.sub.j]), in set OD.

NUMERICAL ILLUSTRATION

This section provides a numerical example of the model and the derivation of the optimal decision set. The example relates to the cost variance investigation problem which, as alluded to earlier, involves fuzzy states of nature, imprecise probabilities and payoffs, and varying levels of precision and, thus, it does not lend itself to treatment by the traditional decision theory.

Assume a cost variance investigation problem with a reported variance (x) of $800. Assume that the problem is formulated in terms of two (fuzzy) states of nature such as in-control ([s.sub.1]) and out-of-control ([s.sub.2]) and two decisions such as investigate ([d.sub.1]) and do-not-investigate ([d.sub.2]). Assume that the reported variance has compatibilities of .8 and .4 with the two fuzzy states of nature (i.e., [u.sub.s1](x) = .80 and [u.sub.s2](x) = .40). Finally, let probabilities judgments ([Q.sub.i]) and payoffs estimates ([B.sub.ij]) be known only imprecisely and given by: [Q.sub.1] = "unlikely" = {(.6/.2), (1/.3), (.6/.4)}; [Q.sub.2] = "likely" = {(.6/.8), (1/.7), (.6/.6)}; [B.sub.11] = "Very High" = {(.5/900), (1.0/1000)}; [B.sub.21] = "Medium" = {(.7/400), (1.0/500), (.7/600)}; [B.sub.12] = "Very Low" = {(1.0/100), (.4/200)}; and [B.sub.22] = "High" = {(.5/700), (1.0/800), (.5/900)}.

Using Equation (11) one can obtain [EVA.sub.1] and [EVA.sub.2] as follows:

[EVA.sub.1] = {(.24/500), (.24/520), (.28/550), (.28/580), (.24/600), (.4/620), (.24/640), (.4/650), (.24/660), (.24/680), (.28/690), (.24/700), (.28/720), (.24/760)},

[EVA.sub.2] = {(.2/460), (.2/500), (.24/520), (.2/550), (.24/560), (.2/580), (.4/590), (.2/600), (.32/620), (.24/660), (.24/680), (.2/690), (.2/740), (.2/760)}.

The set of all possible [b.sub.Lj] is then given by EVA = {460, 500, 520, 550, 560, 580, 590, 600, 620, 640, 650, 660, 680, 690, 700, 720, 740, 760} and, thus, [b.sub.max] = 760. Then using c = 2 and Equation (12), sets [EVI.sub.j] for [d.sub.1] and [d.sub.2] are given by:

[EVI.sub.1] = {(.43/500), (.47/520), (.52/550), (.58/580), (.62/600), (.67/620), (.71/640), (.73/650), (.75/660), (.8/680), (.82/690), (.85/700), (.9/720), (1/760)},

[EVI.sub.2] = {(.37/460), (.43/500), (.47/520), (.52/550), (.54/560), (.58/580), (.6/590), (.62/600), (.67/620), (.75/660), (.8/680), (.82/690), (.95/740), (1/760)}.

Using Equation (13) one can obtain [EVC.sub.1] and [EVC.sub.2] as follows:

[EVC.sub.1] = {(.1/500), (.11/520), (.15/550), (.16/580), (.15/600), (.27/620), (.17/640), (.29/650), (.18/660), (.19/680), (.23/690), (.2/700), (.25/720), (.24/760)},

[EVC.sub.2] = {(.07/460), (.09/500), (.11/520), (.1/550), (.13/560), (.12/580), (.24/590), (.12/600), (.21/620), (.18/660), (.19/680), (.16/690), (.19/740), (.2/760)}.

Then by Equation (14), the compatibilities (relative merits) of [d.sub.1] and [d.sub.2] with OD are .29 and .24, respectively. That is, OD = {(.29/[d.sub.1]), (.24/[d.sub.2])}. The optimal decision is therefore [d.sub.1].

THE PROPOSED FRAMEWORK AND DECISION MAKING UNDER RISK

Decision theory, although not adequate to deal with ambiguity, continues to be the most widely recommended model for decision making under risk. The proposed framework does not discard decision theory but incorporates it and decision making under risk as a special case. The framework is flexible in that it provides, as explained in a previous section, for precise probability judgments, precise payoffs, and well-defined states of nature. Moreover, when there is no ambiguity in the decision situation, the relative merit index is reduced to (and can be calculated by) the previously-mentioned expected value index. Thus, for decision making under risk, the results obtained by the proposed framework will be identical to those obtained by maximizing the expected value index and by decision theory.

Mathematical Proof

The relationship between the framework and decision theory in the absence of ambiguity can be mathematically proven as follows. In the absence of ambiguity, the expected value of each decision [d.sub.j] will be precise. Let the expected value of [d.sub.j] be denoted by [b.sub.j] and let the expected value index of [d.sub.j] be denoted by evi([d.sub.j]) and given, as defined before, by evi([d.sub.j]) = [b.sub.j]/[b.sub.max], where [b.sub.max] is the maximum possible expected value over j.

Using fuzzy set theory notation, the expected value of each decision can be represented by a fuzzy set similar to Equation (8) as follows:

[EVA.sub.j] = {[u.sub.EVAj]([b.sub.j])/[b.sub.j]}. (15)

Since [b.sub.j] is precise, [u.sub.EVAj]([b.sub.j]) = 1 and, thus, set [EVA.sub.j] is reduced to:

[EVA.sub.j] = {1/[b.sub.j]}. (16)

Substituting [b.sub.j] for [b.sub.Lj] in Equation (12) and using c = 1 (since the states of nature are precise), one obtains set [EVI.sub.j] as:

[EVI.sub.j] = {[u.sub.EVIj]([b.sub.j]])/[b.sub.j]},

where,

[u.sub.EVIj]([b.sub.j]) = [b.sub.j]/[b.sub.max] = evi([d.sub.j]). (17)

That is, [u.sub.EVIj]([b.sub.j]) = the expected value index of [d.sub.j]. Then, by substituting [b.sub.j] for [b.sub.Lj] in Equation (13), one can obtain set [EVC.sub.j] as follows:

[EVC.sub.j] = {[u.sub.EVCj]([b.sub.j])/[b.sub.j]}, (18)

where,

[u.sub.EVCj]([b.sub.j]) = [[u.sub.EVAj]([b.sub.j]). [u.sub.EVIj]([b.sub.j])];

Similarly, by substituting [b.sub.j] for [b.sub.Lj] in Equation (14), one can obtain the optimal decision set OD as follows:

OD = {[u.sub.OD]([d.sub.j])/[d.sub.j]},

where,

[u.sub.OD]([d.sub.j]) = [u.sub.EVCj]([b.sub.j]). (19)

Using [u.sub.EVAj]([b.sub.j]) = 1 and [u.sub.EVIj]([b.sub.j]) = evi([d.sub.j]) in Equation (18) results in [u.sub.EVCj]([b.sub.j]) = evi([d.sub.j]). It follows that [u.sub.OD]([d.sub.j]) = evi([d.sub.j]) in Equation (19) and the optimal decision set is given by:

OD = {evi([d.sub.j])/[d.sub.j]}. (20)

That is, the relative merit index of [d.sub.j] is reduced to the expected value index of [d.sub.j]. Then, the optimal decision is the decision with the highest evi([d.sub.j]) or the highest expected value index.

Example

As an illustration of the applicability of the framework to decision making under risk and the relationship between the framework and decision theory, consider the decision to select a research and development project. Assume two possible projects ([d.sub.1] and [d.sub.2]) and two states of nature (success, [s.sub.1], and failure, [s.sub.2]) which are assumed to be well defined. Assume the probabilities are known to be 50% for the two states of nature. Finally, assume the payoffs resulting from [d.sub.1] and [d.sub.2] to be $100 and $50 if [s.sub.1] and $0 and $10 if [s.sub.2].

By using decision theory, the expected value of [d.sub.1] and [d.sub.2] are $50 and $30. Moreover, the expected value index for [d.sub.1] and [d.sub.2] are 1.0 and .60, respectively. Thus, the optimal decision is [d.sub.1] by both maximizing the expected value and maximizing the expected value index. The following decision table summarizes the problem and its solution.

States of Nature Actions [s.sub.1] s2 [d.sub.j] P([s.sub.1]) = .5 P(s2) = .5 [d.sub.1] 100 0 [d.sub.2] 50 10 Actions EV([d.sub.j]) evi([d.sub.j]) [d.sub.j] [d.sub.1] 50 * l * [d.sub.2] 30 .6

Using the proposed framework, the exact probability for the states of nature (.5) is defined by the set {(1/.50)}. Similarly, the exact payoffs of $100, $0, $50, and $10 can be defined by the fuzzy sets {(1/100)}, {(1/0)}, {(1/50)}, and {(1/10)}, respectively. Then, using Equation 11, the expected values of the two options are given by:

[EVA.sub.1] = "Exactly .5" * "Exactly 100" + "Exactly .5" * "Exactly 0" = {(1/.5)} * {(1/100)} + {(1/.5)} * {(1/0)} = {(1/50)}.

[EVA.sub.2] = "Exactly .5" * "Exactly 50" + "Exactly .5" * "Exactly 10" = {(1/.5)} * {(1/50)} + {(1/.5)} * {(1/10)} = {(1/30)}.

The set of all possible [b.sub.Lj] is given by EVA = {30, 50} and the maximum possible [bL.sub.j] = 50. Then, using c = 1 and Equation (12), sets [EVI.sub.j]'s are given by [EVI.sub.1] = {(1/50)} and [EVI.sub.2] = {(.6/30)}. Moreover, sets [EVC.sub.1] and [EVC.sub.2] are given by Equation (13) as: [EVC.sub.1] = {(1/50)} and [EVC.sub.2] = {(.6/30)}. Then, the optimal decision set OD is given by OD = {(1/[d.sub.1]), (.6/[d.sub.2])} and the relative merit index for [d.sub.1] and [d.sub.2] are 1 and .6 (which are identical to the expected value index for [d.sub.1] and [d.sub.2]). Thus, as obtained by decision theory, the optimal decision is [d.sub.1].

THE FRAMEWORK AND THE EFFECT OF AMBIGUITY ON DECISION MAKING

As noted earlier, the effect of ambiguity on decision making has been noted by accountants and other researchers. Moreover, empirical research dealing with Ellsberg's paradox showed that when faced with ambiguous decisions people display ambiguity avoidance or seeking behavior. This ambiguity avoidance and seeking behavior and, more generally, the impact of ambiguity on decisions are better understood using the following two problems.

The first problem involves four pairs of choices concerning four investments. The first two ([D.sub.1] and [D.sub.2]) are in State A and the second two ([D.sub.3] and [D.sub.4]) are in State B. The two states are considering changes to their environmental laws that may affect the payoffs from the four investments. The probabilities that A will change ([s.sub.A1]) and keep ([s.sub.A2]) the old environmental law are known to be 50%. The probabilities that B will change ([s.sub.B1]) and keep ([s.sub.B2]) the old environmental law are unknown. The payoffs for [D.sub.1] and [D.sub.2] are 100 and zero, if A changes the old law, and zero and 100, if A keeps the old law. The payoffs for [D.sub.3] and [D.sub.4] are 100 and zero, if B changes the old law, and zero and 100, if B keeps the old law. The four pairs of choices concerning the four investments are:

Pair 1: [D.sub.1] or [D.sub.2].

Pair 2: [D.sub.3] or [D.sub.4].

Pair 3: [D.sub.1] or [D.sub.3].

Pair 4: [D.sub.2] or [D.sub.4].

The above problem is similar to the problem used by Ellsberg (1961) to examine the effect of ambiguity on decision making. More specifically, in the case of the above choices, people are indifferent to the choice between [D.sub.1] and [D.sub.2] and between [D.sub.3] and [D.sub.4]. People, however, differ in their preferences concerning the last two pairs of choices. The majority of people prefer [D.sub.1] to [D.sub.3] and [D.sub.2] to [D.sub.4] (prefer investments with known probabilities over investments with unknown probabilities). In addition to this majority group of ambiguity averse individuals, a small number of people (called ambiguity seekers) prefer investments with unknown probabilities indicating ambiguity seeking behavior. Another small number of people (called ambiguity neutral) displays indifference to the ambiguous and nonambiguous options.

Both ambiguity avoidance and seeking behavior violate the axioms of decision theory, in particular, Savage's sure thing axiom and the additivity of probabilities principle. Consider, for example, the preferences of [D.sub.1] to [D.sub.3] and [D.sub.2] to [D.sub.4] (ambiguity avoidance) and the violation of the additivity of probabilities principle. The preference of [D.sub.1] to [D.sub.3] implies [Q.sub.A1] = .5 > [Q.sub.B1] and the preference of [D.sub.2] to [D.sub.4] implies [Q.sub.A2] = .5 > [Q.sub.B2]. The implication is that [Q.sub.B1] + [Q.sub.B2] < 1 (subadditivity). Alternatively, preferring [D.sub.1] to [D.sub.3] implies [Q.sub.A1] > [Q.sub.B1] = .5 and preferring [D.sub.2] to [D.sub.4] implies [Q.sub.A2] > [Q.sub.B2] = .5. The implication is that [Q.sub.A1] + [Q.sub.A2] > 1 (superadditivity).

The second problem involves a choice between two different tax treatments ([D.sub.1] and [D.sub.2]) of a possible tax deduction. The two tax treatments differ in their degree of ambiguity; the probability of being allowed by the IRS is known precisely to be .001 for [D.sub.1] and unknown for [D.sub.2]. The taxpayer gets $100 if the deduction is allowed by the IRS, otherwise nothing.

The above problem is similar to the example used by Ellsberg (reported by Becker and Brownson (1964, pp. 63-64, fn. 4)) to show that, when given a choice between options with unknown probability and options with known (but very small) probability, the majority of people would display ambiguity seeking behavior (prefer the ambiguous options). That is, in the case of the above problem, the majority of people would prefer the tax treatment with vague probability.

As noted earlier, Ellsberg's paradox has received wide empirical support in behavioral and expected utility literature. To examine the effect of ambiguity on accounting decisions and the pattern of choices suggested by Ellsberg's paradox and the above two problems, 25 subjects (graduate students) were asked to make the choices presented in the two problems. The choices made by the 25 subjects confirmed the patterns of choices suggested by Ellsberg's paradox and the above two problems. Regarding the first problem, all subjects were indifferent between D1 and D2 and between D3 and D4. However, the subjects differ in their preferences concerning the last two pairs of choices; 18 subjects (the majority) showed ambiguity avoidance (prefer [D.sub.1] over [D.sub.3] and [D.sub.2] over [D.sub.4]), 6 subjects displayed ambiguity seeking behavior (prefer [D.sub.3] over [D.sub.1] and [D.sub.4] over [D.sub.2]), and only 1 subject showed indifference between [D.sub.1] and [D.sub.3] and between [D.sub.2] and [D.sub.4]. Regarding the second problem, 24 subjects displayed ambiguity seeking behavior (prefer the tax treatment with unknown probability over the tax treatment with known but small probability) and only 1 subject showed indifference to the two tax treatments.

The framework can explain the effect of ambiguity on decision making and the pattern of choices suggested by the above two problems and Ellsberg's paradox. The first problem involves two options. For the first option, the probabilities are exact. That is, [Q.sub.A1] = [Q.sub.A2] = .50, which is defined, using the proposed model, by the fuzzy set {(1/.50)}. For the second option, the probabilities are inexact. Based on the amount and quality of the available information, the decision makers may rule out all but 3 probability distributions, (.2, .8), (.5, .5), (.8, .2), as possible distributions. The reliabilities or possibilities of these distributions may be estimated differently by different people. Using the scale [0,1], some decision makers (ambiguity neutral or individuals who display indifference to ambiguity) may judge the possibilities of the three distributions as .3, 1, .3, respectively. That is, those individuals may define QB1 by the fuzzy set {(.3/.2), (1/.5), (.3/.8)} and [Q.sub.B2] by the fuzzy set {(.3/.8), (1/.5), (.3/.2)}. In this case, using Equation (11), one can obtain sets [EVA.sub.j] as follows:

[EVA.sub.1] = {(1/.5)} * {(1/100)} + {(1/.5)} * {(1/0)} = {(1/50)}.

[EVA.sub.2] = {(.3/.2), (1/.5), (.3/.8)} * {(1/100)} + {(.3/.8), (1/.5), (.3/.8)} * {(1/0)}. = {(.3/20), (1/50), (.3/80)}.

Note that the max [b.sub.Lj] = 80. Then, using c = 1 and Equation (12), sets [EVI.sub.j] for the two decisions are given by [EVI.sub.1] = {(.625/50)} and [EVI.sub.2] = {(.25/20), (.625/50), (1/80)}. Then, sets [EVC.sub.j]'s are obtained by Equation (13) as: [EVC.sub.1] = {(.625/50)} and [EVC.sub.2] = {(.075/20), (.625/50), (.3/80)}. Finally, using Equation (14) one can obtain the optimal decision set OD as follows: OD = {(.625/[d.sub.1]), (.625/[d.sub.2])}. Thus, the decision maker is indifferent.

A second group of decision makers (ambiguity averse) may judge the possibilities of the three possible distributions as .3, .3, and .3. That is, those individuals may define [Q.sub.B1] by the fuzzy set {(.3/.2), (.3/.5), (.3/.8)} and [Q.sub.B2] by the fuzzy set {(.3/.8), (.3/.5), (.3/.2)}. In this case, the optimal decision set OD is given by OD = {(.625/[d.sub.1]), (.3/[d.sub.2])} and the decision maker would prefer decision one with exact probabilities. A third group of decision makers (ambiguity seeker) may estimate the possibilities of the three possible distributions as .8, .8, .8, respectively. That is, those individuals may define QB1 by the fuzzy set {(.8/.2), (.8/.5), (.8/.8)} and [Q.sub.B2] by the fuzzy set {(.8/.8), (.8/.5), (.8/.2)}. In this case, the optimal decision set OD is given by OD = {(.625/[d.sub.1]), (.8/[d.sub.2])} and the decision maker would prefer decision two with vague probabilities.

The second problem also involves two options. For the first option, the probabilities are exact. That is, [Q.sub.1] = .001 and [Q.sub.2] = .999, which can be defined by the fuzzy sets {(1/.001)} and {(1/.999)}, respectively. The probabilities for the second option are not exact and may be estimated differently by different people. Consider the ambiguity averse decision makers who may accept 3 distributions, (.2, .8), (.5, .5), (.8, .2), as possible distributions and estimate the possibilities of these distributions as .3, .3, .3, respectively. In this case, the optimal decision set is given by OD = {(.00125/[d.sub.1]), (.625/[d.sub.2])}. Thus, the decision maker may select [d.sub.2] (the ambiguous option). That is, even the ambiguity averse decision makers may select the ambiguous option when the exact probabilities are small.

SUMMARY, LIMITATIONS, AND RECOMMENDATIONS FOR FUTURE RESEARCH

The concern of this study is the problem of ambiguity in accounting and the use of decision theory as a general framework for accounting decision making. The study shows vague probability judgments, imprecise payoffs, fuzzy states of nature, and varying degrees of precision are characteristics of accounting decisions. The study also shows that decision theory fails to deal with the ambiguity in accounting and the failure to deal with ambiguity may lead to incomplete representation of accounting problems. Moreover, ambiguity should not be ignored for, as noted by accountants and other researchers, it affects decision making. The empirical research dealing with Ellsberg's paradox and the effect of ambiguity on decision making also casts doubt about the usefulness of the theory as a model for decision making under ambiguity.

Earlier ambiguity models extended decision theory to allow for only some aspects of the ambiguity in accounting. The study, thus, provides a general framework for decision making under ambiguity. The framework generalizes decision theory and expands previous ambiguity models to allow for the different sources of ambiguity in accounting and, therefore, it provides for more complete representation of accounting problems. Moreover, the model incorporates decision making under risk and decision theory as a special case. Finally, the model explains the behavioral evidence on Ellsberg's paradox and the impact of ambiguity on decision making.

Like much of decision theory literature, the framework assumes a single outcome for each of the different combinations of decisions and states of nature and requires the probabilities of the payoffs to be independent of which alternative is selected. Moreover, the framework assumes the relationship between decision variables to be stochastic (stochastic environment). Extensions of the framework that allow for multiple payoffs, dependent probabilities, and fuzzy environment represent avenues for future research. Empirical examination of the effect of ambiguity on accounting decisions represents another avenue for future research.

Appendix A

The Basic Elements of Fuzzy Set Theory

This appendix presents the basic elements of fuzzy set theory. The reader is referred to Zadeh (1965; 1968; 1972; 1973; 1975; 1976; 1978) and Kaufmann & Gupta (1985) for discussions of fuzzy set calculus; Bellman & Giertz (1973) and Gaines (1976) for discussions of the axioms of fuzzy set theory, and Dubois & Prade (1980) and Zimmermann (1990) for reviews of some of the theory's applications including applications to decision making.

The three basic operations of fuzzy sets are the intersection, union, and complementation, which represent the operators And and Or and the negation Not, respectively. Let E be a universal set and let A and B be two fuzzy subsets of E described by the membership functions [u.sub.A](x) and [u.sub.B](x). Then, the intersection (A[intersection]B), union (A[union]B) and complementation ([A.sub.c]) are given by the following membership functions:

[u.sub.A[intersection]B](x) = min [[u.sub.A](x), [u.sub.B](x)], (A-1)

[u.sub.A[union]B](x) = max [[u.sub.A](x), [u.sub.B](x)], (A-2)

[u.sub.A.sup.c] (x) = l - [u.sub.A](x). (A-3)

As an illustration, let E = {[x.sub.1], [x.sub.2], [x.sub.3], [x.sub.4]} be the set of audit procedures. Moreover, let A (set of efficient procedures) and B (set of inexpensive procedures) be two fuzzy subsets of E defined by A = {(.2/[x.sub.1]), (0/[x.sub.2]), (.8/[x.sub.3]), (l/[x.sub.4])} and B = {(.8/[x.sub.1]), (.5/[x.sub.2]), (.1/[x.sub.3]), (0/[x.sub.4])}. Then, the sets of efficient [and.bar] inexpensive procedures (A[intersection]B), efficient [or.bar] inexpensive procedures (A[union]B), and inefficient procedures ([A.sup.c]) are given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It should be noted that some of the properties of ordinary sets (commutativity, associativity, distributivity, idempotence, involution) and DeMorgan's Theorem hold in the case of fuzzy sets. However, the calculus of fuzzy set theory is different from that of ordinary set theory for, as can be verified by the above example, the intersection and union operations are not equivalent to the Boolean product and Boolean sum in the case of fuzzy sets. Moreover, the negation is not complementary in the case of fuzzy sets. That is, A[union][A.sup.c] [not equal to] E and A[intersection][A.sup.c] [not equal to] [empty set]. The remainder of the appendix presents some of fuzzy sets operations that will be used in the study.

Extension Principle: Extension principle is a principle that "allows the domain of the definition of a mapping or a relation to be extended from points in [the universe] to fuzzy subsets of [the universe]" (Zadeh, 1975a, 236]. The principle is called Extension Principle I when the mapping is an unary function and Extension Principle II when the mapping is an n-ary function.

Extension Principle I: Let f be a mapping from X to W such that w = f(x). Let A be a fuzzy set of X described by a membership function [u.sub.A](x). Then, the image of A under f is a fuzzy set of W given by: f(A) = {[u.sub.f(A)](w)/w}, where w = f(x) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (A-4)

Extension Principle II: Let f be a mapping from the Cartesian product X x Y to a space W such that w = f(x,y). Also, let A and B be fuzzy sets of X and Y, respectively. Finally, let [u.sub.A](x) and [u.sub.B](y) be the membership functions of sets A and B, respectively. Then, the image of (A,B) under f is a fuzzy set of W defined as: f(A,B) = {[u.sub.f(A,B)](w)/w}, where w = f(x,y), and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (A-5)

Extension principle II allows for arithmetic operations to be performed on fuzzy sets. For example, let LOW and HIGH OUTCOMES be defined by L = {(.5/100), (1/200), (.5/300)} and H = {(.5/700), (1/800), (.5/900)}. Then, by extension principle II, (L + H)/2 = {(.5/400), (.5/450), (1/500), (.5/550), (.5/600)}.

The operation of fuzzification: The operation of fuzzification transforms "a fuzzy (or a nonfuzzy) set A into an approximating set A which is more fuzzy than A" (Zadeh, 1972a, 17). That is, the operation of fuzzification provides a means for handling situations in which the elements of a set are fuzzy sets of another set. Let A be a fuzzy set of X with a membership function [u.sub.A](x). If each x in X is a fuzzy set of Y with a membership function [u.sub.x](y), then A can be expressed as a fuzzy set of Y as: A = {[u.sub.A](y)/y}, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (A-6)

As an example, let X = {[x.sub.1], [x.sub.2], [x.sub.3]} and Y = {[y.sub.1], [y.sub.2]}. Let A be a fuzzy subset of X defined by A = {([a.sub.1]/[x.sub.1]), ([a.sub.2]/[x.sub.2]), ([a.sub.3]/[x.sub.3])}. Let [x.sub.1], [x.sub.2], and [x.sub.3] be defined as fuzzy subsets of Y:

[x.sub.1] = {([b.sub.11]/[y.sub.1]), ([b.sub.12]/[y.sub.2])}.

[x.sub.2] = {([b.sub.21]/[y.sub.1]), ([b.sub.22]/[y.sub.2])}.

[x.sub.3] = {([b.sub.31]/[y.sub.1]), ([b.sub.32]/[y.sub.2])}.

Then, by the operation of fuzzification, one can express set A as a fuzzy subset of Y as follows:

A = {([c.sub.1]/[y.sub.1]), ([c.sub.2]/[y.sub.2])},

where,

[c.sub.1] = max {[b.sub.11][a.sub.1], [b.sub.21][a.sub.2], [b.sub.31][a.sub.1]}

[c.sub.2] = max {[b.sub.12][a.sub.1], [b.sub.22][a.sub.2], [b.sub.32][a.sub.1]}

The concept of maximizing set: The concept of a maximizing set for a set was introduced by Jain (1976) as an extension of the concept of a maximizing set for a function (Zadeh, 1972b) which represents an approximation to the concept of maximizing value of an objective function. According to Jain (1976, 700), "the maximizing set M(Y) of a set Y [is] a fuzzy set such that the grade of membership of a point y [member of] Y in M(Y) represents the degree to which y approximates to Sup Y in some specified sense." As an example, let Y = {100, 200, 300, 400}. Note that Sup Y = 400. Then, the maximizing set M(Y) of set Y is given by:

M(Y) = {(.25/100), (.5/200), (.75/300), (1/400)}.

REFERENCES

Alsina, C., & Trillas, E. (1987). Additive Homogeneity of Logical Connectives for Membership Functions. In Bezdek, J. (ed.), Analysis of Fuzzy Information, Volume I, Mathematics and Logic. Boca Raton, Florida: CRC Press, 179-183.

Anthony, R., Dearden, J., & Bedford, N. (1984). Management Control Systems. 5th ed. Homewood, IL.: Richard D. Irwin, Inc.

Ashton, R. (1982). Human Information Processing in Accounting. Sarasota, Florida: American Accounting Association.

Becker, S., & Brownson, F. (1964). What Price Ambiguity? Or The Role of Ambiguity in Decision-Making. Journal of Political Economy, 72, 62-73.

Bellman, R., & Giertz, M. (1973). On the Analytical Formalizm of The Theory of Fuzzy Sets. Information Sciences, 5, 149-156.

Bellman, R., & Zadeh, L. (1970). Decision-Making in a Fuzzy Environment. Management Science, 17, B141-B164.

Bernasconi, M., & Graham, L. (1992). Failures of the Reduction Principle in an Ellsberg-Type Problem. Theory and Decision, 32, 77-100.

Bierman, H., Jr., & Dyckman, T. (1971). Management Cost Accounting. New York: The MacMillan Company.

Black, M. (1963). Reasoning with Loose Concepts. Dialogue, 2:, 1-12.

Boritz, J., Gaber, B., & Lemon, W. (1987). Managing Audit Risk. CA Magazine, (January), 36-41.

Boucher, T., & Gogus, O. (2002). Reliability, Validity, and Imprecision in Fuzzy Multicriteria Decision-Making. IEEE Transactions on Systems, Man, and Cybernetics, Part C, Applications and Reviews, 32 (3), 190-202.

Chesley, G. (1985). Interpretation of Uncertainty Expressions. Contemporary Accounting Research, 2, 179-199.

Chew, S., Karni, E., & Safra, Z. (1987). Risk Aversion in the Theory of Expected Utility with Rank-Dependent Probabilities. Journal of Economic Theory, 42, 370-381.

Cooley, J., & Hicks, J., Jr. (1983). A Fuzzy Set Approach to Aggregating Internal Control Judgments. Management Science, 29:\, 317-334.

Curley, S., & Yates, J. (1985). The Center and Range of The Probability Interval As Factors Affecting Ambiguity Preferences. Organizational Behavior and Human Decision Processes, 36, 273-287.

de Campos, L., & Huete, J. (2001). Measurement of Possibility Distributions. International Journal of General Systems, 30 (3), 309-346.

Dolan, P., & Jones, M. (2004). Explaining attitudes Towards Ambiguity: An Experimental Test of The Comparative Ignorance Hypothesis. Scottish Journal of political Economy, 51, 281-301.

Du, N., & Budescu, D. (2005). The Effects of Imprecise Probabilities and Outcomes in Evaluating Investment Options. Management Science, 51(12), 1791-1803.

Dubois, D., & Prade, H. (1980). Fuzzy Sets and Systems: Theory and Application. New York: Academic Press.

Einhorn, H., & Hogarth, R. (1985). Ambiguity and Uncertainty in Probabilistic Inference. Psychological Review, 92, 433-461.

Einhorn, H., & Hogarth, R. (1986). Decision Making Under Ambiguity. Journal of Business, 59, S225-S250.

Ekenberg, L., & Thorbiornson, J. (2001). Second-Order Decision Analysis. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 9 (1), 13-38.

Ekenberg, L., Thorbiornson, J., & Baidya, T. (2005). Value Differences Using Second-Order Distributions. International Journal of Approximate Reasoning, 38, 81-97.

Ellsberg, D. (1961). Risk, Ambiguity, and The Savage Axioms. Quarterly Journal of Economics, 75, 643-669.

Fine, T. (1988). Lower Probability Models for Uncertainty and Nondeterministic Processes. Journal of Statistical Planning and Inference, 20, 389-411.

Freeling, A. (1980). Fuzzy Sets and Decision Analysis. IEEE Transactions on Systems, Man, and Cybernetics, SMC-10, 341-354.

Freeling, A. (1984). Possibilities Versus Fuzzy Probabilities--Two Alternative Aids. TIMS/Studies in the Management Sciences, 20, 67-81.

Gaines, B. (1976). Foundations of Fuzzy Reasoning. International Journal of Man-Machine Studies, 8, 623-668.

Gajdos T., Hayashi T., Tallon J.-M., & Vergnaud J.-C. (2008). Attitude Toward Imprecise Information. Journal of Economic Theory, 140, 27-65.

Gardenfors, P., & Sahlin, N-E. (1982). Unreliable Probabilities, Risk Taking, and Decision Making. Synthese, 53, 361-386.

Gardenfors, P., & Sahlin, N-E. (1983). Decision Making with Unreliable Probabilities. British Journal of Mathematical and Statistical Psychology, 36, 240-251.

Georgescu, I. (2009). Possibilistic Risk Aversion. Fuzzy Sets and Systems, 60, 2608-2619

Ghosh, D., & Ray, M. (1992). Risk Attitude, Ambiguity Intolerance and Decision Making: An Exploratory Investigation. Decision Sciences, 23, 431-444.

Ghosh, D., & Ray, M. (1997). Risk, Ambiguity, and Decision Choice: Some Additional Evidence. Decision Sciences, 28(1), 81-104.

Guess, A., Louwers, T., & Strawser, J. (2000). The Role of Ambiguity in Auditors' Determination of Budgeted Audit Hours. Behavioral Research in Accounting, 12, 119-138.

Hagan, J., de Korvin, A., & Siegel, P. (1996). Developing Decision Rules to Aid Tax Professionals in Ambiguous Planning Situations. Managerial Finance, 22, 1-17.

Hayashi, T., & Wada, R. (2010). Choice with Imprecise Information: An Experimental Approach. Theory and Decision, 69, 355-373.

Horngren, C., Datar, S., Foster, G., Rajan, M., & Ittner, C. (2009). Cost Accounting: A Managerial Emphasis. 13th ed. Upper Saddle River, New Jersey: Prentice-Hall.

Ijiri, Y., & Jaedicke, R. (1966). Reliability and Objectivity of Accounting Measurements. The Accounting Review, 41, 474-483.

Jain, R. (1976). Decision Making in the Presence of Fuzzy Variables. IEEE Transactions on Systems, Man, and Cybernetics, SMC-6, 698-703.

Jensen, R. (1976). Phantasmagoric Accounting: Research and Analysis of Economic, Social and Environmental Impact of Corporate Business. Sarasota, Florida: American Accounting Association.

Kahneman, D., & Tversky, A. (1979). Prospect Theory: An Analysis of Decision Under Risk. Econometrica, 47, 263-291.

Kaplan, R. (1975). The Significance and Investigation of Cost Variances: Survey and Extensions. Journal of Accounting Research, 13, 311-337.

Kaplan, R., & Atkinson, A. (1998). Advanced Management Accounting. Upper Saddle River, New Jersey: Prentice-Hall.

Kaufmann, A. & Gupta, M. (1985). An Introduction to Fuzzy Arithmetic. New York: Nosfrand Reinhold Co.

Kwak, W., Shi, Y., & Jung, K. (2003). Human Resource Allocation in a CPA Firm: A Fuzzy Set Approach. Review of Quantitative Finance and Accounting, 20, 277-290.

Kyburg, H., Jr. (1988). Higher Order Probabilities and Intervals. International Journal of Approximate Reasoning, 2, 195-209.

Larsson, S. (1976). Studies in Utility Theory. Unpublished Dissertation, University of British Columbia, Vancouver, Canada.

Larsson, S., & Chesley, G. (1986). An Analysis of the Auditor's Uncertainty About Probabilities. Contemporary Accounting Research, 2, 259-282.

Lenard, M., Alam, P., & Booth, D. (2000). An Analysis of Fuzzy Clustering and a Hybrid Model for the Auditor's Going-Concern Assessment. Decision Sciences, 31(4), 861-884.

Lilian, Y., & Yuan, Y. (1990). Dealing with Fuzziness in Cost-Volume-Profit Analysis. Accounting and Business Research, 20, 83-95.

MacCrimmon, K., & Larsson, S. (1979). Utility Theory: Axioms Versus 'Paradoxes'. In Allais M. & Hagen, O. (eds.), Expected Utility Hypotheses and The Allais Paradox. Boston: D. Reidel.

Machina, M. (1982). 'Expected Utility' Analysis without the Independence Axiom. Econometrica, 50, 277-323.

Maher, M. (1981). The Impact of Regulation on Controls: Firms' Response to the Foreign Corrupt Practices Act. The Accounting Review, 56, 751-770.

Marschak, J. et al. (1975). Personal Probabilities of Probabilities. Theory and Decision, 6, 121-153.

McEacharn, M., Zebda, A., & Calloway, J. (1995). A Fuzzy Expert System for Audit Materiality Judgments. Applications of Fuzzy Sets and the Theory of Evidence to Accounting, 135-157.

McIvor, R., McCloskey, A., Humphreys, P., & Maguire, L. (2004). Using a Fuzzy Approach to Support Financial Analysis in the Corporate Acquisition Process. Expert Systems with Applications, 27, 533-547.

Mukerji, S. (2000). A Survey of Some Applications of the Idea of Ambiguity Aversion in Economics. International Journal of Approximate Reasoning, 24, 221-234.

Narazaki, H., & Ralescu, A. (1994). An alternative Method for Inducing a Membership Function of a Category. International Journal of Approximate Reasoning, 11, 1-27.

Nelson, M., & Kinney, W., Jr. (1997). The Effect of Ambiguity on Loss Contingency Reporting Judgments. The Accounting Review, 22, 257-274.

Omer, K., Leavins, J., & Chandra, A. (1998). The Peer Review Process: A Fuzzy Decision Model. Applications of Fuzzy Sets and the Theory of Evidence to Accounting, 245-261.

Pan, Y., & Yuan, B. (1997). Bayesian Inference of Fuzzy Probabilities. International Journal of General Systems, 26 (1-2), 73-90.

Pathak, J., Vidyarthi, N., & Summers, S. (2005). A Fuzzy-Based Algorithm for Auditors to Detect Elements of Fraud in Settled Insurance Claims. Managerial Auditing Journal, 20 (6), 632-644.

Reventos, V. (1999). Interpreting Membership Functions: A Constructive Approach. International Journal of Approximate Reasoning, 20, 191-207.

Ro, Ryung. (1982). An Analytical Approach to Accounting Materiality. Journal of Business Finance and Accounting, 9, 397-412.

Saaty, T. (1974). Measuring the Fuzziness of Sets. Journal of Cybernetics, 4, 53-61.

Savage, L. (1954). The Foundations of Statistics. New York: John Wiley & Sons.

Schultz, J., & Reckers, P. (1981). The Impact of Group Processing on Selected Audit Disclosure Decisions. Journal of Accounting Research, 19, 482-501.

Shafer, G. (1976). A Mathematical Theory of Evidence. Princeton, N.J.: Princeton University Press.

Spicer, M., & Thomas, J. (1982). Audit Probabilities and the Tax Evasion Decision: An Experimental Approach. Journal of Economic Psychology, 2, 241-245.

Stephens, R., Dillard, J., & Dennis, D. (1985). Implications of Formal Grammars for Accounting Policy Development. Journal of Accounting and Public Policy, 4, 123-148.

Tanaka, H., Okuda, T., & Asai, K. (1976). A Formulation of Fuzzy Decision Problems and Its Application to An Investment Problem. Kybernetes, 5, 25-30.

Thole, U., Zimmermann, H., & Zysno, P. (1979). On the Suitability of Minimum and Product Operations for the Intersection of Sets. Fuzzy Sets and Systems, 2, 167-180.

Toth, H. (1992). Probabilities and Fuzzy Events: An Operational Approach. Fuzzy Sets and Systems, 48, 113-127.

Utkin, L. (2003). Imprecise Second-Order Hierarchical Uncertainty Model. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 11 (3), 301-317.

von Neumann, J., & Morgenstern, O. (1944). The Theory of Games and Economic Behavior. Princeton, N.J.: Princeton University Press.

Walley, P., & de Cooman, G. (2001). A Behavioral Model for Linguistic Uncertainty. Information Sciences, 134, 1-37.

Wallin, D. (January 1992). Legal Recource and the Demand for Auditing. The Accounting Review, 67 121-147.

Wallsten, T., Budescu, D., Rapoport, A., Zwick, R., & Forsyth, B. (1986). Measuring the Vague Meanings of Probability Terms. Journal of Experimental Psychology: General, 115, 348-365.

Watson, S., Weiss, J., & Donnell, J. (1979). Fuzzy Decision Analysis. IEEE Transactions on Systems, Man, and Cybernetics, SMC-9, 1-9.

Weber, S. (1979). A General Concept of Fuzzy Connectives, Negations and Implications Based on t-Norms and t-Conorms. Fuzzy Sets and Systems, 2, 115-134.

Weichselberger, K. (2000). The Theory of Interval-Probability as a Unifying Concept of Uncertainty. International Journal of Approximate Reasoning, 24 (2-3), 149-170.

Witteman, C., & Renooij, S. (2003). Evaluation of a Verbal-Numerical Probability Scale. International Journal of Approximate Reasoning, 33, 117-131.

Wright, M., & Davidson, R. (2000). The Effect of Auditor Attestation and Tolerance for Ambiguity on Commercial Lending Decisions. Auditing: A Journal of Practice & Theory, 19, 67-81.

Yager, R. (1979). Possibilistic Decisions. IEEE Transactions on Systems, Man, and Cybernetics, SMC-9, 338-342.

Yager, R. (1980). On a General Class of Fuzzy Connectives. Fuzzy Sets and Systems, 4, 235-242.

Yager, R. (1987). Optimal Alternative Selection in the Face of Evidential Knowledge. In Kacprzyk, J. and Orlovski, S. (eds.), Optimization Models Using Fuzzy Sets and Possibility Theory. Dordrecht, Holland: D. Reidel Publishing Company, 123-140.

Yates, J., & Zukowski, L. (1976). The Characterization of Ambiguity in Decision Making. Behavioral Science, 21, 19-25.

Zadeh, L. (1965). Fuzzy Sets. Information and Control, 8, 338-353.

Zadeh, L. (1968). Probability Measures of Fuzzy Events. Journal of Mathematical Analysis and Applications, 23, 421-427.

Zadeh, L. (1972a). A Fuzzy Set Theoretic Interpretation of Linguistic Hedges. Journal of Cybernetics, 2, 4-34.

Zadeh, L. (1972b). On Fuzzy Algorithm. Electronics Research Laboratory, Memo # ERL-M235, University of California, Berkeley.

Zadeh, L. (1973). Outline of a New Approach to the Analysis of Complex Systems and Decision Processes. IEEE Transactions on Systems, Man, and Cybernetics, SMC-3, 28-44.

Zadeh, L. (1975a). The Concept of a Linguistic Variable and Its Application to Approximate Reasoning, Part I. Information Sciences, 8, 199-249.

Zadeh, L. (1975b). The Concept of a Linguistic Variable and Its Application to Approximate Reasoning, Part II. Information Sciences, 8, 301-357.

Zadeh, L. (1975c). The Concept of a Linguistic Variable and Its Application to Approximate Reasoning, Part III. Information Sciences, 9, 43-80.

Zadeh, L. (1975d). Fuzzy Logic and Approximate Reasoning. Synthese, 30, 407-428.

Zadeh, L. (1976). The Linguistic Approach and Its Application to Decision Analysis. In Ho, Y. and Mitter, S. (eds.), Directions in Large-Scale Systems. New York: Plenum Press, 339-370.

Zadeh, L. (1978). Fuzzy Sets as a Basis for a Theory of Possibility. Fuzzy Sets and Systems, 1, 3-28.

Zadeh, L. (2002). Toward a Perception-Based Theory of Probabilistic Reasoning with Imprecise Probabilities. Journal of Statistical Planning and Inference, 105, 233-264.

Zebda, A. (1984). The Investigation of Cost Variances: A Fuzzy Set Theory Approach. Decision Sciences, 15, 359-389.

Zimmer, A. (1984). A Model for the Interpretation of Verbal Predictions. International Journal of Man-Machine Studies, 20, 121-134.

Zimmerman, H. (1990). Fuzzy Set Theory and Its Applications. 2nd ed. Dordrecht, Holland: D. Reidel Publishing Company.

Zimmermann, H. & Zysno, P. (1983). Decisions and Evaluations by Hierarchical Aggregation of Information. Fuzzy Sets and Systems, 10, 243-260.

Awni Zebda

Texas A&M University-Corpus Christi

Awni Zebda is a Regents Professor and Professor of Accounting at Texas A&M University-Corpus Christi. He received his Ph.D. from Virginia Tech. His research interests are in the areas of fuzzy set theory and the problem of ambiguity in accounting, decision theory, and behavioral research. His prior work has been published in Decision Sciences, Behavioral Research in Accounting, Journal of Accounting Literature, International Journal of Business, Accounting, and Finance, Advances in Accounting, Accounting and Business Research, Managerial Finance, and Applications of Fuzzy Sets and the Theory of Evidence to Accounting. Prior to his appointment at Texas A&M University-Corpus Christi, Dr. Zebda taught at University of Kansas, Louisiana Tech University, and The University of Alabama. Recently, Dr. Zebda has received the Minnie Stevens Piper Professor Award and the Texas Society of Certified Public Accountants' Outstanding Accounting Educator Award.

Printer friendly Cite/link Email Feedback | |

Author: | Zebda, Awni |
---|---|

Publication: | International Journal of Business, Accounting and Finance (IJBAF) |

Geographic Code: | 7IRAN |

Date: | Sep 22, 2011 |

Words: | 15137 |

Previous Article: | Preface. |

Next Article: | Determinants of saving in Lebanon: 1980-2009. |

Topics: |