A Fuzzy Risk Attitude Classification Based on Prospect Theory

Yang Li Department of Automation Science and Electrical

Engineering, Beihang University

Beijing, China e-mail: jamesscottliyang@gmail.com

Edy Portmann Department of Electrical Engineering and Computer

Science, University of California at Berkeley

Berkeley, CA, USA e-mail: portmann@eecs.berkeley.edu

Abstract—Traditional methods do not actually measure peoples’ risk attitude naturally and precisely. Therefore, a fuzzy risk atti-tude classification method is developed. Since the prospect theory is usually considered as an effective model of decision making, the personalized parameters in prospect theory are firstly fuzzified to distinguish people with different risk atti-tudes, and then a fuzzy classification database schema is ap-plied to calculate the exact value of risk value attitude and risk behavior attitude. Finally, by applying a two-hierarchical classification model, the precise value of synthetical risk at-titude can be acquired.

Keywords—Prospect Theory Application, Fuzzy Classification, Risk Attitude, Parameters.

I. INTRODUCTION

The level of risk attitude is commonly used in various fields of human decision making, including wealth ac-cumulation, human capital investment, portfolio alloca-tion, insurance, as well as policy decisions [1]. There are several methods to measure individual’s risk attitude. Let c(x, p) be the subjective money equivalent to the simple prospect (x, p)-receiving $x with a probability of p or nothing with probability of (1-p). According to the tra-ditional method, a person is said to be risk averse if c/x<p, risk neutral if c/x=p and risk seeking if c/x>p. However, the transition of this method from one group to another is too sharp and unnatural, which may cause the problems of overestimation and underestimation and then lead to an inappropriate decision making and unfair treatment.

The interaction of value and weighting function in prospect theory will neatly resolve this problem [2]. Parameter α, for example, can reflect the relative position between the curve c/x=g(p) (g reflects the relationship between c/x and p) and the straight line c/x=p, which manifests the extent of risk attitude. Nevertheless, this method fails to quantify the risk attitude, which limits a more precise estimation of people’s risk attitudes.

Fuzziness can be found in a large number of areas of daily lives from engineering to medicine. In front of fuzzy situations, precise mathematical expressions will lose its gloss since real world never lack ambiguity and vagueness,

which cannot be described by formulas. E.g., the reason why traditional mathematical method fails to determine whether the letter y should belong to the vowel or the consonant is because the two clusters do not ac-tually lead to mutually exclusive classifications at all. Similarly, human reasoning is ambiguous and uncertain and people should not be restricted into one group as well. Therefore, an optimal method to handle people’s risk attitude is to introduce the fuzzy classification theory. Unlike the traditional sharp classification, which dis-tributes people in exact one class or another, fuzzy clas-sification allows people to belong to different classes at the same time, with different membership degrees to each class [3]. This method avoids the unnatural transi-tion in traditional classification and makes possible the quantification of the vagueness of human reasoning.

In this paper, through the fuzzification to the pa-rameters in prospect theory, a two-hierarchy fuzzy classification model is developed to describe the risk attitude of people. The risk attitude is separated into two parts: Risk Value Attitude (RVA), which represents people’s value in face of risky assets or choices, and Risk Behavior Attitude (RBA), which reflects the behavioral incentive of certain risky choices to people.

In section 2, a brief review of prospect theory is in-troduced, and in section 3, the meaning of the parameters in prospect theory is illustrated. Section 4 contains a method to fuzzy the parameters and then applies them into our prototype of risk attitude model. At last, in sec-tion5, the main ideas of this paper are briefly concluded and an outlook to the future work is given.

II. PROSPECT THEORY

For many years, economists and theorists are persistently striving for an optimal method to describe the risk attitude of people. The original attempt of decision theories is the probability theory, which asserted that people ought to choose the option that offers the highest expected value. Yet this method fails to explain why people would prefer $49 over a 50-50 chance of acquiring $100 or nothing [2]. The expected utility theory can explain this contradiction, in which a concave utility function of people is justified, and it believes that people should always

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maximize the expected utility. However, expected utility theory is still called into question when the powerful challenge comes as “Allais paradox” [4, 5].

Perhaps the leading and most widely used behavior model of decision making under risk is the prospect theory [9, 10]. According to prospect theory, the subjective value V of a simple prospect (x, p) is given as follows:

( , ) ( ) ( )V x p v x pω= (1)

where v is the value function that measures the subjective value of money amount x for gains or losses, and ω, the probability weighting function, illustrates the subjective importance of the prospect with probability p.

A. Value Function

The value function in prospect theory inherits the basic idea of utility function, but it replaces the states of wealth on the horizontal axis with pure gains or losses relative to a reference point-status quo. It is assumed to be concave for gains and convex for losses as shown in Fig.1a, which reflects the diminishing margin sensitivity of people, and steeper for losses than for gains, exhibiting the loss aversion [6]. This special shape contributes to the risk aversion attitude of gains and risk seeking attitude of losses.

Up to now the most popular paramerization of value function is Kahneman and Tversky’s power function:

( )( )

xv x

x

α

βλ=

− −

0

0

x

x

≥<

(2)

α and β measure the curvature of subjective value function when people are dealing with pure gains and losses and λ>1 is the coefficient of loss aversion.

B. Loss Aversion

In value function, the curve is steeper for losses than gains – a property known as loss aversion. Loss aversion manifests that people will usually generate more negative emotion for losses than gains when the amount of gains and losses are the same. Kahneman and Tversky [7] defines loss aversion as –v(–$1)/v($1) (–v($1)>v($1)), while Wakker and Tversky [8] expresses loss aversion as v’(–x)/v’(x), meaning that the slope of value function for any amount of money loss is larger than the corresponding amount of gain.

Fig.1. (a) Value function (b) Probability weighting function

There are many manifestations of loss aversion. The phenomenon that the minimum amount of money that a person is Willing To Pay (WTP) is lower than the minimum amount of money that he Willing To Accept (WTA), for example, is thought to relate to loss aversion. What’s more, loss aversion is believed to be responsible for the trade aversion, an inertial tendency to stick with status quo options [9].

C. Weighting Function

Value function in prospect theory is weighted not by probability p, but instead by a probability weighting function ω(p). Weighting function reflects the impact or subjective importance of a monetary event in people’s mind. In probability theory, there are two natural extreme: impossibility and certainty. Similar to value function, weighting function captures the diminishing sensitivity as well, gradually flattening from the two extreme to the middle, which gives rise to the inverse-S shaped curve as shown in Fig.1b.

Several functional forms have been designed to express the weighting function. Prelec’s model [10,11] is shown in formula (3), which complies with three main principles: overweighting of low probabilities and underweighting of high probabilities; sub-proportionality of decision weights; and sub-additivity of decision weights.

( ) exp[ ( ln ) ]p p γω δ= − − (3)

This function implies a weighting function that crosses the identity at 1/e. δ measures the elevation of the weighting function and γ>0 measures its degree of curvature. The function is more curved as the decrease of γ, exhibiting a more rapidly diminishing sensitivity to probabilities near 0 and 1.

III. PARAMETERIZATION

Traditional prospect theory literatures depend more on the patterns of value and weighting function with fixed variables α, β, λ and γ to evaluate people’s risk attitude. But these personalized parameters have already offered a relatively valid and complete resource to assess individual risk attitudes and expected decision attitudes under risk. E.g., Tversky and Kahneman [7] estimates median values of α=0.88, β=0.88 and λ=2.25 through a sample survey of 25 graduate students, while in a study with 420 undergraduate students, Wu and Gonzalez [12] fixes that α=0.49, γ+=0.68, etc. The difference of these parameters provides us a justification to apply these parameters into the estimation of people’s risk attitude.

A. Risk Value Attitude (RVA)

The parameters α and β in power function (2), both with a range of value (0, 1), justify a concave shape of value function for gains and a convex shape for losses. Fig.2 shows the value functions with different values of α=0.9, 0.7, 0.4, 0.1. Suppose that the four values of α represent

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four people (Mr. A, B, C and D), with varying risk attitudes, respectively. When facing a same amount of money, say, $100, it is quite obvious that the subjective value of the four, i.e., vA(100), vB (100), vC(100) and vD(100), are significantly different. Moreover, such inconsistency will be more remarkable as the increase of gains. The value function of Mr. D (α=0.1) is the farthest from v(x)=x, which means Mr. D holds the strongest risk aversion attitude when facing a $100 risky asset, while Mr. A (α=0.9) expresses a comparatively moderate risk aversion, whose value function is the nearest to v(x) =x of the four.

From another perspective, the diminishing margin sensitivity of value function reflects a diminishing subjective incentive of pursuing another money amount x0

when one has already gained x0. Formula (4) measures the descend rate of such incentive of receiving an extra x0:

0 0

0

(2 )2 1

0

x x

x

α αα

α

−= −

− (4)

With the decrease of α, 2 1α − also decreases, which means the ratio of the additional incentive generated from the extra x0 to the initial incentive acquired from the former x0 is dropping. Such situation implies that people with a lower α will be easier to content with status quo and express a reluctance to take risk, which can be interpreted as a higher level of risk aversion.

In summary, it can be clearly concluded from the analysis that a smaller α represents a higher level of risk aversion while a larger α represents a lower level of risk aversion. In the same way, although β is considered as a reflection of risk seeking attitude, it shares the same principle and analysis methods with α, suggesting an inverse proportion relationship with people’s risk seeking attitudes. In order to assess the contribution of α and β on people’s risk attitude, we naturally propose the concept of RVA to combine the two parameters. A relatively large α and small β reveals a RVA of risk seeking, while a relatively small α and large β embodies a RVA of risk averse.

B. Risk Behavior Attitude (RBA)

One of the manifestations about loss aversion is that it is thought to be consistent with the inertia to stick with status quo, known as status quo bias [9] and the unwillingness to trade [13]. People usually tend to weight loss more than weight gain, so if status quo is the reference point, we would be biased in favor of status quo rather than stepping forward. Such an attribute that reveals people’s trade aversion can be measured by the parameter λ in the value function (λ>1). Considering loss aversion is not a value that only related to a specific amount of gain or loss, but a kind of personalized behavioral inertia that is suitable for all the gains and losses, we treat it as a standard to appraise the Risk Behavior Attitude (RBA) of people. With the increase of λ, people will show an increasing negative emotion to trade, i.e., increasing trade aversion, which is just another kind of performance of risk aversion.

Fig.2. Different value functions with different parameter α.

Fig.3. Different weighting functions with different parameter γ.

Another parameter that is assumed to be responsible for RBA is γ, a parameter that reflect the curvature of the weighting function. Prelec’s single–parameter weighting function with different γ are plotted in Fig.3, in which the curve stands closer to ω(p)=p when γ approaches to 1, reflecting a diminishing degree of underweighting large probability and overweighting small probability. Suppose that the linear function ω(p)=p is the most indifferent weighting attitude toward risk, with which a possible outcome is weighted by its probability, then a much smaller γ manifests an individual with a higher level of risk aversion and sensitivity, and a larger γ reflects a lower level of risk aversion and sensitivity.

As above, the combination of the two parameters, λ and γ, provides us a method to assess people’s RBA. A person with a pretty large λ and small γ shows a relatively strong aversion to make changes as well as an excessive response to risk - excessively overweighting low probability and underweighting high probability. This suggests a RBA of strong risk aversion. Contrarily, a person with smaller λ and larger γ will be more likely to show a RBA of moderate risk aversion.

C. Elicitation of Parameters

The methods to elicit the parameters can be broadly classified into four categories: statistical methods; non-parametric methods; semi-parametric methods; and parametric methods. In comparison, the parametric estimation of parameters has several advantages over the others such as less time expense, more convenience and higher reliability.

Craig et al. [2] mentioned a simple parametric method over traditional one, assuming the power function and

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Prelec’s single–parameter weighting function. By eliciting a number of certainty equivalents cij for prospects that pays $xi with probability pj, we can get the following functions after several transformations of prospect theory:

ln[ ln( / )] ln( ) [ ln( ln )]

ln[ ln( / )] ln( ) [ ln( ln )]

ij i j

ij i j

c x p

c x p

α γ

β γ

+

−

− − = + − −

− − = + − − (5)

This equation evidently leads to a linear regression to get exact parameters α, β, γ+ and γ-.

IV. PROTOTYPE

This section firstly introduces a method to fuzzify the parameters in prospect theory with fuzzy classification database schema. Then people are applied into a two-hierarchical classification model, which attempts to group them into different risk attitude classes according to their performance. Finally, a comparison between this proposed model and the traditional risk assessment method is shown.

A. Fuzzification of Parameters

Fuzzification is a practical approach because it allows to quantify the parameters for fuzzy classification use.

1) Fuzzy Classification Database Schema The relational database schema [14] displays data in

two-dimensional tables and determines the attributes in the table columns, which makes the data easier to interpret. A proposed classification approach extends the relational database schema by a context model proposed by Chen [15], assigning each attribute a domain D and a descriptive context C.

In order to better fuzzy classification use and the determination of the membership degrees in different classes, Wero [3] extends the context model with a fuzzy classification database schema. In fuzzy classification schema, each attribute is considered as a linguistic variable and verbal terms are assigned to each equivalence class. By definition, the pattern of a fuzzy classification schema is like R(A,C,X,T), which contains a set of attributes A, a set of associated contexts C, a set of linguistic variables X and a set of corresponding terms T. Consider the attribute α as a linguistic variable, and the definition range of α is D (α). We give the linguistic variable α two equivalent terms, say, T (α) = {high risk averse, low risk averse}, describing two equivalence classes (0, 0.5) and (0.5, 1) (see Fig.5). Besides, every term of a linguistic variable represents a fuzzy set that is determined by a membership function μ over the domain of the corresponding attribute [16]. In the same way, we can also execute the procedure to β, λ and γ, then we get each fuzzy classification schema of these parameters (see Fig.4).

2) Fuzzy Classification According to the analysis in Section3, the combination

of α and β is a measure of RVA, while the combination of λ and γ can be used to describe RBA. Empirically, several if-then rules with the terms of linguistic variables are shown as follows, based on which we are able to assign a person into

several fuzzy classes from C1 to C8 and calculate the degrees of RVA and RBA according to his or her personalized parameters (see Fig.5a, 5b).

RVA:

If high risk averse for gains & low risk seeking for losses, then C1, labeled RISK AVERSE for RVA.

If low risk averse for gains & low risk seeking for losses, then C2, labeled CONSERVATIVE for RVA.

If high risk averse for gains & high risk seeking for losses, then C3, labeled RADICAL for RVA.

If low risk averse for gains & high risk seeking for losses, then C4, labeled RISK SEEKING for RVA.

RBA:

If high trade averse & high sensitivity, then C5, labeled RISK AVERSE for RBA.

If low trade averse & high sensitivity, then C6.

If high trade averse & low sensitivity, then C7.

If low trade averse & low sensitivity, then C8, labeled RISK SEEKING for RBA.

B. Aggregation Operator and Values of RVA and RBA

In this subsection, a method is chosen to aggregate the membership degrees of parameters and to calculate the values of RVA and RBA based on the aggregation.

1) Aggregation of Attributes Since every class Ci (i=1,2,...,8) is defined by the terms of

linguistic variables, the membership degree M(Ok| Ci) of an object Ok can be calculated by an aggregation of the membership degrees over all the terms which define the class Ci. The class C1 for instance is labeled by the terms “high risk averse for gains” and “low risk seeking for losses” of the linguistic variables α and β, then M(Ok| C1) should be the aggregation of the membership functions, μhigh

risk averse and μlow risk seeking.

Several operators have been developed to calculate the aggregation of membership values, but the γ-operator was empirically tested and proven to be the most approximate to human decision making in real world [17]. The γ-operator of m fuzzy sets A1... Am defined over a reference set U with membership functions μ1... μm is defined as

(1 )

1 1

( ) ( ( )) (1 (1 ( )))

(0,1) and

m m

aggregation i ii i

x x x

x U

γ γμ μ μ

γ

−

= =

= − −

∈ ∈

∏ ∏ (6)

According to γ-operator, the membership degree of Mr. X (Fig.6a) in the class C1, i.e., M (X|C1) is given as (using γ-value=0.5): M (X|C1) = ((0.6*0.75) ^0.5)*(1-(1-0.6) (1-0.75)) ^ 0.5 =0.63.

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Fig.4. Fuzzy classification database schemas for each parameter in prospect

theory. The terms of α are labeled {high risk averse, low risk averse}, while β - {high risk seeking, low risk seeking}, λ-{high trade aversion, low trade aversion} and γ- {high sensitivity, low sensitivity}.

In order to maintain that the sum of every membership degrees of an object to all classes is exactly equivalent to 1, all the membership degrees that are calculated from the γ-operator should be normalized depending on the following formula:

1

( / ) ( / ) / ( / )n

real k i k i k jj

M O C M O C M O C=

= (7)

Fig.5. Fuzzy classification for RVA (a) and RBA (b).

Therefore, the real membership degrees of Mr. X to the four classes can be calculated as Mreal(X|C1) =0.372, Mreal(X|C2)=0.298, Mreal (X|C3) =0.191 and Mreal(X|C4)=0.139. The normalization takes place in order

to derive the final membership degrees in each class, and meanwhile it also provides us a precise method to gauge the real extent of people’s risk value or behavior attitude.

2) Values of RVA & RBA Although the membership degrees to every fuzzy class can be calculated from the method mentioned in 4.2.1, people usually want to figure out the values of RVA and RBA to precisely evaluate a person’s risk attitude. The acquisition of the values of RVA and RBA can be achieved with the help of a similar method that is used to calculate the personalized value of online customers by Werro [3]. Since RVA and RBA can be interpreted as personalized values as well, according to Werro, the value of RVA and RBA of the object Ok are given in (9,10):

4

1

( ) ( / ) ( )RVA k real k i ii

V O M O C gr C=

= (9)

8

5

( ) ( / ) ( )RBA k real k i ii

V O M O C gr C=

= (10)

where gr(Ci) is a given concept, representing the specific extent of importance of the class Ci to RVA or RBA. By supposing that gr(C1)=10, gr(C2)=7, gr(C3)=4 and gr(C4)=1, the value of risk value attitude of the person X can be computed as: VRVA(X)=10*0.372+7*0.298+ 4*0.191+1*0.139=6.71.

This result is relatively reasonable because the nearly centered location in Fig.5a reveals a moderate attitude toward risk, which ought to be assigned a medium RVA value of 6.71 (max 10).

C. Two-Hierarchical Classification Model

In order to provide practical and profitable financial advices and to better allocate personal portfolio, it is necessary for individuals and financial planners to precisely estimate personal risk attitude as a whole. As discussed in former sections, the risk attitude of people encompasses all the psychological and behavioral contributions, i.e., risk value attitude and risk behavior attitude. These two contributions can further be separated into α and β, λ and γ, respectively. Since the combination of these four parameters in just one fuzzy classification would make no sense, a method has been introduced above to determine the value of RVA (VRVA) from α and β and RBA (VRBA) from λ and γ. Then the remaining problem is to aggregate VRVA and VRBA in a new model for the estimation of a synthetical risk attitude. By implementing a two-hierarchical classification model (Fig.6), such an aggregation can be clearly achieved.

In a two-hierarchy model, the VRVA and VRBA that have been calculated in lower hierarchy are considered as two attributes in upper hierarchy. Moreover, the risk attitudes of customers in financial markets typically fall into five categories: conservative, moderately conservative, moderate, moderately aggressive and aggressive. For classification purpose, four of them are extracted as four classes in the upper hierarchy. Thus, based on the proposed fuzzy classification method, both the membership degrees of a customer in each class and the value of risk attitude can be

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exactly calculated. The two-hierarchical model bridges our theoretical method with the real world financial activities. But it meanwhile transcends it by giving different membership degrees to different risk attitude classes, which is quite conducive to a much more precise estimation.

D. Development of Membership function

The field of developing membership functions is rapidly blossoming in recent years, and many methods have been proposed for better estimations. Broadly speaking, six mainstream methods are most commonly used in real life [18] including Intuition, Inference, Rank Ordering, Neural Networks, Genetic Algorithms and Inductive Reasoning.

The Genetic Algorithms (GA) application of membership function [19] is relatively suitable for determining our risk attributes, i.e. α, β, λ and γ, because it translates the theory of evolution into algorithms to search for solutions in a more ‘natural’ way. More importantly, GA method can reduce the workload of people by distributing the large amount of computation work to computers, and can also improve the accuracy of estimation through choosing the optimal chromosome with the highest fitness under given limitations.

Fig.6. Two-hierarchical classification model.

E. Test of Model

To test the validity of the two-hierarchical model, a comparison is made between this model and a traditional risk attitude assessment used by commercial banks. In the survey, 24 respondents are randomly chosen from the teachers and students in Beihang University, and each of them is given two questionnaires: one with 10 traditional risk assessment questions and the other with several specifically designed prospect questions. Through calculations to their answers, the percentages of respondents in each risk attitude class are shown in Table 1. It is can be seen that although the overall results of the two methods are similar to each other, the two-hierarchical method actually sets no class boundaries and is able to provide more accurate risk attitude values.

V. CONCLUSION AND OUTLOOKS

In this paper we have tried to show how to go beyond the traditional classification methods of risk attitude for a more natural estimation of people’s decision making under risk. A two-hierarchical classification model based on the parameters in prospect theory turns out to be an adequate tool. With this model, a person can be assigned to several risk attitude classes that are currently used in financial activities at the same time, holding different membership degrees over these classes, rather than only belonging to one sharp class or another. After that, a synthetical risk attitude value can be calculated to provide a more precise index of risk attitude to individuals, financial planners and policy makers.

This paper is considered as a preliminary attempt of the fuzzy theory application to people’s risk attitude. In our future works, the combination modes of the four parameters will be further discussed, the extent of every class gr(C) will be determined and the membership function of each attribute will be estimated with GA.

REFERENCES [1] Sherman D. Hanna, Michael S. Gutter and Jessie X. Fan (2001). A

Measure of Risk Tolerance Based On Economic Theory, Association for Financial Counseling and Planning Education.

[2] Craig R. Fox and Russel A. Poldrack (2009), Prospect Theory and the Brain, Neuroeconomics: Decision Making and the Brain, Elsevier Inc., 145-173.

[3] Nicolas Werro (2008). Fuzzy Classification of Online Customers. Received by the Faculty of Economics and Social Sciences at the University of Fribourg, Switzerland, 27, 41-43, 86-92.

[4] Allais, M. (1953). Le comportement de l’homme rationel devant lerisque, critique des postulates et axioms de l’ecole americaine. Economica 21, 503-546.

[5] Allais, M. and Hagen, O. (1979). The so-called Allais paradox and rational decisions under uncertainty. In: O.H.M. Allais (ed.). Expected Utility Hypothesis and the Allais Paradox. Dordrecht: Reidel Publishing Company, pp.434-698.

[6] Kahneman, D. and Tversky, A. (1979). Prospect theory: an analysis of decision under risk. Economica 4, 263-291.

[7] Tversky, A. and Kahneman, D. (1992). Advances in prospect theory – cumulative representation of uncertainty. J. Risk Uncertainty, 5, 297-323.

[8] Wakker, P. and Tversky, A. (1993). An axiomatization of cumulative prospect theory. J. Risk Uncertainty 7, 147-176.

[9] Samuelson, W. and Zeckhauser, R. (1988). Status que bias in decision making. J. Econ. Behav. Org. 1, 7-59.

[10] Prelec, D. (1988). The probability weighting function. Economica 66, 497-527.

[11] Prelec, D. (2000). Compound invariant weighting function in prospect theory. In: D. Kahneman and A. Tversky (eds), Choices, Values, and Frames. Cambridge: Cambridge University Press, pp. 67-92.

[12] Wu, G. and Gonzalez, R. (1996). Curvature of the probability weighting function. Management Sci. 42, 1676-1690.

[13] Knetsch, J.L. (1989). The endowment effect and the evidence of non-reversible indifference curves. Economic Rev. 79, 1277-1284.

[14] Codd, E.F (1970). A Relational Model of Data for Large Shared Data Banks. Communications of the ACM, 13(6): 377-387.

[15] Chen, G. (1998). Fuzzy Logic in Data Modeling – Semantics, Constraints and Database Design, Kluwer Academic Publishers, London.

[16] Zadeh, L.A. (1975). The concept of a linguistic variable and its application to approximate reasoning-1. Information Science, Volume 8, issue 3, 199-249.

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[17] Zimmermann, H.-J. and Zysno, P. (1980). Latent connectives in human decision making, Fuzzy sets and systems, 4:37-51.

[18] Ross, T.J. (2010). Fuzzy logic with engineering applications, 3rd edition, John Wiley & Sons, Ltd, 6:174-207.

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Table 1.A comparison between commercial bank risk attitude assessment and two-hierarchical classification method. The larger the synthetical risk attitude value, the higher level of risk aversion.

conservative moderately

conservative moderate

moderately aggressive

aggressive Commercial bank customers risk attitude assessment

62.5% 20.8% 16.7%

10~8 8~6 6~4 4~2 2~0 Synthetical risk attitude value

8.3% 58.3% 16.7% 16.7%

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