Proposed Problem and Model
We consider individuals as taxpayers in the scope of knowledgesharing as shown in Figure 7.1. These individuals transfer information about paying their taxes with each other. They receive this information from the three categories shown in Figure 7.1. The smallest box is the information about taxes, which consists of an individual’s observations, experiences and activities. The second largest box, or the explicit knowledge, is an individual’s obvious experiences. In other words, in this box the individual’s information about taxes is documented, which consists of information being collected, stored, and archived about the previous year’s taxes and the effects
FIGURE 7.1 Tax knowledge sharing system.
of these paid taxes on improvements in facilities, services and so on. The largest box is tacit knowledge or the individual’s unobvious experiences about taxes that consist of decreasing unemployment, increasing employment, implementing the law and performing public facilities.
Now the question is how we can fill these boxes? One way is surveying taxpayers and collecting information about the way they share their knowledge.
We have i categories (five for the time being) that knowledge is sharing among them. Individuals transfer their knowledge and experiences about these categories with each other and by this means play an important role in tax paying. These categories are collected from surveying academics and studying some articles and economic reports.
1. Justice.
Unjust tax system, lack of identification of tax resources, inconsistency between tax level and offered services and between the amount of determined tax and type of job.
People believe that taxpayers with higher incomes compared to those with lower incomes must have more responsibility to pay taxes.
2. Proportionality between the amount of tax received and the services offered.
People believe that there is no correlation between the amount of taxes received and the value of public services offered by the government. In other words, people cannot see the effect of taxes on their lives.
3. Individuals’ ignorance of the tax mechanism.
The government must clarify which projects are funded with taxes paid by the people. Another problem is that these are longterm projects. People do not know why they pay taxes and how these taxes will be used.
4. The role of social culture in paying taxes.
Not transferring information about taxes, ignorance of the importance of paying tax, nonexistence of a culture of taxation, not controlling tax evasion and so on are the most important factors in this category. Some people think that it is government’s duty to supply all their needs and because of this, there is no reason for them to pay taxes.
5. Trust in the government to prevent inappropriate expenditure.
People mostly are distrustful of the government in spending their taxes.
At first, the government should spend taxes for public goods and avoid wasting money by spending on luxuries. If the government spends taxes on nonessential items, people cannot trust them and will not make a correct declaration about their income.
Risk Modeling Structure
We introduced five categories earlier of how individuals transfer their knowledge about taxes and related issues in these categories. We apply the information transfer as tax knowledge sharing. Each of these factors causes a degree of risk in receiving taxes. We compute the quality with a risk function. We aim to compute the effectiveness quality of each factor in sharing information [16]. Risk function is an interval function that is defined below:
where L, is loss function and is given by
and the distribution function fix) shows the behavior of each variable that follows a uniform distribution. The values of x,are five determined factors and are random variables. We want to compute the risk of x, values at first and then to compute L,. After that, we can compute Ri. Next, we will introduce interval computations, functions and integrals that will be used in our research.
Interval Programming
Linear programming is among the most widely and successfully used decision tools in the quantitative analysis of practical problems where rational decisions have to be made. The conventional linear programming model requires the parameters to be known as constants. In the real world, however, the parameters are seldom known exactly and have to be estimated. Interval programming is one of the tools to tackle uncertainty in mathematical programming models.
A closed real interval [x,,x_{5}] denoted byx, is a real interval number which can be defined completely by
where x, and x_{s} are called infimum and supremum, respectively.
Two interval numbers x = (x_{;}, x_{s}] and у = [у,, y_{5}] are called equal if and only if x, = y, and x_{s} = y_{s} [15].
Let x = [x,,x_{s}] and у = [y,,y_{s}], then
 1. x + y= x,+y,,x_{s}+y_{s} (Addition)
 2. xy = x, y_{s},x_{s} + y, (Subtraction)
 3. xv = [min{x_{;}V/,x,y_{s},x_{s}y,,x_{s}y_{s}},max{x_{/}y_{;},x_{;}y_{s},x_{s}y_{/},x_{s}y_{5}}]
 (Multiplication)
Let
 5. x + у = у + x, xy = yx (Commutativity)
 6. (x + y) + z = x + (y + z) ,(xy)z = x(yz) (Associativity)
 7. x(y+z)e(xy + xz) (Subdistributivity)
 8. a (x + у) = ax + ay, a c R.
The construction of the interval integral, given in the general case, can be simplified drastically in the case that the interval of integration is finite and the integrand is a bounded interval function. (Definitions of the necessary concepts will be given below.) In particular, the use of the extended real number system is not required, so that all computations can be done by ordinary interval arithmetic.
Following the definitions, an interval function Y defined on an interval X = [a,b] assigns the interval value,
to each real number x e X, where у, у are real functions called, respectively, the lower and upper boundary functions"(or endpoint functions) of Y.
In general, the interval integral of an interval function Y over the interval X = a,b] is the interval.
where _[y(x)dx denotes the lower Darboux integral of the lower endpoint function у over the interval X and I у (,v)dx gives the upper Darboux integral of the upper endpoint function у over X. As these Darboux integrals always exist in the extended real number system, it follows that all interval (and hence all real) functions are integrable in this sense.
We consider the below algorithm for the explained parts:
1. Identifying the effective factors in tax paying.
We mean collected categories from surveying academics and studying some articles and economic reports that are explained above.
 2. Effective factors in tax paying are estimated by knowledge sharing.
 3. Computing the risk function of each factor in tax knowledge sharing.
A risk is the effect of each factor in tax knowledge sharing. In this part, we will use the presented formulas for computing risk. The reason for using the risk function here is the uncertainty in sharing taxpayers’ opinions. This uncertainty is caused by the fact that a factor could be effective as a in a case study while as (1 in another one. It is also possible to have both opponents and proponents about a factor. As a result, there is no definitive opinion about knowledgesharing factors.
Now we draw a flowchart for the above algorithm (see Figure 7.2).
Case Study
As stated before, we considered five categories of ways in which individuals share their tax knowledge. These categories were collected from surveying academics and studying some articles and economic reports. Briefly, the categories are justice, the proportionality between the amount of received tax and services offered, people’s ignorance of the tax mechanism, the role of social culture in paying taxes and trust in the government to prevent inappropriate expenditure. To gather the results of knowledge sharing between individuals we decided to survey the taxpayers. So a survey form was prepared (see Figure 7.3).
We asked 65 academics and experts to complete the survey form. The results of this survey are collected in Table 7.1. In Table 7.1, the first factor expresses justice in the tax system and shows the individual’s ideas about justice, that one person has chosen option one, five persons have chosen option two, 18 persons agree with option four and one person has chosen option five. Respectively, the second factor expresses proportionality between the amount of tax received and services offered by the tax system; the third one expresses individuals’ knowledge about the tax mechanism; the fourth factor is the role of social culture in paying taxes and finally the last one
FIGURE 7.2 An algorithm.
expresses the role of trust in the government to prevent inappropriate expenditure the of taxes received.
Computing Mean Values (ti)
tj is the mean value of the achieved response for each factor. We compute t, for the factors. The solution method, in this case, is that we equal interval (0,0.25) to the first option, interval (0.25,0.5) to the second option, interval (0.5,0.75) to the third option and interval (0.75,1) to the last option. Then we multiply the number of individuals who had chosen the first option by interval (0.0.25) and multiply the number of individuals who had chosen the second option by interval (0.25,0.5) and so on to intervals (0.5,0.75) and (0.75,1). Next, we sum the above four achieved intervals and
 1) How would you rate justice by tax authorities?
 1) very high □ 2) high □ 3) moderate □ 4) low □
 2) How would you rate proportionality between the amount of received tax and offered services in the tax system?
 1) very high □ 2) high □ 3) moderate □ 4) low □
 3) How would you rate individual’s knowledge of tax mechanism?
 1) very high □ 2) high □ 3) moderate □ 4) low □
 4) How would you rate the role of social culture in paying taxes?
 1) very high □ 2) high □ 3) moderate □ 4) low □
 5) How would you rate the role of trust to the government inappropriate expenditure of received taxes?
 1) very high □ 2) high □ 3) moderate □ 4) low □
FIGURE 7.3 Survey form.
then divide this summation by 65, that is the number of individuals who took part in the survey. After receiving qualitative responses, for applying to formulas we use them in the normal distance (N(0,1)). We divided this distance by four because the questions consist of four options. It means that the first option “very high”, is equal to (0,0.25), the second option “high”, is equal to (0.25,0.5), and respectively options three and four are equal to (0.5,0.75) and (0.75,1).
TABLE 7.1
The Results of the Survey
Option One 
Option Two 
Option Three 
Option Four 

First factor 
l 
5 
18 
41 
Second factor 
0 
8 
20 
37 
Third factor 
l 
2 
18 
44 
Fourth factor 
11 
12 
15 
27 
Fifth factor 
0 
7 
21 
37 
Computing Loss Function
In this section, by means of t, values that were computed in the previous part we can compute loss functions. We use the loss formula that was defined in the ‘Risk Modeling Structure’ section for computing loss function for all factors:
.v, values are variable. The computed values for t, are as below:
Now we compute L, functions by putting these intervals in a loss function formula:
Computing f(xi) for the Factors
/(a,) follows a uniform distribution. Random variables x, is said to distribute uniformly in the interval [a,b] if it has the probability density function as below:
The support is defined by the two parameters, a and b, which are its minimum and maximum values, respectively.
Here a and b will be achieved by the survey. For each factor, b is obtained by multiplication of the second component of attributed interval (0.75,1) to the fourth option that is number 1 with the number of individuals who had chosen the fourth option for each factor. Also, a is obtained by multiplication of the second component of attributed interval (0,0.25) to the first option, that is number 0.25, with the number of individuals who had chosen the first option for each factor.
According to Table 7.1, for the factors we compute a and b and after that/fv,) for each factor:
For.*,:
For x_{2}:
For x_{}}:
For x_{4}:
For x_{5}:
Computing Risk Function (R(xi))
All of the five factors create a different degree of risk in tax paying. We have to compute the risk function to control the effectiveness quality of each factor in tax knowledge sharing among individuals.
As stated before for computing risk we use the formula below:
L, is loss function that was computed before for the factors, also /(a,) has been introduced before.
For integral bounds, we do as follows:
For each factor, we consider the first option that has a nonzero answer (it means that at least one person has chosen the option), then, the supremum value of the attributed interval to this option is chosen as the lower bound (m), and the next nonzero answer for the same factor is chosen as the supremum value of the attributed interval to this option as the upper bound (v). In the following computations the upper bound and lower bound for each factor will compute. Then, using loss functions and/(x,) we will compute risk function for each factor.
First factor (x,):
Integral bounds:
We have no zero answers among the answers for the first factor, so we consider the interval (0,0.25) attributed to the first option. The supremum value of this interval is 0.25. So, we choose и = 0.25 as the lower bound. The attributed interval to the fourth option is (0.75,1) that the supremum value of it is 1, so we choose v = 1 as the upper bound.
Also, we computed L_{:} and/(x,) as below:
So, we compute fi(xt) as below:
TABLE 7.2
A Summary of Risk Computations
U 
/(*«) 
Ri 

Second factor 

Third factor 

Fourth factor 

Fifth factor 
Then we compute the other factors as above. The results of computations are in Table 7.2.
Concluding Remarks
In recent decades, tax policy has been at the center of public debate. The economic, social and cultural effects of tax payment is one of the most important topics that needs extra attention from every government. By focusing on the role of tax knowledge sharing in tax payments that have not been mentioned previously in other studies, this chapter adds knowledge sharing as an important factor in tax systems. Since taxation is an instrument of economic growth, policies to improve a tax system’s productivity and efficiency should be implemented and sustained. This conclusion points to the need for additional attempts by governments in educating individuals in tax knowledge and encouraging them to share and transfer that knowledge with others to increase tax compliance by taxpayers, and as a result to increase tax revenues for better facilities and for growth.
Here, we introduced five categories in which taxpayers share their knowledge. We considered individuals as taxpayers in knowledge sharing. They transfer their knowledge and experiences about these categories with each other and play an important role in tax raising. These categories are collected from a survey which was extracted from studying some articles and economic reports. We apply this transferring of information as tax knowledge sharing. Each of these factors caused a degree of risk in receiving taxes. We computed risk for each factor using a loss function.
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