The modeling of the static game between the tax administration and taxpayers is as follows:

| Taxpayers | |

TAO | NC | C | |

**NC** **C** | \({U}_{1} , {U}_{2}\) | \({U}_{3} , {U}_{4}\) | |

\({U}_{5} , {U}_{6}\) | \({U}_{7} , {U}_{8}\) | |

Now, the economic utility function of each player will be examined based on mathematical relationships in different situations:

$${U}_{1}=E+{(D-E)}^{2}-c$$

$${U}_{2}=R-{(D-E)}^{2}-E$$

$${U}_{3}=E+{(D-E)}^{2}-c-\alpha f$$

$${U}_{4}=R-{(1-\alpha )(D-E)}^{2}-E$$

$${U}_{5}=E+{(1-\theta )(D-E)}^{2}-c$$

$${U}_{6}=R-{\left(D-E\right)}^{2}-E-\theta k$$

$${U}_{7}=E+{(1-\theta )(D-E)}^{2}-c-(1-\theta )\alpha f$$

$${U}_{8}=R-(1-\alpha ){\left(D-E\right)}^{2}-E-(1-\alpha )\theta k$$

The (NC, NC) strategy means that none of the players (the tax authorities and the taxpayers) are willing to collude. The outcome of \({U}_{1}\) is obtained according to the utility function 1, in which none of the players have a willing to collude, and therefore the values of \(\alpha\) and \(\theta\) are considered equal to zero. The outcome of \({U}_{2}\)is also obtained according to the utility function 2, in which none of the players have a willing to collude, and therefore the values of \(\alpha\) and \(\theta\) are considered equal to zero.

The strategy (NC, C) means that tax administration (tax auditors) do not have a tendency to collude with the other player (taxpayers) (\(\theta =0\)), while taxpayers willing to collude with tax auditors (\(\alpha =1\)). The outcome of \({U}_{3}\) is obtained according to the utility function 1, where the values of \(\alpha =1\) and \(\theta =0\) are considered equal to zero. The outcome of \({U}_{4}\) is obtained according to the utility function of 2 and the values of \(\alpha =1\) and \(\theta =0\).

The strategy (C, NC) means that tax administration (tax auditors) willing to collude with the other player (taxpayers) (\(\theta =1\)), while taxpayers do not tend to collude with tax auditors (\(\alpha =0\)). The outcome of \({U}_{5}\) is obtained according to the utility function 1, where the values of \(\alpha =0\) and \(\theta =1\)are considered equal to zero. The outcome of \({U}_{6}\) is also obtained according to the utility function 2 and the values of \(\alpha =0\) and \(\theta =1\).

The strategy (C, C) means that both players, i.e. the tax auditors and taxpayers, willing to collude. The outcome of \({U}_{7}\) is obtained according to the utility function 1, in which both players willing to collude, and therefore the values of \(\alpha\) and \(\theta\) are considered equal to one. The outcome of \({U}_{8}\) is also obtained according to the utility function 2, in which both players tend to collude, and therefore the values of \(\alpha\) and \(\theta\) are considered equal to one.

Now the behavior strategies of the game players will be examined. In other words, all four equilibrium cells will be discussed and we are looking to achieve the equilibrium points at which time and under what conditions the equilibrium will fall in a specific cell. In the following, all four possible equilibrium states will be discussed.

**4.1. The first state: equilibrium in the first cell of the game matrix (the unwilling of tax auditors to collude and the unwilling of taxpayers to collude)**

In order to equilibrium occur in the first cell (the unwilling of tax auditors to collude and the unwilling of taxpayers to collude), the following conditions must be met:

$${U}_{1}\ge {U}_{5 }\& {U}_{2}\ge {U}_{4 }$$

$$\Rightarrow \left\{\begin{array}{c}{U}_{1}\ge {U}_{5 }or {U}_{1}-{U}_{5 }\ge 0\iff (E+{\left(D-E\right)}^{2}-c)-(E+{\left(1-\theta \right)\left(D-E\right)}^{2}-c)\ge 0\\ {U}_{2}\ge {U}_{4 }or{ U}_{2}-{U}_{4 }\ge 0\iff (R-{\left(D-E\right)}^{2}-E)-(R-{\left(1-\alpha \right)\left(D-E\right)}^{2}-E)\ge 0\end{array}\right.$$

3

⇒\(\left\{\begin{array}{c}E=D \vee \theta \ge 0\\ E=D \vee \alpha \le 0\end{array}\right.\) (4)

Since both results must hold simultaneously, therefore, for the second case (\(\alpha \le 0\)), only \(\alpha = 0\) holds. Also, since it is considered according to the first state of this game (the unwilling of the players to collude), therefore, for the first state (\(\theta \ge 0\)), only the state \(\theta =0\) is established.

⇒\(\left\{\begin{array}{c}E=D \vee \theta =0\\ E=D \vee \alpha = 0\end{array}\right.\)

The results show that only in the conditions of equilibrium in the first cell (the unwilling of tax auditors to collude and the unwilling of taxpayers to collude) occurs that the expressed tax of taxpayers is equal to the diagnostic tax. These results are completely in line with the real world because if the taxpayers express their statement to be true (equal to diagnosis) then there will be no collusion motivation for the players. In other words, if taxpayers adopt the strategy of unwilling to collude, then the best strategy for tax auditors will also be unwilling to collude. For the second case, only \(\alpha = 0\) will be significant. This means the equilibrium in the first cell occurs that the tendency of taxpayers to collude is zero, and in this case, the tendency of tax auditors to collude is also zero, which has no effect on the equilibrium. In the following, the equilibrium state in the second cell will be investigated.

**4.2. The second state: equilibrium in the second cell of the game matrix (the unwilling of tax auditors to collude and the willing of taxpayers to collude)**

In order for the equilibrium to occur in the second cell (the unwilling of tax auditors to collude and the willing of taxpayers to collude), the following conditions must be met:

$${U}_{3}\ge {U}_{7 }\& {U}_{4}\ge {U}_{2 }$$

$$\Rightarrow \left\{\begin{array}{c}{U}_{3}\ge {U}_{7 }or {U}_{3}-{U}_{7 }\ge 0\iff (E+{(D-E)}^{2}-c-\alpha f)-(E+{(1-\theta )(D-E)}^{2}-c-(1-\theta \left)\alpha f\right)\ge 0\\ {U}_{4}\ge {U}_{2 }or{ U}_{4}-{U}_{2 }\ge 0\iff (R-{(1-\alpha )(D-E)}^{2}-E)-(R-{(D-E)}^{2}-E)\ge 0\end{array}\right.$$

5

⇒\(\left\{\begin{array}{c}{if \theta \ge 0\Rightarrow (D-E)}^{2}>f\alpha or \frac{{(D-E)}^{2}}{f}>\alpha \vee {if \theta \le 0\Rightarrow (D-E)}^{2}<f\alpha or \frac{{(D-E)}^{2}}{f}<\alpha \\ D=E \vee \alpha \ge 0\end{array}\right.\) (6) Since according to the assumptions set in the second part of the relationship, only the state \(\theta =0\)is established, therefore we have:

\(\left\{\begin{array}{c}{if \theta \ge 0\Rightarrow (D-E)}^{2}>f\alpha or \frac{{(D-E)}^{2}}{f}>\alpha \vee {if \theta =0\Rightarrow (D-E)}^{2}<f\alpha or \frac{{(D-E)}^{2}}{f}<\alpha \\ D=E \vee \alpha \ge 0\end{array}\right.\) (7) According to 7 relationship, the conditions in which both relationships are established can be examined in the form of the following relationships:

$$\left\{\begin{array}{c}{if \theta \ge 0\Rightarrow (D-E)}^{2}>f\alpha or \frac{{(D-E)}^{2}}{f}>\alpha \\ D=E \vee \alpha \ge 0\end{array}\right.$$

8

$$\left\{\begin{array}{c}{if \theta =0\Rightarrow (D-E)}^{2}<f\alpha or \frac{{(D-E)}^{2}}{f}<\alpha \\ D=E \vee \alpha \ge 0\end{array}\right.$$

9

As stated earlier, the equilibrium in the second cell of the game matrix, which shows the unwilling of tax auditors to collude and the willing of taxpayers to collude, therefore: \(\alpha \ge 0\) and \(\theta =0\). Therefore, relations 8 and 9 will be as follows:

$$\left\{\begin{array}{c}{if \theta =0\Rightarrow (D-E)}^{2}>f\alpha or \frac{{(D-E)}^{2}}{f}>\alpha \\ D=E \vee \alpha \ge 0\end{array}\right.$$

10

$$\left\{\begin{array}{c}{if \theta =0\Rightarrow (D-E)}^{2}<f\alpha or \frac{{(D-E)}^{2}}{f}<\alpha \\ D=E \vee \alpha \ge 0\end{array}\right.$$

11

Since \(\alpha\) is defined in the range of 0 and 1, therefore, if the gap between the expressed tax and the taxpayer's diagnostic tax is significant and large enough, there will be a collusive incentive for taxpayers. Because if the gap between the expressed tax and the diagnostic tax is small and close to zero, collusion will not be formed in any way. Therefore, only relation 10 will be maintained:

$$\left\{\begin{array}{c}{if \theta =0\Rightarrow (D-E)}^{2}>f\alpha or \frac{{(D-E)}^{2}}{f}>\alpha \\ D=E \vee \alpha \ge 0\end{array}\right.$$

Therefore, according to this relationship, since the condition \(\alpha \ge 0\) is established, \(\frac{{(D-E)}^{2}}{f}>\alpha \ge 0\) should be. This means that the willing of the taxpayer to collude is less than the square root of the difference between the expressed and diagnostic taxes divided by f (the benefit lost for the tax organization if the taxpayers are willing to collude). In other words, if the obtained utility by the tax auditors through collusion with the taxpayers (the lost utility of the Tax Administration) is high (so if the amount proposed by the taxpayers for collusion is high enough), the total deduction is reduced and \(\alpha\) will decrease as much as possible and equilibrium will be established in this cell.

These results show that equilibrium in the second cell (unwilling of tax auditors to collude and the willing of taxpayers to collude) occurs only in conditions where tax expressed tax of the taxpayers is equal to the diagnostic tax and \(\frac{{(D-E)}^{2}}{f}>\alpha\). As discussed, either the difference in expression and diagnosis should be low, or the amount proposed by the taxpayers should be high enough for collusion, the equilibrium will be established in this cell. The following theorem expresses the clearer meaning of these results:

**Theorem 1**

*If the amount proposed by the taxpayers is high enough for collusion, then the tax auditors will recognize the diagnostic tax as equal to the expressed tax of the taxpayers!!*

In the following, the equilibrium state in the third cell will be investigated.

**4.3. The third state: equilibrium in the third cell of the game matrix (the willing of tax auditors to collude and the unwilling of taxpayers to collude)**

In order to equilibrium occur in the third cell (tax auditors' willing to collude and taxpayers' unwilling to collude), the following conditions must be met:

$${U}_{5}\ge {U}_{1 }\& {U}_{6}\ge {U}_{8 }$$

$$\Rightarrow \left\{\begin{array}{c}{U}_{5}\ge {U}_{1 }or {U}_{5}-{U}_{1}\ge 0\iff \left(E+{\left(1-\theta \right)\left(D-E\right)}^{2}-c\right)-\left(E+{\left(D-E\right)}^{2}-c\right)\ge 0\\ {U}_{6}\ge {U}_{8 }or{ U}_{6}-{U}_{8 }\ge 0\iff \left(R-{\left(D-E\right)}^{2}-E-\theta k\right)-\left(R-\left(1-\alpha \right){\left(D-E\right)}^{2}-E-\left(1-\alpha \right)\theta k\right)\ge 0\end{array} \right.\left(12\right)$$

⇒\(\left\{\begin{array}{c}D=E \vee \theta \le 0\\ {if \alpha \ge 0\Rightarrow -(D-E)}^{2}>\theta k or-\frac{{\left(D-E\right)}^{2}}{k}>\theta \vee {if \alpha \le 0\Rightarrow -\left(D-E\right)}^{2}<k\theta or-\frac{{(D-E)}^{2}}{k}<\theta \end{array}\right.\) (13)

According to the assumptions determined in the first part of the relationship, only the state \(\theta =0\) and in the second part \(\alpha =0\) is established, so we have:

$$\left\{\begin{array}{c}D=E \vee \theta = 0\\ {if \alpha \ge 0\Rightarrow -(D-E)}^{2}>\theta k or-\frac{{\left(D-E\right)}^{2}}{k}>\theta \vee {if \alpha =0\Rightarrow -\left(D-E\right)}^{2}<k\theta or-\frac{{(D-E)}^{2}}{k}<\theta \end{array}\right.$$

14

According to relationship 14, conditions in which both relationships are established can be examined in the form of the following relationships:

$$\left\{\begin{array}{c}D=E \vee \theta = 0\\ {if \alpha \ge 0\Rightarrow -(D-E)}^{2}>\theta k or -\frac{{\left(D-E\right)}^{2}}{k}>\theta \end{array}\right.$$

15

$$\left\{\begin{array}{c}D=E \vee \theta = 0\\ {if \alpha =0\Rightarrow -\left(D-E\right)}^{2}<k\theta or -\frac{{(D-E)}^{2}}{k}<\theta \end{array}\right.$$

16

Here, too, the equilibrium in the third cell of the game matrix, which shows the willing of tax auditors to collude and the unwilling of taxpayers to collude, so we have: \(\alpha = 0\)and \(\theta \ge 0\). Therefore, relations 15 and 16 will be as follows:

$$\left\{\begin{array}{c}D=E \vee \theta = 0\\ {if \alpha =0\Rightarrow -(D-E)}^{2}>\theta k or -\frac{{\left(D-E\right)}^{2}}{k}>\theta \end{array}\right.$$

17

$$\left\{\begin{array}{c}D=E \vee \theta = 0\\ {if \alpha =0\Rightarrow -\left(D-E\right)}^{2}<k\theta or -\frac{{(D-E)}^{2}}{k}<\theta \end{array}\right.$$

18

Since \(\theta\) is defined in the range of 0 and 1, and also in accordance with the third state, which shows the tax auditors' tendency to collude, Therefore, due to the negativity of relation 17, only relation 18 can be discussed, and this relation is always established, which is interpreted as relation 19:

$$\left\{\begin{array}{c}D=E \\ {if \alpha =0\Rightarrow -\left(D-E\right)}^{2}<k\theta or -\frac{{(D-E)}^{2}}{k}<\theta \end{array}\right.$$

19

Therefore, according to this relationship, if the rate of willing of taxpayers to collude is zero, then the level of willing of auditors to collude (which is always established according to relation 18) has no effect on collusion. These results show that the equilibrium in the third cell (tax auditors' willing to collude and the taxpayers' unwilling to collude) occurs only in conditions that expressed tax of taxpayers is equal to the diagnostic tax and \(-\frac{{\left(D-E\right)}^{2}}{k}<\theta .\) The very important point of this result like the Granger causality test is expressed as the following theorem:

**Theorem 2**

*The willing of taxpayers to collude is the main condition for collusion in the game between taxpayers and tax auditors.*

In the following, the equilibrium state in the fourth cell will be examined.

**4.4. The fourth state: equilibrium in the fourth cell of the game matrix (the willing of tax auditors to collude and the willing of taxpayers to collude)**

In order to equilibrium occur in the fourth cell (tax auditors' willing to collude and taxpayers' unwilling to collude), the following conditions must be met:

$${U}_{7}\ge {U}_{3 }\& {U}_{8}\ge {U}_{6 }$$

$$\Rightarrow \left\{\begin{array}{c}7\ge {U}_{3 }or {U}_{7}-{U}_{3}\ge 0\iff (E+{(1-\theta )(D-E)}^{2}-c-(1-\theta \left)\alpha f\right)-(E+{(D-E)}^{2}-c-\alpha f)\ge 0\\ {U}_{8}\ge {U}_{6 }or{ U}_{8}-{U}_{6 }\ge 0\iff (R-(1-\alpha ){\left(D-E\right)}^{2}-E-(1-\alpha \left)\theta k\right)-(R-{\left(D-E\right)}^{2}-E-\theta k)\ge 0\end{array} \left(20\right)\right.$$

⇒\(\left\{\begin{array}{c}{if \theta \ge 0\Rightarrow (D-E)}^{2}<f\alpha or \frac{{(D-E)}^{2}}{f}<\alpha \vee {if \theta \le 0\Rightarrow (D-E)}^{2}>f\alpha or \frac{{(D-E)}^{2}}{f}>\alpha \\ {if \alpha \le 0\Rightarrow -(D-E)}^{2}>\theta k or-\frac{{\left(D-E\right)}^{2}}{k}>\theta \vee {if \alpha \ge 0\Rightarrow -\left(D-E\right)}^{2}<k\theta or-\frac{{(D-E)}^{2}}{k}<\theta \end{array}\left(21\right)\right.\)

Here, too, the equilibrium in the fourth cell of the game matrix, which shows the willing of tax auditors and the willing of taxpayers to collude, is therefore: \(\alpha \ge 0\) and \(\theta \ge 0\). So:

$$\left\{\begin{array}{c}{if \theta \ge 0\Rightarrow (D-E)}^{2}<f\alpha or \frac{{(D-E)}^{2}}{f}<\alpha \vee {if \theta =0\Rightarrow (D-E)}^{2}>f\alpha or \frac{{(D-E)}^{2}}{f}>\alpha \\ {if \alpha =0\Rightarrow -(D-E)}^{2}>\theta k or-\frac{{\left(D-E\right)}^{2}}{k}>\theta \vee {if \alpha \ge 0\Rightarrow -\left(D-E\right)}^{2}<k\theta or-\frac{{(D-E)}^{2}}{k}<\theta \end{array}\right.$$

22

Relationship 22 can be examined as the following relationships (relationships 23 to 26):

$$\left\{\begin{array}{c}{if \theta \ge 0\Rightarrow (D-E)}^{2}<f\alpha or \frac{{(D-E)}^{2}}{f}<\alpha \\ {if \alpha =0\Rightarrow -(D-E)}^{2}>\theta k or-\frac{{\left(D-E\right)}^{2}}{k}>\theta \end{array}\right.$$

23

$$\left\{\begin{array}{c}{if \theta \ge 0\Rightarrow (D-E)}^{2}<f\alpha or \frac{{(D-E)}^{2}}{f}<\alpha \\ {if \alpha \ge 0\Rightarrow -\left(D-E\right)}^{2}<k\theta or-\frac{{(D-E)}^{2}}{k}<\theta \end{array}\right.$$

24

$$\left\{\begin{array}{c}{if \theta =0\Rightarrow (D-E)}^{2}>f\alpha or \frac{{(D-E)}^{2}}{f}>\alpha \\ {if \alpha =0\Rightarrow -(D-E)}^{2}>\theta k or-\frac{{\left(D-E\right)}^{2}}{k}>\theta \end{array}\right.$$

25

$$\left\{\begin{array}{c}{if \theta =0\Rightarrow (D-E)}^{2}>f\alpha or \frac{{(D-E)}^{2}}{f}>\alpha \\ {if \alpha \ge 0\Rightarrow -\left(D-E\right)}^{2}<k\theta or-\frac{{(D-E)}^{2}}{k}<\theta \end{array}\right.$$

26

According to the previous explanations, relations 23 and 25 are unacceptable according to the interval conditions of θ. For relation 24

$$\left\{\begin{array}{c}{if \theta \ge 0\Rightarrow (D-E)}^{2}<f\alpha or \frac{{(D-E)}^{2}}{f}<\alpha \\ {if \alpha \ge 0\Rightarrow -\left(D-E\right)}^{2}<k\theta or-\frac{{(D-E)}^{2}}{k}<\theta \end{array}\right.$$

Which shows the willing of the game players to collude, considering that the second relationship is always established, equilibrium will occur when the first relationship \(\frac{{(D-E)}^{2}}{f}<\alpha\) is also established. The establishment of this relationship happens when either the taxpayers' proposed amount for collusion is high enough to maximize the auditors' utility, or the expression tax and diagnostic tax are very close (\(D-E<\sqrt{f}\)) which will be the basis for collusion due to the low probability of collusion error detection and the fact that the consequences of detection are not high. Of course, it is reminded again that the probability of this state is very, very low because \(\alpha\) is defined in the range of 0 and 1.

Finally, relationship 26

$$\left\{\begin{array}{c}{if \theta =0\Rightarrow (D-E)}^{2}>f\alpha or \frac{{(D-E)}^{2}}{f}>\alpha \\ {if \alpha \ge 0\Rightarrow -\left(D-E\right)}^{2}<k\theta or-\frac{{(D-E)}^{2}}{k}<\theta \end{array}\right.$$

It shows that the second relation always holds and the other conditions will be similar to the theorem 1 which has been discussed before.

### 5. نتیجهگیری و پیشنهادات

Collusive corruption refers to tax auditors requesting bribes for fully or partially overlooking the incidence of underreported income by private taxpayers, while coercive corruption or extortion refers to tax auditors extracting bribes against the threat of over-assessing the private taxpayer's income and falsely accusing them of tax evasion. The first type of corruption is cooperative while the latter is antagonistic. The encounter between a private taxpayer and a public tax auditor can contain both kinds of corruption.