Tax auditing models with the application of theory of search Текст научной статьи по специальности «Математика»

Suriya Sh. Kumacheva

St.Petersburg State University,

Faculty of Applied Mathematics and Control Processes, Bibliotechnaya pl. 2, St.Petersburg, 198504, Russia E-mail: s_kumach@mail.ru

Abstract. One of the most important aspects of modeling of taxation — tax control — is studied in the network of game theoretical attitude. Realization of systematic tax audits is considered as the interaction of two players - hiding (T - taxpayer) and seeking (A - authority) — in different statements of the search game.

In the problem statement it is assumed, that tax authority knows due to indirect signs, that some taxpayer T evaded from taxation in the given period (it put down lower than real income in the tax declaration). Also it is assumed, that this taxpayer is quite a large company, having many branches. Company can distribute the hidden income among all or some of its branches or can hide it in the income of one of its branches. Different game cases, which conditions are determined by various possibilities of the evading taxpayer and the tax authority, are considered. In the conditions of the games considered, tax authority A solves the problem of maximization of its income, using mixed strategies, which allow making optimal (in sense of evasion search) tax audits. Taxpayers T aspire to hide their income with the help of mixed strategies, allowing to reduce a probability of revealing of their evasion.

Along with search games, in which tax authority is defined as a search unit or, at best, as several search units, the task of tax audits using the theory of search of immobile chain with the help of big search system is considered.

This model takes into account, that practically tax authority is hierarchic structure and is solving tasks on different levels (federal, regional, distirct etc.).

In this model tax authority solves the task of maximizing the probability of revealing of evasion with the help of optimal (in given sense) distribution of search efforts on the given period of time. Company solves the opposite task: to distribute evasion in the way, that minimizes probability of revealing of evasion.

Thus, different possibilities of solving the task of tax evasion search are considered.

Keywords: tax control, taxpayers, tax authority, tax evasions, search games, theory of search.

1. Introduction

One of the most important aspects of tax control is revelation of tax evasions. Gathering and analysis of indirect information about taxpayers’ income (see, for example, Vasin and Agapova, 1993, Macho-Stadler and Perez-Castrillo, 2002 and others) allows to realize systematic tax audits to reveal tax evasions.

Another important task of tax authority is to distribute its resources to improve the net income, which is realized to the state budget (see, for example, Sanchez and Sobel, 1993).

Tools for solving both of these problems is given by the theory of search, in which the probability of revealing of the subject in finite time interval is defined as the function of the search strategy or search efforts (spent search resources).

At the end of the given tax period each taxpayer declares income and pays tax in compliance with them.

Let’s assume, that tax authority knows due to indirect signs, that some firm evaded from taxation in the given period (it put down lower than real income in the tax declaration). Let’s also assume, that this firm is quite a large company, having many branches. The company can distribute the concealed income among all or some or one of its branches. The tax authority doesn’t know in which.

It should be noted that the solution of mentioned problem can be considered in another statement:

— the evasion of big company can be distributed among its different business fields, but not among different branches (it is spoken about some total tax payments, including, for example, transport tax, property tax, value-added tax, profit tax, excises etc.);

— a set of taxpayers, acting together in the network of a coalition agreement, can be considered as evader.

A problem of optimal distribution of resources on tax audit was studied before by such authors as Sanchez and Sobel, 1993. Tax authority has hierarchical structure and solves tasks on different levels (federal, regional, distirct etc.). Taking this feature into account, use of the described by Hellman, 1985, theory of search of immobile chain with the help of a big search system becomes acceptable to the task of tax audits.

2. The application of theory of search for a big search system to the task of revelation of tax evasions

Let’s suppose, that the firm conceals income only in one of n branches. Let k be the number of this branch; the tax authority doesn’t know it. As opposed to the theory, described by Hellman, 1985, the search of the object (concealed income) is realized in the discrete space of n branches. Speaking, that the tax authority is a big search system, following Hellman, 1985, let’s assume, that the activity of the system’s separate unit can be defined with the help of function Xi(t) with the next characteristics:

Aj (t) >0, i = 1, n, t > 0; (1)

n

^Aj(t) = L(t),t> 0; (2)

i=1

Ai(t)At + o(At) (3)

- is the probability of revelation of the evasion in the i-th branch in time interval

(t, t + At) in the condition that in it there is a real evasion, which was not revealed

until the moment t. Aj(t) is called as density or strategy of search.

Consider the vector u = (u1,...,un), which characterizes a distribution of a tax evasion among branches. Each its component ui is the probability of evasion of the i-th branch.

It should be noted that the distribution u can be improper (see Feller, 1967).

n

Then the inequality ^ ui < 1 is fulfilled. It is due to the information about the tax

i=1

evasion (which stimulates the tax authority to audit the taxpayer's branches) has an estimation character and can turn out false with some nonnegative probability. (In the paper by Macho-Stadler and Perez-Castrillo, 2002, such information called

n

as signal). If this information is true, then the condition ^ ui = 1 is hold. So, in

i=1

the common case (when the tax authority a priori doesn’t know how the signal corresponds to the reality) let's suppose that

< 1.

1

Let k be the number of the branch, where the income concealment was. Let’s define {(t1,t2)} as the event consists in that there was no revelation of the evasion in the time interval (t1,t2). Let for fixed k

Pi(t) = P{(i = k)|(0,t)}

be the probability of the revealing of the tax evasion in the i-th branch in the condition that there was the evasion and it was unrevealed until the time t. From the definition of search density obtain:

P {(t, t + At)l(i = k)(0, t)} = 1 — Xi (t)At + o(At)

- is the probability, that in time interval (t, t + At) there is no revelation of the evasion in the branch in which this evasion was in the condition that it was not revealed until the time t. Then the probability of unrevealing of the evasion of the whole company in time interval (t,t + At):

n

P{(t,t + At)|(0,t)} = 1 — Aty 'uiXi(t) + o(At).

i=1

Let’s denote by P(t) the probability of that the evasion was revealed in time interval (0;t): P(t) = 1 — P{(0,t)}. Then, similar to obtained in the book by Hellman, 1985, the equality for the probability of revealing of the object in time

T, let’s define the probability of the existing tax evasion of the company in (0; T),

which is a period of limitation for tax crimes:

P(T) = 1 — ui exp( f Xi(r)dr). (4)

i=i Jo

The strategy of search Xi must be defined such way as it fulfills the conditions (1) - (3) maximizes the probability (4) of the revealing of the evasion in the given time period (0; T).

Let’s take into consideration a function ^(t), which fulfills the conditions

and call it as the number of search resources, spent on the revelation of the tax evasion in the i-th branch during the time interval (0; T). Then

The tax authority’s aim is to distribute the number of search resources ^(T) on the given tax period (0; T) in order to fulfill the conditions (5) - (7) and maximize the probability (8).

The company solves the opposite task: to minimize probability of revealing of evasion P(T), i.e. to distribute evasion in the way, that the function probability of revealing of evasion get its maximum.

Each proper task requires an individual solution, which depends on a lot of reasons: a number of branches, involved in the evasion, a geography of branches, resources (time, finances, search units) and the ways helping to realize the auditing etc.

Consider several examples of the search tasks depending on the ways of distribution of the concealed income and the ways of searching it by the tax authority. These models are based on the search games, studied by Petrosyan and Garnaev, 1992, Petrosyan and Zenkevich, 1987. Let’s consider the tax authority (player A - authority ) as a searcher and taxpayer (taxpayers) (player T - taxpayers) as a concealer.

The model of search of evasion, which is concentrated in the one branch.

The taxpayer T conceals its income in one of its n branches. The tax authority A also searches in one of them. Both of the players — A and T — have n pure strategies, identifiable with numbers of the branches, chosen for evasion or auditing correspondingly. If there is an evasion in the k-th branch and audit is in the i-th branch, the A's benefit is equal to

where k is the number of the branch, where the evasion was, i is the number of the branch, where the audit was. The quantity <jj (0 < <jj < 1, k = l,n) is interpreted as the probability of revelation of the evasion in the k-th branch.

In the terms of this game the event consists in the revelation of the total evasion of the company, is equal to the revelation of existing evasion in the one (the k-th) branch; uk = 1, that’s why with (8):

(5)

fi(T) > 0, i = l,n;

(6)

(7)

n

(8)

i=1

ak = 1 — e-Lpk (T},

(9)

where T can be interpreted as the time interval, given for realization of the audit. As in the terms of this model the audit can be realized only in one branch, it is possible to suppose that the tax authority will use all of its resources, i. e. total

T

search effort j L(t)dr. Then (9) will turn out

-f l(t)dT ak = 1 — e 0 .

(10)

The game considered is the matrix game, and the matrix of the A's benefits is

o\ 0 ... 0 0 72 ... 0

0 0 ... ar,

This game’s value v and optimal mixed strategies are i* = (i\,...,i*n) v* =

K ,...,<):

1

v = —-----,

E —

k = 1 ak

li = vi =

i = 1, n.

E —

k = 1 ak

If 7 = for i = j, then the choice of the branches, in which there were evasion and audit, are equiprobables for each player. Then the optimal strategies and the game value get the form

i

* t1 1 ^ 1

n n n n

Then let s consider the situations in which the company can evade in several branches.

The model of search of evasion, which is concentrated in one or two branches. Let ’ s suppose, that the company either concealed its income in one branch or divide it into two parts and distribute on two branches.

The set of T s pure strategies contains n2 elements and consists of various pares (i, j), where i, j are the numbers of the branches, in which there exist an evasion.

Player A can make an audit in one of n branches. Therefore, he has n pure strategies. Player A s aim is to maximize the probability of revealing of even one of the parts of evasion. In this case the probability of revelation of existing evasion in the fc-th branch during time T is defined similar to the game of search of the one evasion from (9).

The tax authority ’s profit K(k, (i,j)) in the situation (k, (i,j)) is:

K(k, (i,j))

^ if i = j, i = k;

7i, if i = k,i = j;

7j, if j = M = j;

7i(2 — 7i), if i = j = k;

0, if i = k,j = k.

Using the solution of corresponding game (see Petrosyan and Garnaev, 1992), obtain the optimal mixed strategies i* = (i\,..., i*n) and v* = {v*} of the players A and T correspondingly and the game value v:

Pi vi,i

i(2 — <7i) ■ r ■ 1

----------, if 1=1,71]

Ej <Ti(2-<T!) j = 0, if i = j; 1

E

1

l_1 <Ti(2-<Ji)

It should be noted that from the statement of T's strategy it is reasonable for this player to concentrate total concealed income in one of its branches.

The model with the distribution of the concealed income among m branches. Assume, that the company T evades in m branches (m < n). It means that concealed income is divided into m equal parts and distributed among corresponding number of T’s branches. The tax authority A can make audits in l branches at the same time.

Let pi be the probability of revelation of the evasion in the i-th branch when it is audited. First let’s consider an ideal situation, when

f 1, if there is an evasion in this branch; . ^—

'P'1 \ 0, if there is no evasion in this branch. i ,n

The number of the taxpayer’s pure strategies is Cm (various samples without replacement m of n branches to conceal income there), the number of the tax authority’s pure strategies is Cln (various samples without replacement l of n branches to audit them simultaneously).

The player A's profit is defined as a tax from total income revealed in one or several branches (or a number of revealed parts of the income). This player's aim is to maximize his income. The game matrix has dimension Cm x Cln.

The solution of this game is found using the results, obtained by Petrosyan and Garnaev, 1992: the player A’s (T’s) optimal mixed strategies i* (v*) consist in the choice of one of the Cln{C™) possible pure strategies with probability The

game value is v =

The model with the distribution of equal or different parts of the concealed income among m branches. Consider another situation: the taxpayer T divides an income, he wants to conceal, into m different parts. But, distributing this income among its branches, T can conceal either one or several parts of it in each branch.

First, let's suppose that player A can make an audit only in one of T's branches (this fact can be reasoned by time, financial and official restrictions for the unit of tax authority), but this audit reveals total tax evasion in the branch. In this game players T and A have nm and n pure strategies correspondingly. So, the A’s profit matrix has dimension n x nm.

Using the results, obtained for the base game by Petrosyan and Garnaev, 1992, define the tax authority’s optimal mixed strategy i* as auditing one of n branches

1

a

v=

of the company with the probability The taxpayer’s optimal mixed strategy v* is to choose one of nm possible placements of m parts of the evasion into n branches with the probability The game value is equal to

The generalization of considered situation is the case when the tax authority can make simultaneous audit in l branches, revealing every part of existing evasion in them. The A's profit is the tax from total concealed income of the company, revealed as the result of auditing. In this game player T (A) has nm(Cln) pure strategies, therefore, the A’s profit matrix has a dimension Cln x nm.

The tax authority's optimal mixed strategy i* is to choose one of Cnn samples without replacement l of n company's branches for simultaneous auditing with the probability The taxpayer’s optimal mixed strategy v* is to choose one of nm possible placements of m parts of the evasion into n branches with the probability The game value is equal to It should be noted that in the case of distribution of the evasion among m branches the game value is the same as in the last case (when there can be revealed one or several parts of the evasion in audited branch). That is, the possibility to conceal in one branch grater or smaller part of the evasion does not influence on player A’s profit.

To approach the models with distribution of concealed income among several branches to the practical tax auditing it is necessary to decline an assumption that the audit is always effective. That is, the probability of revelation of the tax evasion in the i-th branch during time T, given for the tax audit, can be calculated with use of (10):

This probability depends on the search efforts, spent in the i-th branch on revelation of the tax evasion, if it exists there.

There are two principally different ways of the searching of tax evasion in one of l branches — in consecutive order and simultaneous realizing of l audits.

The tax authority doesn’t know in which branches there is concealed income. It makes the estimation of spent resources more difficult and, therefore, it becomes more difficult to choose the most preferable way of auditing in this sense.

It is also obviously that the consecutive audits have no preferences in comparison with parallel auditing in sense of spending time: simultaneous search in l branches takes as long as an audit of one branch when the consecutive search is realized.

Everything told above lets draw a conclusion that parallel auditing in l branches is more rational in comparison with consecutive audits.

Total search resources, spent by the tax authority on l simultaneous tax audits, is defined as the amount of search resources in each branch. Then, with (8), the probability of revelation of the tax evasion when there realized a parallel auditing of l branches in time T is defined as:

Pi(T)

1 — e if there is an evasion; ^ _ j—^

0, if there is no evasion,

j+i-1

If the company's evasion is really distributed among l branches, then the vector u = (ui,..., un) is such as

j+—1

ui = 1.

i=j

In this case the statement (11) becomes:

P(T) = 1 — exp(— f L(t)dr),

J 0

where L(T) is the total cost of simultaneous search of evasion in l branches.

Thus, the application of theory of search to the task of tax auditing is considered. The results, obtained above, allows to solve tasks of tax evasion search and optimal distribution of tax authority’s resources to improve its net income.

References

Vasin, A. and Agapova, O. (1993). Game Theoretic Model of the Tax Inspection Organization. International Year-Book of Game Theory and Applications, Novosibitsk: Nauka,

1, 83-94.

Macho-Stadler, I. and Perez-Castrillo, J. (2002). Auditing with signals. Economica, 02, 1-20.

Sanchez, I. and Sobel, J. (1993). Hierarchical design and enforcement of income tax policies. Journal of Public Economics, 1993, 50, 345-369.

Hellman, O. (1985). Introduction to the theory of optimum search. Optimization and operation research. M: Nauka (in Russian).

Feller, V. (1967). Introduction to the theory of probability and its applications. 2. M.: Mir (in Russian).

Petrosyan, L. and Garnaev, A. (1992). The search games. SPb: SPbSU (in Russian). Petrosyan, L. and Zenkevich, N. (1987). The optimum search in the conditions of conflict.

Leningrad: LSU (in Russian).

Kumacheva, S. and Petrosyan, L. (2009). A game theoretical model of interaction between taxpayers and tax authority. St. Petersburg, ’’Processes Of Conrtol And Stability: the 40-th International Scientific Conference Of Post-graduate And Graduate Students”, 634-637 (in Russian).