A game theoretical model of interaction between taxpayers and the tax authority 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 The hierarchical game, in which the tax authority and finite number of taxpayers are players, is considered. The tax authority interacts to each taxpayer as ”principal-to-agent”. The players are supposed to be risk neutral. Every taxpayer can declare his income’s level as equal or less, than the true level of his income. Tax and penalty rates are defined as constants.

The tax authority makes audit with individual probability for every taxpayer. It is assumed that these probabilities are known by the taxpayers.

Audit is supposed to reveal 100% of evasions. The taxpayer must pay a tax on his evasion level and a penalty, which depends on this level, as a result of audit, that revealed a tax evasion. Different cases of penalties are considered. Players’ payoff fuctions and strategies are defined. The aim of every player is to maximize his payoff function.

Thus, the game theoretical model of taxation for finite number of taxpayers is considered. The Nash equilibrium in the corresponding game is found.

Keywords: tax control, taxpayers, tax authority, penalties, hierarchical game, optimal strategies, Nash equilibrium.

1. Introduction

One of the most important aspects of modeling of taxation is the tax control. Due to the mathematical tradition, founded in such works as (Chander and Wilde, 1998) and (Vasin and Vasina, 2002) this aspect is considered as the interaction between taxpayers and tax authority and is studied in the network of game theoretical attitude. This paper is devoted to a consideration of the simple game theretical model of this interaction and (as in previous models) to solving of such a problem as the search of the equilibrium and optimal strategies of players.

The hierarchical game (Petrosyan and Zenkevich and Semina, 1998), in which the tax authority (high level) and the finite number of taxpayers (low level) are players, is considered. Following (Boure and Kumacheva, 2005), (Chander and Wilde, 1998) and (Vasin and Morozov, 2005), it is considered that the tax authority interacts with each taxpayer as ”principal-to-agent”. The players’ behaviour is supposed to be risk neutral.

It is also interesting to study this model in different cases of the profit functions of players. These functions depend on taxes and penalties. Such sources as (Vasin and Vasina, 2002) or (Vasin and Morozov, 2005) give us information about four cases of penalties (known from the economic practice in different countries). All of them are considered in the model described. Also one specific case of penalty (interesting from the mathematical point of view) is considered.

2. The model

n taxpayers are considered, each of them has income level equal to ik, where k = 1, n. The income of the taxpayer rk is declared at the end of a tax period, where rk < ik for each k = l,n. As in earlier models, such as (Chander and Wilde, 1998) or (Vasin and Vasina, 2002), it is considered here that the audit of the fc-th taxpayer is made by the tax authority with probability pk. Model is built following the assumption, that taxpayers are aware of these probabilities. It is supposed, that the tax authority’s audits reveal tax evasions always.

3. Players’ strategies and their profit functions

In the game considered the first move is made by the tax authority (the central player), choosing the vector p = (p\,p2, ...,pn). Then the taxpayers’ moves are made (low level players), choosing the values of declared income rk.

If the evasion is revealed by the tax audit, then the evaded taxpayer should pay the underpaid tax and the penalty; both of which depend on the evasion’s level. Let t be the tax rate, n be the penalty rate. Four cases of penalties are known from (Vasin and Vasina, 2002):

1. the penalty is proportional to difference between true and payed tax:

F(I,Ir) = (1 + Sa)(T(I) - T(Ir));

2. the net penalty is proportional to evasion:

F(I, Ir) = T(I) - T(Ir) + Sb(I - Ir);

3. the penalty is restricted due to the given level of the agent’s minimal income I(< Imin) in the case of his nonoptimal behaviour:

0 < F(I,Ir) <I-T(Ir) - I;

4. the post-audit payment is proportional to the revealed evaded income:

F(I, Ir) = Sd(I - Ir),

where (in terms of (Vasin and Vasina, 2002)) I and Ir are taxpayer’s true and declared incomes correspondingly, T (Ir) is a tax function, 6 is a penalty coefficient and F(I, Ir) is a penalty function. It should be noticed that in (Vasin and Vasina, 2002)) a post-audit payment is understood under F(I,Ir).

The net penalty is proportional to evasion Let’s consider the model in the simplest case, when the penalty is proportional to evasion (the second case in (Vasin and Vasina, 2002))). (Analogical model was considered in (Kumacheva and Petrosyan, 2009)). Let’s define the penalty as (t + n)(ik — rk) for the A;-th taxpayer (where k = 1, n). The expected tax payment of the taxpayer k is:

Uk = trk + Pk(t + n)(ik - rk),

(1)

where the first summand is always paid by the taxpayer (pre-audit payment), and the second - as the result of the tax auditing, made with probability pk (post-audit payment). The expected payoff bk of the taxpayer k is:

bk = ik - uk = ik - trk - pk(t + n)(ik - rk). (2)

Function bk depends on values of the audit’s probability and on the declared income, i.e.

bk = bk(pk ,rk).

Every taxpayer’s aim is to maximize his payoff function

max bk (pk,rk), rk

or (which is equivalent aim in this case) to minimize his expected tax payments

min uk (pk,rk). rk

Let ck be the cost of the audit of the taxpayer k. It is different for each taxpayer, because the costs of auditing of each taxpayer are different. As in previous models (for example, (Chander and Wilde, 1998), (Vasin and Vasina, 2002) or (Boure and Kumacheva, 2005)), the tax authority’s net income consists of taxation (taxpayers’ payments corresponding to the declared income), taxes paid on the difference between true and declared levels of income and penalties (last two are the audit results) less total audit cost. Being the sum of payments got from every taxpayer, the expected tax authority’s net income can be calculated as the difference between expected tax payments of n taxpayers and expected cost of audit of n taxpayers:

n n

R ^ ^Rk ^ ^(uk pkck). (3)

k=i k=i

The tax authority’s aim is to maximize its expected income

maxR(p, ri,r2,..., rn), p

where p = (pi,p2, ...,pn).

4. A search of the optimal strategies

Proposition 1. The optimal strategy of the tax authority (in order to maximize its income) isp= ~j~^; the optimal strategy of the taxpayers (in the same meaning) is rl(Pk) = when pk <p, and r*k{pk) = ik when pk >p.

Proof. If the k-th taxpayer’s aim is considered as the minimization of the function uk , it should be taken into account, that in terms of this model uk is the linear function of variable rk . Therefore, the function gets its minimum on one of the ends of the segment [0, ik].

Figurel. The case of pk(t + 7r) < t, that is equivalent to pk < p■ The evasion is profitable for the fc-th taxpayer. Thus, his optimal strategy is rk = 0.

Figure2. The case of pk(t + tt) > t, that is equivalent to pk > p■ The evasion is not profitable for the fc-th taxpayer. Thus, his optimal strategy is rk = ik.

When declared income is defined as rk = ik, then function uk = tik, which corresponds to the true tax payment. When declared income is defined as rk = 0, then function uk = pk(t + n)ik, which corresponds to the expected post-audit tax payment.

The value of the probability of audit p = is critical for each taxpayer’s decision whether to evade or not.

So, the £;-th taxpayer’s optimal strategy is r*k(pk) = 0, whenpfc < p, andr^pk) = ik, when pk > p.

Remark 1. The value of the probability of audit pk =p is the point of indifference,

i.e. every taxpayer can declare rk = 0 or rk = ik. In both cases the taxpayer’s expected tax payment is uk = tik. That’s why it is possible to join the case of equality with the case of inequality pk > p.

If every taxpayer plays in optimal way, i.e. = r*k(pk) for each k = 1 ,n, then the expected tax authority’s income depends only on vector p, chosen by the tax authority, i.e.

R = R(pi,p2, ...,pn).

If pk > p, the taxpayer doesn’t evade and declares = ik for each k = 1 ,n. Then Rk = tik ~ PkCk, function Rk is decreasing on the segment \p, 1] and gets its maximum in the point pk = p\

maxi?^ Rkijp) tik t £k •

Pk t + n

If pk <p then the taxpayer evades and declares his minimal possible income as rk = 0 for each k = 1, n. Then

Rk pk(t + n)ik pk ck.

Depending on correlation between (t + n)ik and ck, function Rk can decrease or increase on the half-interval [0,p) and correspondingly can get its maximum in either 0:

max Rk = Rk (0) = 0,

pk

or when pk —>■ p-

lim Rk Rki.p') tik , c-k

- t + 7T

Pk ^P-

Remark 2. Due to the economic meaning of the parameters t, n and ck, it is rational to suppose that (t + n)ik > ck (penalties and taxes from the true income are not less, then the audit cost). Different situations, in which this condition is not hold, exist in practice. Among them there is expensive audit (because of its technical dificulties). But the case (t + n)ik < ck is not profitable for the tax authority in terms of this model (the audit gives only dead loss, i.e. Rk < 0). That’s why in the following reasonings it will be supposed that the condition (t + n)ik > ck is hold.

The maximal total expected income from taxation of n taxpayers is defined as a sum of maximal incomes from taxation of each taxpayer:

n

max R =y max Rk. (4)

p pk

k=i

Thus, the optimal (in order to maximize of income) strategies of taxpayers and tax authority were found. □

%

lk

i + 71

Figure3. Dependence the fc-th taxpayer’s expected profit bk on the probability of audit

Pk

Proposition 2. The situation (r^,p) is the unique Nash equilibrium in this game.

Proof. Obviously, there is no profit for any player in the case of deviation from their optimal strategies.

For example, consider the situation, when the tax authority chooses p as a strategy and one of the taxpayers chooses rk = r* — k. When the taxpayer’s strategy is r*, then his expected tax payment is uk = tik and his profit function is bk =

ik — tik. When the k-th player does not play optimal, then rk = r* — k, uk increases and bk decreases. Therefore the unilateral deviation of the situation (>’l,p) is not profitable for every taxpayer.

Analogical situation appears, when the tax authority plays nonoptimal, i.e. pk = p. In this situation R(pk) > R(p)-

These considerations correspond to the next defenition (Petrosyan and Zenkevich and Semina, 1998):

Figure4. Dependence the tax authority’s expected profit Rk on the probability of audit

Pk

Definition 1. Situation x* = (x*,.. .,x*,...,x‘n) in the game r is the situation of Nash equilibrium, if for every x* G X.,, and i = 1, n the next equation is hold:

Hi(x*,...,x*,...,x*n) > Hi(x*,...,xi,...,x*n),

where H is the profit function, xi is the i-th player’s strategy and Xi is the set of the i-th player’s strategies.

Thus, (r*k,p) is the Nash equilibrium. Its uniqueness follows from the uniqueness of the players’ optimal strategies. □

5. Other cases of penalties

5.1. The penalty is proportional to the difference between true and payed tax

As the penalty is proportional to the difference between true and payed tax, in the considered model it is defined as (1 + n)t(ik — rk). Then, the expected tax payment (1) of the fc-th taxpayer is

Uk = trk + Pk (1 + n)t(ik — rk), the expected payoff (7) of the taxpayer k is

bk = ik — Uk = ik — trk — Pk (1 + n)t(ik — rk).

The expected tax authority’s net income (3) in this case gets the form

R = ^2 Rk = ^2(trk + Pk(1 + n)t(ik - Tk) - PkCk).

k=1 k=1

While searching for the optimal players’ strategies and their profit functions, reasoning and results, got in the previous case, remain valid for p =

5.2. The penalty is restricted due to the given level of the agent’s minimal income in the case of his nonoptimal behavior

In this case (the third case in (Vasin and Vasina, 2002) the post-audit payments obviously belong to the segment [0, / — T(Ir) — /]. In terms of this model it means that

Uk € [trk, ik ~ *]•

Then the minimal income i should be defined. When it’s done, there is a possibility to search opimal strategies.

5.3. The post-audit payment is proportional to the revealed evaded income

In this case the expected tax payment (1) can be defined as

uk = trk + &d(ik Tk

where the second summand is the post-audit payment. If put t + n as 5d, then this case is absolutely analogical to the first case of the penalty taking considered.

5.4. Specific case: penalty is proportional to the square of difference between true and declared levels of the taxpayer’s income

As far as in this model the players are supposed to be risk-neutral, it is interesting how the players’ strategies will change if the penalty function changes significantly. As an example let’s consider the case of the penalty defined as n(ik — rk)2. It is the most interesting case from the mathematical point of view. The expected tax payment (1) of the taxpayer k is:

Uk = tTk + Pk (t + n)(ik — Tk)2, (5)

The players’ profit function is defined analogically to the previous cases. The expected payoff bk of the taxpayer k is:

bk = ik — Uk = ik — tTk — Pk (t + n)(ik — Tk )2.

The expected tax authority’s net income is:

n n

R =^2 Rk = ^2(uk — PkCk). (6)

k=1 k=1

Proposition 3. The optimal strategy of the tax authority (in order to maximize its income) is p = —, 1 ; the k-th taxpayer’s optimal strategy (in the same

2^/(t+n)ck

meaning) is r%{pk) = ik ~ 2pk(t+-x)-

Proof. The optimal strategy of the taxpayer k is the solution of the task of the minimization of the function (10). This strategy is found from the equation:

Uk(Tk) = °,

which is equivalent to

t — 2Pk(t + n)(ik — Tk) = 0. The solution of the last equation is

Tk (Pk) = ik —

2Pk (t + n)'

When every taxpayer’s behaviour is optimal, i.e. rk = r*k(jpk) for every k = 1, n, the tax authority’s expected income depends only on vector p, i.e.

R = R(p1,p2, ...,Pn).

Then (11) becomes

t2

R = J2Rk = J2(u(rl(pk)) -pkck) = J2(tik - ~PkCk).

k = 1 k=1 k = 1 Pk( + n)

The optimal strategy of the tax authority is the solution of the task of the maximization of function (11). This strategy is found from the equation:

Rk = 0,

that is

t2

— ck = 0.

4Pk(t + n)

Then the optimal strategy of the tax authority is

_ t

P = -----)

2s/(t + n)ck

The maximum of the tax authority’s income is defined as the sum of the maximal values of Rk (as in (4)), where

Remark 3. So, big penalty, as considered in this case, can be explained as the psychological pressure on those taxpayers, who evade taxation. They know that if their evasion is revealed they may become bankrupts. Taking this possibility into account it is necessary for the tax authority to set the penalty depending on the level of the agent’s minimal income in the case of his nonoptimal behavior (as it was done in the third case in (Vasin and Vasina, 2002)). This way of penalty taking is reasonable when (t + n)ik < ck or (1 + n)tik < ck (as were considered in the previous cases), i.e. when auditing is a priori not profitable.

Proposition 4. (Analogical to the case, when the net penalty is proportional to evasion) the situation (rk,p) is the unique Nash Equilibrium in this game.

Proof. Absolutely analogical to the proof of 2. □

6. Conclusion

Thus, the different cases of penalties are considered:

1. In the case, when the penalty is proportional to the difference between true and payed taxes, the £;-th taxpayer’s optimal strategy is r*k(pk) = 0, when pk < p, and r*k(pk) = ik, when pk > p\ the optimal strategy of the tax authority is p = -j-pjj:. The situation (rk,p) is the unique Nash equilibrium.

2. In the case, when the net penalty is proportional to the evasion, results, got in the previous case, remain valid for p =

3. In the case, when the post-audit payment is proportional to the revealed evaded income (uk = tTk + 5d(ik — uk)), t + n was put as 5d. Then, this case becomes absolutely analogical to the first case of the penalty taking considered.

4. In the case, when the penalty is proportional to the square of difference between

true and declared levels of the taxpayer’s income, the optimal strategy of the tax authority is p = ^ = S the A:-th taxpayer’s optimal strategy is r*k(jpk) =

ik — 2pk(t+ir) • situation (rk,p) is the unique Nash Equilibrium in this game. The reasonings for the using of this case of penalty taking is the realization of conditions (t+n)ik < ck or (1+n)tik < ck. To stave any taxpayer off bankruptcy the additional restriction on the penalty (as were considered in the third case in (Vasin and Vasina, 2002)) can be used.

Thus, the game theoretical model of interaction between taxpayers and tax authority is considered. The optimal players’ strategies (in order to maximize their income) and the Nash equilibrium are found.

References

Boure, V. and Kumacheva, S. (2005). A model of audit with using of statistical information about taxpayers’ income. St. Petersburg, ”Vestnik SPbGU”, series 10, 1—2, 140-145 (in Russian).

Chander, P. and Wilde, L. L. (1998). A General Characterization of Optimal Income Tax Enfocement. Review of Economic Studies, 65, 165-183.

Vasin, A. and Vasina, P. (2002). The optimization of tax system in condition of tax evasion: the role of penalty restriction. Consorcium of economic research and education (EERC), series Nauchnye doklady (in Russian).

Vasin, A. and Morozov, V. (2005). The Game Theory and Models of Mathematical Economics. - M.: MAKSpress, pp. 143-293 (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).

Petrosyan, L., Zenkevich, N. and Semina, E. (1998) The game theory. - M.: High School (in Russian).