Advances in Systems Science and Applications (2013) Vol.13 No.4 379-391

Purpose and Non-Purpose Resource Use Models in
Two-Level Control Systems

Olga I. Gorbaneva and Guennady A. Ougolnitsky
Southern Federal University, Russia

Abstract

In the paper a two-level control system consisting of one element in top level
and one element in bottom level is considered. Both levels have purpose use and
non-purpose use interests. The model of resource allocation between purpose and
non-purpose use interests for different classes of payoff functions is investigated.
The model is built as a two-person game where the Stackelberg equilibrium is
found. Analytical and numerical results are presented.
Keywords resource allocation, purpose use interests, non-purpose use interests,
Stackelberg equilibrium

1 Introduction

Many problems of the social-economic development are solved due to the federal
financing. The financing has different forms (grants, subsidies, assignments, cred-
its) and is always strictly purpose oriented, i.e. the allocated resources should
be spent for the pre-scribed needs only. There are legislative sanctions for the
non-purpose use of the federal financing. Nevertheless, the non-purpose use of
federal resources is widely spread and may be considered as a variety of the op-
portunistic behavior meeting the private interests of active agents[1].

The non-purpose resource use is closely connected with corruption, especially
with so-called “returns” when federal resources in some programs are assigned
in exchange of a bribe and only partly satisfy the prescribed social destination
being mostly used in the private interests of bribe-givers.

It is natural to consider the problem of non-purpose resource use from the
point of view of the concordance of interests in hierarchical control systems. This
permits to use the mathematical formalism of the hierarchical game theory [2],
the theory of incentives [3] and the theory of organizational systems [4-5]. In the
same time, namely the models of resource allocation in the hierarchical systems
with respect of their non-purpose use are scantily known and are analyzed by the
authors’ methodology [6].

In this paper the emphasis is on the dependence of the distribution of resources
between the purpose and non-purpose use on different classes of payoff functions
characterizing the common (social) interest and the private interests of the re-
source distributor and resource recipient.

In Section 2 the structure of investigation is presented. Sections 3 and 4 include
analytical and numerical results respectively. Section 5 concludes.



380 Olga I. Gorbaneva: Purpose and Non-Purpose Resource Use Models in ...

2 Structure of Investigation

Let us consider a two-level control system which consists of one element on the
top level A1 (resource distributor-she) and one element on the bottom level A2
(resource recipient-he). Without loss of generality we can equate to one the
number of resources on the top level. The distributor delegates a part of her
resources to the recipient for the purpose use, and the other part she keeps for
the non-purpose use. In turn, the bottom level divides the received resource
between the purpose use and his private non-purpose use. Both levels have their
shares in the purpose-use income and have their private payoff functions (Fig.1).

Fig.1 The structure of the modeled system

The model is built as a hierarchical two-person game in which a Stackelberg
equilibrium is sought [2]. The payoff functions of both players include two terms:
a non-purpose income and the respective share of the purpose-use income. So,
the payoff functions are:

g1(u1, u2) = a1(1− u1) + b1(u1, u2)c(u1, u2) → max
u1

,

g2(u1, u2) = a2(u1, 1− u2) + b2(u1, u2)c(u1, u2) → max
u2

subject to 0 ≤ ui ≤ 1.
and the following conditions for the functions a, b and c:

ai ≥ 0,
∂ai
∂ui

≤ 0,
∂ai

∂uj ̸=i
≥ 0,bi ≥ 0,

∂c

∂ui
≥ 0, i = 1, 2.

Here the subscript 1 relates to the parameters of the top level (the leader), and
the subscript 2 relates to the parameters of the bottom level (the follower);

ui - a part of resources assigned by the i-th level for the purpose use (respec-
tively the part 1- ui leaves for the private non-purpose use);



Advances in Systems Science and Applications (2013) Vol.13 No.4 381

gi - the i-th level payoff function;
ai - the i-th level function of his/her private interest;
bi - a share of the purpose-use income received by the i-th level;
c - a function of the purpose-use income of the whole system (society, organi-

zation).
As functions a and c power, exponential and logarithmic functions of the vari-

ables u1 and u2 are considered which are cumulative ones, i.e. a1=a1(1-u1),
a2=a2(u1 (1-u2)), c=c(u1u2). In this case the share u1u2 of resources is assigned
for the purpose use.

The relations a1=a1(1-u1), a2=a2(u1 (1-u2)) reflect the system’s hierarchical
structure. The non-purpose income of the top level does not depend on the part of
resources assigned by the bottom level for the purpose use, but the non-purpose
income of the bottom level depends on the share of resources received by him
from the top level.

The following distributions of the purpose-use income b are considered:
(1) uniform one, in particular, for n = 2

bi =
1

2
, i = 1, 2.

(2) proportional one

b1 =
u1

u1 + u2
, b2 =

u2
u1 + u2

.

In this paper it is assumed that b1 + b2 = 1.
The strategy of a player i is a part ui of his/her resources as-signed for the

purpose use. The top-level player moves first, i.e. she chooses a value u1 and
informs about it the bottom-level player who chooses his optimal reaction u2.

The aim of investigation is study of the influence of the relations between func-
tions a1, a2, b1, b2, c to the solution of the game (Stackelberg equilibrium).

The following types of non-purpose use functions were used:
- power function with an exponent smaller than one (a(x) = axα, 0 < α <

1, a > 0),
- linear function (a(x) = ax),
- power function with an exponent greater than one (a(x) = axk, k > 1, a > 0);
- exponential function (a(x) = a(1− e−λx), λ > 0, a > 0);
- logarithmic function (a(x) = a log2(1 + x), a > 0).
Almost all of the functions satisfy the conditions ∂a/∂x ≥ 0, ∂2a/∂x2≤0 (ex-

cept the second condition for the function a (x) = axk, k > 1). Similarly, the
following types of purpose use functions were used:

- power function with an exponent smaller than one (c (x) = cxα, 0 < α <
1, c > 0);



382 Olga I. Gorbaneva: Purpose and Non-Purpose Resource Use Models in ...

- linear function (c(x) = cx);
- power function with an exponent greater than one (c(x) = cxk, k > 1, c > 0);
- exponential function (c(x) = c(1− e−λx), λ > 0, c > 0);
- logarithmic function (c(x) = c log2(1 + x), c > 0).
Thirteen of the possible twenty five combinations of the functions a and c are

studied analytically, namely:
(1) combinations of the one-type functions (both functions a and c are power,

exponential, or logarithmic ones);
(2) combinations of any non-purpose use function with linear purpose-use func-

tion;
(3) combinations of any purpose-use function with non-purpose use linear func-

tion.
Six of other twelve cases are investigated numerically.

3 Analytical Investigation of the Different Classes of Models

First, let’s consider the following parameterization:

a1(u1, u2) = a1(1− u1), a2(u1, u2) = a2u1(1−u2),

c(u1, u2) = clog2(1 + u1u2), b1 = b, b2 = 1− b.

In this case the payoff functions are

g1(u1, u2) = a1(1− u1) + bclog2(1 + u1u2) (1)

g2(u1, u2) = a2u1(1−u2) + (1− b)clog2(1 + u1u2) (2)

subject to 0 ≤ ui ≤ 1, i = 1, 2.
Omitting the calculations, consider each branch of the Stackelberg equilibrium

separately:
I. u = (0; 0) , if a2 > (1− b) c/ln2 or a1 > bc/ln2 (Fig.2), i.e. for one of the

players the non-purpose resource use is much more profitable than the purpose-
use activity; therefore, it is disadvantageous for him/her to assign resources for
the purpose-use activity, and in this case it is also disadvantageous for the other
player. The payoffs in this case are equal to: g1 = a1, g2 = 0.

II. u = (1; 1) , if a2 < (1− b) c/ln2 and a1 < bc/ln2 (Fig.2), i.e. for both
players the purpose-use activity is much more profitable, and each of them as-
signs all resources for it. The payoffs are equal to: g1 = bc, g2 = (1− b) c.

III. u = (bc/ (a1ln2)− 1; 1), if the conditions bc/2ln2 < a1 < bc/ln2 and
a2 < a1 (1− b) /b are satisfied (Fig.2), i.e. for the top-level player it is profitable
to assign only a part of her resources for the purpose use because her incomes
from both activities are comparable, meanwhile for the bottom-level player it is
profitable to assign all his resources to the purpose-use activity. The payoffs are



Advances in Systems Science and Applications (2013) Vol.13 No.4 383

Fig.2 Equilibrium outcomes in the game(1) - (2)

equal to:

g1 = 2a1 −
bc

ln 2
+ bclog2

(
bc

a1 ln 2

)
,

g2 = (1− b)clog2

(
bc

a1 ln 2

)
IV. u = ((1− b) c/ (a2ln2)− 1; 1), if (1− b) c/2ln2 < a2 < (1− b) c/ln2 and

a2 > a1 (1− b) /b (Fig.2), i.e. for both players it is profitable to assign only a part
of their resources for the purpose-use activity because their incomes from both
types of activities are comparable. But the leader gives to the follower exactly
the fixed number of resources which he planned to allocate for the purpose use,
therefore compel-ling him to assign all his resources for the purpose use. The
players’ payoffs are equal to

g1 = 2a1 −
a1(1− b)c

a2 ln 2
+ bclog2

(
(1− b)c

a2 ln 2

)
,

g2 = (1− b)clog2

(
(1− b)c

a2 ln 2

)
.

Second, consider the following parameterization:

a1(u1, u2) = a1(1− u1)
k, a2(u1, u2) = a2(u1(1−u2))

k,

c(u1, u2) = c(u1u2), b1 = b, b2 = 1− b.



384 Olga I. Gorbaneva: Purpose and Non-Purpose Resource Use Models in ...

Then the payoff functions have the form

g1(u1, u2) = a1(1− u1)
k + bc(u1u2) → max

u1

(3)

g2(u1, u2) = a2(u1(1−u2))
k + (1− b)c(u1u2) → max

u2

(4)

subject to 0 ≤ ui ≤ 1, i = 1, 2.
The Stackelberg equilibrium outcomes are the following :

ū =

{
(1; 1), (a1 < bc)& (a2 < (1− b)c)
(0; 0), (a1 > bc) ∨ (a2 > (1− b)c)

Consider the cases separately (Fig.3):

Fig.3 Equilibrium outcomes in the game(3) - (4)

I. u = (0; 0), if a2 > (1− b) c or a1 > bc, i.e. for one of the players the non-
purpose resource use is much more profitable than the purpose-use one, therefore
it is disadvantageous for her/him to finance the purpose-use activity. The payoffs
are: g1 = a1, g2 = 0.

II. u = (1; 1), if a2 < (1− b) c and a1 < bc, i.e. for both players the pur-pose-
use activity is much more profitable, and each of them assigns all resources for
it. The payoffs are equal to: g1 = bc, g2 = (1− b) c.

When even one of the functions of purpose or non-purpose re-source use is
power with an exponent greater than one (and the other function is the same
or linear) then it is advantageous for both players to allocate resources or only
to the purpose use (altruistic strategy), or only to the non-purpose use (egoistic
strategy).



Advances in Systems Science and Applications (2013) Vol.13 No.4 385

Third, consider the following case:

a1(u1, u2) = a1(1− e−λ(1−u1)), a2(u1, u2) = a2(1− e−λu1(1−u2)),

c(u1, u2) = c(1− e−λu1u2), b1 = b, b2 = 1− b.

Then the payoff functions have the form

g1(u1, u2) = a1(1− e−λ(1−u1)) + bc(1− e−λu1u2) (5)

g2(u1, u2) = a2(1− e−λu1(1−u2)) + (1− b)c(1− e−λu1u2) (6)

subject to 0 ≤ ui ≤ 1, i = 1, 2.
Let’s consider all Stackelberg outcomes separately (Fig.4):

Fig.4 Equilibrium outcomes in the game(5) - (6)

I. u = (0; 0), if
(
a1 > bceλ

)
&(a1 > (1− b) c) or a1 > bceλ

√
a2

(1−b)c , i.e. for

one of the players the non-purpose resource use is much more profitable than
the purpose-use one, therefore it is disadvantageous for her/him to finance the
purpose-use activity. The payoffs are: g1 = a1(1− e−λ), g2 = 0.

II. u = (1; 1), if a1 < bce−λ and a2 < (1− b) ce−λ, i.e. for both players the
purpose-use activity is much more profitable, and each of them assigns all re-
sources for it. The payoffs are equal to:g1 = bc(1− e−λ), g2 = (1− b)c(1− e−λ).

III. u =
(
1; 12 − 1

2λ ln a2
(1−b)c

)
, if a1 < b

2

√
a2c
1−be

−λ
2 and (1− b) ce−λ < a2 <

(1− b) ceλ , i.e. for the top-level player it is profitable to assign all her resources
to the purpose-use activity, meanwhile for the bottom-level player it is advanta-
geous to divide his resources. Payoffs are the following:



386 Olga I. Gorbaneva: Purpose and Non-Purpose Resource Use Models in ...

g1 = bc

(
1− e

−λ
(

1
2
− 1

2λ
ln

a2
(1−b)c

))
= b

(
1− e−

λ
2

)√ a2c

(1− b)
,

g2 = (1− b)
(
1− e−

λ
2

)√ a2c

(1− b)
.

IV. u =
(
1
2 − 1

2λ ln a1
bc ; 1

)
, if bce−λ < a1 < bceλ and a2 < (1 − b)

√
bc
a1
ce

λ
2 , i.e.

the situation is opposite to the previous one. The payoffs are:

g1 = a1 − a1e
−λ

2

√
bc

a1
+ bc− e−

λ
2

√
bca1,

g2 = (1− b)c

(
1− e−

λ
2

√
a1
bc

)
.

V. u =

(
2
3 − 2

3λ ln 2a1
b

√
1−b
a2c

;
λ−ln

2a1a2
b(1−b)c

2
(
λ−ln

2a1
b

√
1−b
a2c

)
)
, if the conditions b

2

√
a2c
1−be

−λ
2 <

a1 <
b
2

√
a2c
1−be

λ and (1− b)
√

2a1
bc e

−λ
2 < a2 <

bc2

2a1(1−b)e
λ are satisfied, i.e. for both

players it is profitable to divide their resources. The payoffs are omitted due to
their tediousness.

At last, consider the following case:

a1(u1, u2) = a1(1− u1), a2(u1, u2) = a2 (u1(1−u2)) ,

c(u1, u2) = c(u1u2)
α, b1 = b, b2 = 1− b.

Then the payoff functions have the form

g1(u1, u2) = a1(1− u1) + bc(u1u2)
α → max

u1

(7)

g2(u1, u2) = a2 (u1(1−u2)) + (1− b)c(u1u2)
α → max

u2

(8)

The Stackelberg equilibrium has the form (Fig.5):

ū =


(1; 1), (a1 < bcα)&(a2 < (1− b)cα),(

1−α

√
αbc
a1

; 1
)
, (a1 > bcα)&(a2 < a1),(

1−α

√
(1−b)cα

a2
; 1

)
, (a2 > (1− b)cα)&(a2 > a1).

In this case the egoistic strategy is disadvantageous for both players. Besides,
the leader is always able to compel the follower to assign all his resources to the



Advances in Systems Science and Applications (2013) Vol.13 No.4 387

Fig.5 Equilibrium outcomes in the game(7) - (8)

purpose use. The payoffs are:

g1 = a1 − a1

(
α(1− b)

a2

) 1
1−α

+ b

(
αα(1− b)αc

a2α

) 1
1−α

,

g2 =

(
αα(1− b)c

a2α

) 1
1−α

.

When the purpose-use function is linear, and the function of non-purpose use is
power with exponent smaller than one, the altruistic strategy is profitable for the
bottom-level player. The egoistic strategy is disadvantageous for both players.

The thirteen analyzed cases are grouped by the structure of equilibrium out-
comes of the game:

I. One outcome when both functions of purpose and non-purpose use are power
with exponent smaller than one. In this case it is profitable for both players to
divide their resources between purpose and non-purpose use.

II. Two outcomes (0; 0) and (1; 1) (Fig.3) when:
• The function of non-purpose use is power with exponent smaller than one,

and the function of purpose use is linear;
• Both functions are linear or power with exponent greater than one, in any

combination.
III. Three outcomes (Fig.5) when the non-purpose resource use function is lin-

ear, and the purpose-use function is power with exponent smaller than one. In
this the altruistic strategy is profitable even for one player.

IV. Four outcomes (Fig.2) in cases when one of the functions is linear, and the



388 Olga I. Gorbaneva: Purpose and Non-Purpose Resource Use Models in ...

other is logarithmic.
V. Five outcomes (Fig.4) when
• Both functions are linear or exponential in any combination except the case

when they are both linear.
• Both functions are logarithmic.

4 Numerical Analysis

Let’s consider an example of the numerical analysis for the following parameter-
ization:

a1(u1, u2) = a1(1− u1)
α, a2(u1, u2) = a2(u1 (1− u2))

α,

c(u1, u2) = c(1− e−λu1u2), b1 = b, b2 = 1− b.

In this case the game has the form

g1(u1, u2) = a1(1− u1)
α + bc(1− e−λu1u2) → max

u1

(9)

g2(u1, u2) = a2(u1 (1− u2))
α + (1− b)c(1− e−λu1u2) → max

u2

(10)

To find an optimal strategy of the bottom-level player let’s calculate the deriva-
tive of the function g2 with respect to the variable u2 and equate it to zero:

∂g2
∂u2

(u1, u2) = − a2αu1
α

(1− u2)
1−α + λu1(1− b)ce−λu1u2 = 0 (11)

Let’s prove that the method of bisection is applicable for the solution of (11).
Note that the second derivative of the function g2 with respect to the variable u2
is negative

∂2g2
∂u22

(u1, u2) =
a2α(1− α)u1

α

(1− u2)
2−α − λ2u1

2(1− b)ce−λu1u2 < 0,

and therefore the function ∂g2/∂u2 is monotone.
Now let’s calculate the signs of ∂g2/∂u2 in the ends of the segment [0; 1].

∂g2
∂u2

(u1, 0) = −a2αu1
α + λu1(1− b)c (12)

∂g2
∂u2

(u1, u2) →
u2→1−

−a2αu1
α

0+
+ λu1(1− b)ce−λu1u2 →

u2→1−
−∞ (13)

If (12) is positive then the equation can be solved by bisection, and the solution
will be the point of maximum due to the negativity of the second derivative. If
(12) is negative then the method of bisection is not applicable but the left side
of the equation is monotone and therefore it is negative in the segment [0; 1], so
the function g2 de-creases and the point of maximum is u2 = 0. Thus,



Advances in Systems Science and Applications (2013) Vol.13 No.4 389

u2
∗ =

{
0, −a2αu1

α + λu1(1− b)c < 0,
∈ (0; 1), −a2αu1

α + λu1(1− b)c > 0.

The top-level player can use the information to impel the bottom-level player to
choose a non-zero strategy u2 > 0. It is necessary for this to assure the condition
−a2αu1

α+λu1 (1− b) c > 0. Solving the inequality with respect to u1 we rewrite

the condition as u1 > 1−α

√
a2α

λ(1−b)c . The top-level player can satisfy the condition

only if 1−α

√
a2α

λ(1−b)c < 1 , or a2 <
(
λ(1−b)c

α

)1−α
. If the top-level player cannot

choose the strategy then the bottom-level player chooses u2 = 0. In this case
g1(u1, 0) = a1(1− u1)

α. As far as the function g1 decreases with respect to u1 we
receive u1 = 0.

Let’s summarize:

I. If a2 >
(
λ(1−b)c

α

)1−α
then the leader cannot influence to the follower and

u2 = 0, therefore u1 = 0. This case takes place when the effect from non-purpose
activity on the bottom level is essentially greater than the effect from his purpose-
use activity.

II. If a2 <
(
λ(1−b)c

α

)1−α
then the leader can impel the follower to assign a part

of his resources for the purpose-use activity by choosing u1 > 1−α

√
a2α

λ(1−b)c . This

case takes place when the effect from purpose-use activity on the bottom level is
essentially greater than the effect from his non-purpose use activity.

The case of parameterization

a1(u1, u2) = a1log2(2− u1), a2(u1, u2) = a2log2(1 + u1(1− u2)),

c(u1, u2) = c(u1u2)
α, b1 = b, b2 = 1− b.

is considered similarly. The payoff functions have the form

g1(u1, u2) = a1log2(2− u1) + bc(u1u2)
α, (14)

g2(u1, u2) = a2log2(1 + u1(1− u2)) + (1− b)c(u1u2)
α (15)

The results of analysis are presented in Fig.6.

If a1 < αbcln2 and a2 > α (1− b) cln2 then u1 =
1−α

√
(1−b)cα ln 2

a2
, u2 = 1.

The payoffs of the players are:

g1

(
1−α

√
(1−b)cα ln 2

a2
, 1

)
= a1log2

(
2− 1−α

√
(1−b)cα ln 2

a2

)
+ bc

(
1−α

√
(1−b)cα ln 2

a2

)α

g2

 1−α

√
(1− b)cα ln 2

a2
, 1

 = (1− b)c

(
(1− b)cα ln 2

a2

) α
1−α

.



390 Olga I. Gorbaneva: Purpose and Non-Purpose Resource Use Models in ...

Fig.6 Equilibrium outcomes in the game(14) - (15)

If a1 < αbcln2 and a2 < α (1− b) cln2 then the altruistic strategy is advanta-
geous for both players: u1 = 1, u2 = 1. The payoffs are:g1 (1, 1) = bc, g2 (1, 1) =
(1− b) c.
If a1 > αbcln2 then it is profitable for the top-level player to allocate for the

purpose use a part of her resource u1 ∈
(
0;min

{
1−α

√
(1−b)cα ln 2

a2
; 1

})
.

5 Conclusion

In this paper the problem of non-purpose resource use is considered from the
point of view of analysis and design of the control mechanisms providing the
concordance of interests in the hierarchical (two-level) systems. Interests of the
agents are described by their payoff functions including two terms: profit from
the purpose and non-purpose resource use respectively. Different classes of the
payoff functions are studied. The top-level control agent (resource distributor) is
treated as leader, and the bottom-level agent (resource recipient) as follower what
results in the concept of Stackelberg equilibrium. The analytical and numerical
investigation allows for the following conclusions.

• When both purpose and non-purpose interests functions are power (k < 1)
then it is advantageous for both players to invest a part of their resources to the
purpose use and the other part to the non-purpose use;

• When even one of the payoff functions is power (k > 1), and the other is also
power (k > 1) or linear then it is advantageous for both players or assign the re-
sources only for the purpose use (the altruistic strategy), or only for non-purpose
use (the egoistic strategy);

• In other cases the following situations are possible:



Advances in Systems Science and Applications (2013) Vol.13 No.4 391

A) When the payoff from the non-purpose activity of a player is much greater
than the payoff from the purpose activity then the egoistic strategy is advanta-
geous;

B) When the payoff of non-purpose activity of both players is much smaller
than the payoff from the purpose activity then the strategy of pure altruism is
advantageous;

C) When the payoffs of the purpose and non-purpose activities are comparable
then it is profitable to divide the resources between the purpose and non-purpose
interests.

Acknowledgements
The work is supported by RFBR (projects 12-01-00017, 12-01-31287) and by
Southern Federal University.

References

[1] Williamson O.E. (1985), The Economic Institutions of Capitalism, Free
Press.

[2] Basar T, Olsder G.Y. (1999), Dynamic Noncooperative Game Theory, SIAM,
Philadelphia, pp.536.

[3] Laffont J.J, Martimort D. (2002), The Theory of Incentives: The Principal-
Agent Model, Princeton University Press, pp.421.

[4] Prof. D. Novikov (2013),Mechanism Design and Management: Mathematical
Meth-ods for Smart Organizations, N.Y: Nova Science Publishers, pp.163.

[5] Novikov D. (2013), Theory of Control in Organizations, N.Y: Nova Science
Publishers, pp.341.

[6] Ougolnitsky G. (2011), Sustainable Management, N.Y: Nova Science Pub-
lishers, pp.285.

Corresponding author
Guennady A. Ougolnitsky can be contacted at: ougoln@gmail.com