Analysis of Incentives Influence on Great Social
Groups’ Behavior in Stackelberg game
Mikhail Geraskin
Department of mathematical methods in Economics
Samara National Research University
Samara, Russia
innovation@ssau.ru
Abstract—We consider the encouragement of the great social rational increase in the volunteer activities. The meta-agent
groups (agents) to the socially optimal behavior by an example of chooses the coefficients of the incentive function from the
the volunteering. We search for the optimal actions vector of following condition: if the incentive is equal to the average
these social groups, i.e., the equilibrium in the incentives wage, then at least half of the available time fund of citizens is
allocation game. On the basis of the game-theoretic model with allocated for volunteering.
Stackelberg leadership, under conditions of the awareness
asymmetry, the possible equilibrium variants are investigated. In On the basis of this model, the equilibrium conditions were
the case of a linear decreasing incentive function and linear cost derived, and the formulas for calculating the socially optimal
functions of the agents, Nash equilibrium conditions in actions vector were obtained. In this case, when choosing
Stackelberg game are proved. For various types of the agents’ actions, the social groups do not take into account each other's
tendency to altruism, the analytical formulas for calculating the behavior. In the game theory, this condition was called Cournot
equilibria are derived. On the basis of the Russian population hypothesis [19], and it expresses the symmetry of the players
statistics, we simulate the behavior of the volunteers groups. due to the a priori information unawareness of the player about
the actions of other players (hereinafter, environment).
Keywords—incentive system, Stackelberg game, Nash However, in reality, some social groups may be informed about
equilibrium, volunteer the activity of other social groups, which leads to a situation of
I. INTRODUCTION the awareness asymmetry, therefore, in the game, the
asymmetry of the equilibrium arises. In the case of the
The encouragement in social systems is used to awareness asymmetry, the game of the social groups describes
purposefully change of the social groups’ behavior patterns. the behavior of agents, who are informed about the optimal
For this purpose, the incentives are calculated from the choice of the environment; such agents become Stackelberg
optimality conditions of the social criteria, which are leaders [20]. In this case, the environment has the followers
established by the governments of these systems. Most often, at status, whose behavior is described by Cournot hypothesis.
the state level, the goal of the incentives is to encourage
citizens to perform actions that maximize the collective utility Further article is structured as follows: the description of
function. Hereinafter, these actions are referred to the socially the agent incentive system according to [8], the analysis of the
optimal actions or the volunteering. This encouragement is principles of choosing the actions in Stackelberg game, the
caused by the need to overcome the trends of individual investigation of the stratifying the agents into leaders and
rationalism [1,2], and it is expressed in the implementation of followers, the formulation of the equilibrium model, the
the social national programs [3,4], including the information development of analytical formulas for calculating the
systems development programs [5]. equilibrium in Stackelberg game
For the practical implementation of the incentive system, II. METHODS
methods and algorithms were developed [6], and the game- We consider as the object of stimulation the social system,
theoretic model of the social groups (hereinafter, agents) for example, citizens of a country or employees of a
behavior was formulated [7] in the form of the non-cooperative corporation, which are divided into K groups (agents). These
game. The model was based on a compensatory linearly agents differ by attribute that affects the effectiveness of
decreasing stimulation function, for which the conditions of the stimulation, which is further called the agent type parameter. In
individual rationality, Pareto efficiency, and non-manipulation other words, all individuals in the group k have a predictable
were proved [8–17]. identical reaction to equal incentives. The number of
The model [7] describes the dependence of the citizen’s individuals in the group k is indicated n k , k K , the symbol K
individual utility function on the distribution of his disposable denotes a set of social groups and the number of elements of
time fund, the degree of propensity towards the altruism and this set.
the incentive, i.e., the price of the socially optimal action. In
turn, the incentive is calculated as a decreasing function of the The agent’s type parameter is determined by his altruism,
total number of all volunteers’ actions. Based on the i.e., the propensity to charity, and it is estimated by the
optimization of the individual utility functions of all citizens, coefficient of the charity time elasticity with respect to the
the model enables us to calculate the vector of socially optimal disposable time fund ak 0 ,1 . The agent is more inclined to
actions, which satisfies the interests of all citizens, i.e., it is altruism, if the coefficient ak is closer to one. Actual values
Nash equilibrium. In addition, the model takes into account the of the agent’s altruism coefficient are estimated from the
interests of the state (meta-agent), which is aimed at the
Copyright © 2020 for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0)
Data Science
ak The effectiveness of the incentive system is evaluated
following function ak D ,k K , which describes the
according to the following individual agent’s utility function:
dependence of the time interval of socially optimal actions a k
in the absence of any stimulation on the available time fund D.
U k a k p a A p d
1 ak
a , k K ,
k (3)
On the basis of this function and taking into account the where U is the continuously differentiable agent’s utility
statistics of the volunteer time, the altruism coefficient is
calculated by the following formula function. Function (3) is used on the basis of the following
hypothesis of the altruism influence on the agent’s behavior: an
ln a k increase in the propensity to altruism leads to a decrease in the
ak a , k K , ak 0 ,1 a k 1 . (1)
ln D utility of wages.
The incentive system includes the subsystem for recording The problem of searching for Nash equilibrium vector A
the actions a k and the subsystem for paying incentive. The from the maximization of function (3) under condition (2) in
the case of a constant number of the social groups (i.e.,
incentive is equal to the product of the incentive price p k and
nk
the action value, i.e., p k A a k . The incentive price is 0 k K ) enable us to obtain the following system of
a k
calculated on the basis of the following incentive function [7]: equilibrium conditions [8]:
p a A b1 b 2 n k a k , k K , b1 , b 2 0 , (2)
1 ak
kK
b 1 b 2 n j a j b 2 n k a k 1 kj p 0, k K , (4)
d
where A a k , k K is the vector of the socially optimal j K j K \ k
actions; b1 , b 2 are constant coefficients that are independent of subject to
the vector A in the current period. These coefficients are kj 2 , (5)
calculated by formulas that depend on the vector j K \ k
A 0 a0k , k K of the agents’ actions in the previous a j
period :1 where k j is the conjectural variation in the equation of
a k
b1 p d
A0
, b2
pd
, A0 a0k , A
D
D
nk , (2а) the agent k, i.e., the expected change in the action of the agent j
D D
A A0 A A0 kK 2 k K in response to a single increase in the action of the agent k.
where p d is the price (tariff rate) of the working time. It The conjectural variation expresses the effect of the agent’s
awareness asymmetry on the resulting equilibrium (i.e., the
should be noted that the coefficients b1 , b 2 are calculated
according to formulas (2a), if the incentive fund is not fixed, actions vector A * ), which is the solution of system (4). The
and the administration (state) is aimed at ensuring a balance symbol «*» indicates the equilibrium values. In the case of
between the working and the volunteer time. In the case of the Cournot game, when kj 0 j , k K , all agents
fixed incentive fund (let is equal to F), the coefficients of the symmetrically do not change the actions in response to the
incentive function are calculated by the following formulas [7]: environment’s actions, therefore, the asymmetry of the
min min
resulting equilibrium [7] depends on the differentiation of the
F n n F n n
b1 1 , b2 ,n nk , (2b) agents by types. Further, we investigate the case of Stackelberg
2
A0 2 A0 2 kK game (i.e., kj 0 j , k K ), when some agents (leaders) may
choose the actions taking into account the principles of
where min is the minimum guaranteed incentive2. choosing actions by other agents (followers). This is another
reason for the asymmetry of the resulting equilibrium, and it is
the research question of our study.
III. RESULTS AND DISCUSSION
1
Formulas (2a) are obtained from the following conditions:
1) with a low level of socially optimal actions A 0 , the administration sets a We introduce the following notation: q k n k a k is the
high incentive price, which is equal to the average wage p d ; 2) if the aggregate action of the social group k; q k n k a k is the
disposable time fund is divided equally between the working time and the aggregate action of the environment; q q k is the
charity time (i.e., D/2), then the price of the incentive is zero. Under these kK
conditions, the system of equations b1 b 2 A 0 p d , b1 b 2 A D 0 leads
to solution (2a). F n
min
2 A0 n A0
A0 u k , A 0 / n u . Therefore, b
2
Formulas (2b) are obtained from formulas [7] kK
1
A0 2 A0 n
F n
min 2u uk F n
min
, u 1 F n
min
u k , as a result of
min
b1
kK
, b2 n F n min
F n
1 , b2
n .
uk 2u 2u u k n kK
k K k K A0 2 A0 2
A0 2
2 A0
the following transformations. The following notation is used: a k u k , n
VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 138
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aggregate action of all agents in the system. The environment qk
includes all agents except the agents of the social group k. In where the following relation is taken into account: a k .
nk
this case, the system of equations (4) may be transformed as
follows: A substitution of formula (9) into the first equation of system
(8) leads to the explicit reaction function of the second agent:
1
b1 b 2 q k q k b 2 q k 1 kj p d ak 0 , 1 q2
. (10)
q 1 L
j K \ k n1
2
1 ak
2n2
b1 p
d
2qk qk qk kj 0 , In formula (10), the index «L» is introduced for the first
b2 j K \ k b2
agent, because according to the accepted assumption, he is the
1 ak leader.
b1 p
d
2qk qk kj q k 0 , Taking into account the introduced notation and the
j K \ k b2
transformations, we write the reaction system (8) in the case of
and we write the system in the following resulting form: the first agent’s leadership as follows:
L qF F qL
qL ,qF . (11)
f k q k , q q k 2 kj q k k 0 , k K , (6) nL 2
2
j K \ k 2nF
1 ak
b1 p d Thus, the process of the agents’ stratification into leaders
where k . The function f k q k , q is the and followers proceeds in accordance with the sequence, which
b2
is demonstrated in Fig. 1.
reaction function of the agent k, because it expresses implicitly
the dependence of the optimal action of the agent k on the Because, in the considered social system, the equilibrium
actions of the environment. action vector q *L , q *F is defined as the intersection point of
We describe the leader appearance process for a social reactions (11) on the plane q L , q F , the ratio of equilibrium
system consisting of two agents: *
qL
actions depends on the ratios of the slopes and free terms
f 1 q 1 , q q 1 2 12 q 2 1 0 , (7) *
qF
f q , q q 2 q 0 .
2 2 2 21 1 2
of reaction (11). An analysis of reactions (11) is illustrated in
The reaction functions may be expressed explicitly from Fig. 2.
system (7) as follows: We introduce the relative indicators of the system state: η is
1 q2 2 q1 the ratio of the leaders group number to the followers group
q1 ,q2 . (8) number, β is the ratio of the constants in equations (6), μ is the
2 12 2 21
ratio of the agents’ type parameters. These indicators are
If the second agent is not informed about the reaction calculated by using the following formulas:
function of the first agent, then, in accordance with the Cournot
nL L aL
hypothesis, in the second equation of system (8), the , , . (12)
nF F aF
conjectural variation is zero (i.e., 21 0 ), therefore this
equation may be written in the form: Given these notations, solving of system (11) allows us to
write the following expressions of the Stackelberg equilibrium
2 q1
q 2F . (8а) vector coordinates:
2
In formula (8a), the index «F» is introduced for the second 2 F L
2 L F 2
agent, because, according to the accepted assumption, he is the *
qL
*
,qF . (13)
follower. 3 3
If at the same time the first agent is informed about the The Stackelberg equilibrium vector is indicated in Fig. 2 by
reaction function (8a) of the second agent, then he calculate the a point E S in contrast to the Cournot equilibrium vector,
conjectural variation 12 as follows: which is indicated by a point E K .
q The following assertion, the proof of which is placed in the
2
a 2 n 2 n1 q 2 n1 appendix, defines the conditions for the equilibrium existence
12 , (9)
a1 q n 2 q1 2n2 in the system.
1
n 1
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The various variants of the approximating function
b1
are investigated in Fig. 3. With p d 100 and 1 ,1 , the
pd
index of power has the following limitation 0 , 3 .
β 2,4
2,2
2,0
1,8
1,6
1,4
1,2
1,0
0,8
0,6
0,4
0 1 2 3 4 5 6 7 8 9 10
μ
b1/pd=2 b1/pd=4 b1/pd=6 b1/pd=8 b1/pd=10 b1/pd=1,1
Fig. 3. Analysis of the approximating function .
On the basis of Assertion 1, we may derive the following
practical conclusion.
Corollary 1: for the equilibrium existence in the social
Fig. 1. Diagram of agents' stratification process. system, the number of the leaders should not exceed the
number of the followers by more than 4 times.
We introduce the indicator of the equilibrium actions
unevenness * , which is determined by the following formula:
qL / qF .
* * *
(15)
The following assertion, the proof of which is placed in the
appendix, estimates the influence of the state parameters ratio
on the equilibrium actions unevenness.
Assertion 2. In the social system, an increase in the ratio of
the leaders number to the followers number η increases
(decreases) the equilibrium actions unevenness for a given
value of β according to the following rule:
0 0 ,5 ,
*
Fig. 2. Graphical analysis of equilibria in social system. (16а)
0 0 ,5 ;
Assertion 1. The social system is in the equilibrium, i.e., the
an increase in the ratio β increases (decreases) the
equilibrium actions are non-negative q *L 0 q *F 0 , if the equilibrium actions unevenness for a given value of η
following conditions are satisfied: according to the following rule:
1 1 0 3,
*
2 3 , and 2 3 , (14) . (16b)
2 2 2 2
0 3;
and if the approximating function of the following form
an increase in the factor η more (less) affects the change in
, 0 1 is exists, then conditions (14) have the the equilibrium actions unevenness than an increase in the
form: factor β, under the following conditions:
ln 0 , 5 ln 2 0 , 5 ln 0 , 5 ln 2 0 , 5
*
*
2 1
*
2 1
e
e
3 , and e
e
3 .(14а) if 3 and if 3 . (16с)
2 2
On the basis of Assertion 2, we formulate the following
practical conclusions.
Corollary 2. In the social system
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b1 TABLE I. CHARACTERISTICS OF SOCIAL GROUPS OF VOLUNTEERS IN 2016
1) under the conditions p d 100 and 1 ,1 , an
pd Groups
Parameter Total
increase in the leaders group number n L in comparison with 1 2
the followers group number n F leads to a shift of the Population n k , thousand 1435 997 438
equilibrium actions unevenness towards the leaders, if the Average duration of volunteer
2.35 23.0
propensity to altruism of the followers exceeds this indicator of activities per week a k , hours 8,64
the leaders by more than 10 times (i.e., 0 ,1 ); Aggregate duration of volunteer
activities per week A 0 , 2343 10055
2) an increase in the leaders’ propensity to altruism L in thousand hours 12398
comparison with this indicator of the followers F leads to a Propensity to altruism ak 0.18 0.66
shift of the equilibrium actions unevenness towards the leaders, Type parameter α 55168 80360
if the number of the followers is more than 3 times the number
of the leaders. q2=qL
60000
We simulate equilibrium (14) and sensitivity indicators (16)
50000
by an example of the social groups of Russian volunteers, the
number of which in 2016 was 1.435 million, or about 1% of 40000
the population3. The volunteers were divided into 9 groups
according to the propensity to altruism [7]. In our case, we 30000
divide the volunteers into 2 groups. The type parameters are
20000
calculated (Table 1) with the following constant values: D=112
hours per week, p d 240 rub. per hour. Into the leaders 10000
group (the second group), we combine the groups 2–9 from the
0
article [7], because the numbers of these groups individually
0 10000 20000 30000 40000 50000 60000
are small in comparison with the first group. The coefficients q1(η=1)=qF(η=0,44)=qF(η=1) qL(η=0,44) q1=qF
q2(η=1) qL(η=1)
of the incentive function calculated by formulas (2a) are
Fig. 4. Analysis of Cournot equilibrium and Stackelberg equilibrium.
b1 284 , b 2 0 , 0035 . The ratio of the leaders number to the
followers number, calculated by formula (12), is η=0.44. ξ
10
In this system, if the number of social groups is equal (i.e.,
when η=1), the Cournot equilibrium is shifted toward the 5
second agent, which has the higher propensity to altruism
(Fig. 4). The Stackelberg equilibrium at η=1 leads to greater 0
unevenness toward the second agent (i.e., the leader), and at
η=0.44 this equilibrium, on the contrary, shifts toward the first -5
agent, the group of which has the predominant number. In all
these cases, the aggregate equilibrium actions significantly -10
exceeds the actual indicator A 0 (Table I), which is a
consequence of the stimulation effect. -15
0 0,25 0,5 0,75 1 1,25 1,5
β
Fig. 5 illustrates features (16) of the Stackelberg equilibria. η=0,5 η=1 η=2 η=4
According to conditions (16a), with an increase in the ratio of Fig. 5. Analysis of Stackelberg equilibrium sensitivity to state parameters.
the leaders number to the followers number η, in the case of
0 , 5 , the equilibrium actions unevenness grows, and in the IV. CONCLUSION
* We investigate the behavior of the volunteers social groups.
case of 0 , 5 , the parameter decreases. According to
conditions (16b), in the case of η <3, an increase in the The study of the game-theoretic model in the framework of the
parameter β causes an increase in the equilibrium actions Stackelberg game leads to the following conclusions. First, the
equilibrium in the social system exists if the number of the
unevenness * , and in the case of η>3, this leads to a decrease leaders group does not exceed the number of the followers
in the parameter * . The values of the parameter * in the group by more than 4 times. Second, in the real conditions, an
increase in the number of the leaders group in comparison with
negative half-plane correspond to the case of non-existence of the number of the followers group leads to an increase in the
the equilibrium according to conditions (14) in the case of η<3 equilibrium actions unevenness towards the leaders, if the
for 0 , 5 or for 2 , and in the case of η>3 for followers’ propensity to altruism exceeds this indicator of the
2 leaders by more than 10 times. Third, an increase in the
0 ,5 . leaders’ propensity to altruism in comparison with this
indicator of the followers leads to an increase in the
3
Labor and Employment in Russia 2017: Stat. Sat. / Rosstat M., 2017.
equilibrium actions unevenness towards the leaders if the
http://www.gks.ru/free_doc/doc_2017/trud_2017.pdf
VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 141
Data Science
number of the followers is more than 3 times the number of the follows. A comparison of the modulus of these expressions
leaders. * *
2 1
demonstrates that if 3 , therefore,
Proof of Assertion 1. Equilibrium (13) exists in the first 2
2 F L we write conditions (16c).
* 2 L F * 2
orthant if qL 0 qF 0; Proof of Corollary 2. As an analysis of the approximating
3 3
then, taking into account (12), we may write the following function demonstrates (Fig. 3), for p d 100 and
system of inequalities: b1
1 ,1 there is a restriction 0 ,3 . A comparison of
1 1 pd
2 3, 2 3 . (А1) inequalities (14) and (14a) leads to the conclusion that
2 2 2 2
ln 0 , 5 ln 0 , 5
L 0 ,5 e
3 and 0 ,5 e
3 .
Taking into account the notation (6), the ratio is
F ln 0 , 5
associated with the ratio of the agents’ type parameters as 0 ,3
0 ,1 ,
Because e it follows from formula (16a) that
follows:
0 0 ,1 ,
*
1 aL i.e. the first part of the corollary is correct.
b1 p d 0 0 ,1 ,
. (А2)
1 aF The second part is derived from formula (16b), in which we
b1 p d
* * * *
replace to , because if 0 , then 0.
aL
The ratio (A2) is the dependence on the ratio .
aF
Taking into account that b1 p d , according to (2a), and REFERENCES
[1] G. Roland “Transition and Economics. Politics, Markets, and Firms,”
because 0 ,1;10 taking into account (1), dependence (A2), Cambridge: MIT Press, 2000, 840 p.
as shown by the numerical experiment in Fig. 3, may be [2] S. Braguinsky and G. Yavlinsky, “Incentives and Institutions. Transition
approximated by the following function: to a Market Economy in Russia,” Princeton. NJ.: Princeton University
Press, 2000, 420 p.
[3] RF Government Decree of 30.12.2015 N 1493 “On State program"
0 ,1;10 , 0 1 . (А3) Patriotic Education of Citizens of the Russian Federation for 2016-2020.
In the case of approximation (A3), inequalities (A1) may be [4] RF Government Decree of December 27, 2012 N 2567-r “On the state
program of the Russian Federation" Development of Culture and
written in the form (14a). Tourism 2013-2020.
Proof of Corollary 1. It follows from formulas (13) that for [5] I.N. Khaimovich, V.M. Ramzaev and V.G. Chumak, “Data modelling to
η=3 the equilibrium is not defined. It follows from formulas analyze how the cities in the Volga region correspondent to the digital
state format,” CEUR Workshop Proceedings, vol. 2212, pp. 46-55,
2018.
(14a) that 2 0 , because the logarithm function is defined
2 [6] V.M. Ramzaev, I.N. Khaimovich and V.G. Chumak, “Big data analysis
only with a positive sub-logarithmic expression. Therefore, for demand segmentation of small business services by activity in
there is the limitation 4 . region,” CEUR Workshop Proceedings, vol. 1903, pp. 48-53, 2017.
[7] M.I. Geraskin, “Game-theoretic model of wide social groups’ behavior
Proof of Assertion 2. A substitution of formulas (14) into with stimulation of volunteering activities,” CEUR Workshop
Proceedings, vol. 2416, pp. 43-49, 2019.
formula (15) and transformation taking into account formulas
(12) allows us to obtain the following expression: [8] V.N. Burkov, B. Danev, A.K. Enaleev, T.B. Nanev, L.D. Podvalny and
B.S. Yusupov, “Competitive mechanisms in problems of distribution of
scarce resources,” Avtomatika i telemekhanika, no. 11, pp. 142-153,
L
*
2 1 1988.
* qL 2 L F F 2 1 [9] V.N. Burkov, A.K. Enaleev and V.F. Kalenchuk, “Coalition with the
. (А4)
*
qF L competitive mechanism of resource distribution,” Avtomatika i
2 F L 2 2 telemekhanika, no. 12, pp. 81-90, 1989.
2 2 F 2
[10] V.N. Burkov, A.K. Enaleev and Y.G. Lavrov, “Synthesis of optimal
A differentiation (A4) with respect to η leads to the planning and incentive mechanisms in the active system,” Avtomatika i
telemekhanika, no. 10, pp. 113-120, 1992.
*
expression
2 1
, from which inequality (16a) [11] V.N. Burkov, M.B. Iskakov and N.A. Korgin “Application of
2 generalized median schemes for the construction of non-manipulable
2 2 mechanism multicriterion active expertise,” Automation and Remote
2 Control, vol. 71, no. 8, pp. 1681-1694, 2010.
follows. A differentiation (A4) with respect to β leads to the [12] N.A. Korgin, “Equivalence of non-manipulable and non-anonymous
* priority resource distribution mechanisms,” Upravleniye bol'shimi
expression
3
, from which inequality (16b) sistemami, vol. 26.1, pp. 319-347, 2009.
2
2
2
VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 142
Data Science
[13] V.N. Burkov, I.I. Gorgidze, D.A. Novikov and B.S. Yusupov, “Models networks,” Computer Optics, vo. 43, no. 5, pp. 886-900, 2019. DOI:
and cost and revenue distribution mechanisms in the market economy,” 10.18287/2412-6886-900179-2019-43-5-886-900.
Moscow: Institut problem upravleniya, 1997, 356 p. [17] O.O. Evsutin, A.S. Kokurina and R.V. Meshcheryakov, “A review of
[14] N.A. Korgin, “Use of intersection property for analysis of feasibility of methods of embedding information in digital objects for security in the
multicriteria expertise results,” Automation and Remote Control, vol. internet of things,” Computer Optics, vol. 43, no. 1, pp. 137-154, 2019.
71, no. 6, pp. 1169-1183, 2010. DOI: 10.18287/2412-6179-2019-43-1-137-154.
[15] V.N. Burkov, N.A. Korgin and D.A. Novikov, “Problems of aggregation [18] A.A. Cournot, “Researches into the Mathematical Principles of the
and decomposition mechanisms of management of organizational and Theory of Wealth,” London: Hafner, 1960.
technical systems,” Problemy upravleniya, no. 5, pp. 14-23, 2016. [19] H. Stackelberg, “Market Structure and Equilibrium,” Bazin, Urch &
[16] Yu.V. Vizilter, V.S. Gorbatsevich and S.Y. Zheltov, “Structure- Hill, Springer, 2011.
functional analysis and synthesis of deep convolutional neural
VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 143