=Paper=
{{Paper
|id=Vol-2416/paper6
|storemode=property
|title=Game-theoretic model of wide social groups' behavior with stimulation of volunteering activities
|pdfUrl=https://ceur-ws.org/Vol-2416/paper6.pdf
|volume=Vol-2416
|authors=Mikhail Geraskin
}}
==Game-theoretic model of wide social groups' behavior with stimulation of volunteering activities ==
https://ceur-ws.org/Vol-2416/paper6.pdf
Game-theoretic model of wide social groups’ behavior with
stimulation of volunteering activities
M I Geraskin1
1
Samara National Research University, Moskovskoe Shosse, 34, Samara, Russia, 443086
e-mail: innovation@ssau.ru
Abstract. The problem of developing tools for the stimulation system of socially optimal
actions (volunteering) is considered. Based on the study of the population’s differentiation
according to the propensity to an altruism, the game-theoretic model of the social group’s
behavior is formed, accounting for the incentives for volunteering. In the cases of the linear
decreasing incentive function and the linear cost functions of agents, the Cournot-Nash
equilibrium mechanism in the corresponding game is proved. An existence of the equilibrium
actions and an impact of incentives on the volunteers’ time distribution are confirmed by the
simulation of the volunteers’ behavior in Russia.
1. Introduction
In Russia, in recent years, volunteering has been developing, because in 2016 a number of the
volunteers was 1.435 million 1, i.e., 1% of the population. The volunteering is realized in the
performance of actions that maximize the collective utility function, that is, socially optimal actions.
The volunteers do not receive income, therefore, these actions do not correspond to the criterion of
individual rationality. The Russian statistics demonstrates that moral motivation is effective for a
narrow social group of altruistic people. Among the broad masses of the population, trends of
individual rationalism [1,2] are exist. The state programs [3,4] implement for overcoming of these
trends. In addition, the expansion of volunteering can be provided by the system of stimulation of
these actions on the base of the state information system [5].
The interconnection algorithms of the stimulation system and the information system [6,7] enable
to solve the problems of personalized registration of citizens' actions, the distribution of the state
incentive fund, the monitoring of the of the stimulation effectiveness.
The methodological basis for the development of incentive systems includes the following
mechanisms: the competitive mechanism is Pareto-efficient and optimal by the criterion of the
additive utility function under non-coalition [8] and coalition [9] agents' behavior; the mechanism of
sequential resource distribution (MSRD) [10,11]; the mechanisms of direct and reverse priorities [12].
For MSRD the existence and the uniqueness were proved [13]; according to MSRD, the incentive is
distributed [14] as a minimum of the agent's message about its action and the average undistributed
rest of incentives. MSRD satisfies [15] the conditions of individual rationality, Pareto efficiency and
1
Labor and employment in Russia 2017: Statistic compilation / Rosstat. Moscow. 2017. 261 p. http://www.gks.ru/free_doc/doc_2017/
trud_2017.pdf
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019)
Data Science
M I Geraskin
non-manipulability. MSRD is not applicable in a system with independent and simultaneous agents’
actions, because the MSRD implies the sequential registration of the agent’ actions and the
distribution of incentives. Therefore, we use the compensatory linearly decreasing stimulation
function, for which these conditions were proved [6,7]. The simulation of the social management
[16,17] on the basis of the large groups of population demonstrated the effectiveness of this approach.
The utility functions of agents can take into account the symmetry and asymmetry of their awareness
[18], which is provided by analyzing the correlation of information flows between social groups
[19,20].
The stimulation system of socially optimal actions provides the following results:
- an increase in a number of the volunteers and the time fund of these actions;
- the cross-impact of the altruism and the individual rationality on the behavior of the population
groups;
- an emergence of the contradictions between the interests of large social groups, differentiated by a
degree of the propensity to the altruism.
Consequently, the model of the population behavior can be formed as the non-cooperative game of
the social groups (hereinafter, the agents). The utility functions of these agents include both the
incentives for performing socially optimal actions and the loss of the income due to the redistribution
of available time, which is a constraint.
2. Methods
We assume that, for the kth agent, a continuous function ζ k (•) is defined, and we denote the socially
optimal actions time interval in the absence of stimulation as ak . The function ζ k (•) expresses the
dependence of ak on the available time fund, and the function has the following form:
ak = ζ k (D ) = D δ ak , δ ak ∈ [0,1], D >> 1, k ∈ K , (1)
where D is the disposable time fund, i.e., the physical time fund with the exception of the rest time; δ a
is the elasticity coefficient of the “charitable” time to D, which characterizes the propensity to the
altruism; K is the set of agents; ak is the component of the socially optimal actions vector
A = {ak , k ∈ K }.
Definition: the altruism (the propensity to the charity) of the kth agent is called the type of agent,
for which δ ak > 0,5 in function (1).
We introduce the hypothesis of the influence of the propensity to the altruism on the agent's
behavior: an increase in the propensity to the altruism leads to a decrease in the utility of the wage,
i.e.,
U δ/ a ( pd ) < 0 , (2)
where d is the working time interval; pd is the price (the tariff rate) of the working time; U (•) is the
continuously differentiated utility function of the agent.
In the case of the stimulation, the model of the agents’ actions choice includes the utility function
and the stimulation function. We describe these components of the model.
The agent’s utility function is the difference between the sum of the incentives and the costs of the
working time loss:
U k (ak ) = pa ak − p1d− δ ak ak , k ∈ K , (3)
where pa is the sum of the insensitive, i.e., the price of a unit of “charitable” time. Formula (3)
corresponds to hypothesis (2), because, with a growth of δ a , the influence of the working time price
and the working time fund on the agent’s utility decreases.
The stimulation function determines the price of the insensitive as follows [7]:
pa (A ) = b1 − b2 ∑ nk ak , k ∈ K , b1 , b2 > 0 , (4)
k ∈K
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 44
Data Science
M I Geraskin
where nk is the number of agents in kth social group, which is constant in the current period; b1 ,b2
are the constant coefficients, calculated by the following formulas
A0 p D
b1 = pd 0 , b2 = 0 d D , A0 = ∑ ak0 , AD = ∑ nk . (4а)
A −A D
A −A k ∈K 2 k ∈K
The coefficients b1 ,b2 depend on the of agents’ actions vector in the previous period
{ }
A 0 = ak0 , k ∈ K . Formulas (4a) are obtained on the basis of the following conditions 1: 1) in the
absence of incentives (i.e., for pa (A ) = 0 ), the total number of the socially optimal actions is equal to
∑ a ; 2) when the price of the incentive is equal to the average wage p , the disposable time fund is
k∈K
0
k d
D
divided equally between the working time and the “charitable” time (i.e., ).
2
We consider the problem of searching for the Nash equilibrium vector А from the maximum of the
utility function (3) under condition (4) in the case of the constancy of the agents’ number in all social
∂n
groups (i.e., k = 0∀k ∈ K ).
∂pa
3. Results and discussion
The equilibrium conditions are formulated as the following assertion.
Assertion 1 2. The actions vector А, satisfying the following conditions
∂a j
b1 − b2 ∑ n j a j − b2 nk ak 1 + ∑ − p1d− δ ak = 0 ,
(5)
j∈K j ∈ K \ k ∂ak
∂a j
∑ > −2, k ∈ K ,
j∈K \ k ∂ak
(6)
is the Nash equilibrium in problem (3), (4).
In the case of the Cournot hypothesis [21], all agents symmetrically do not change the selected
actions in response to the environment’s actions, i.e.
∂a j
= 0, j ∈ K \ k , (7)
∂ak
which corresponds to the simultaneous and independent choice of actions.
Under condition (7), system (5) has the following form
b + p1d− δ ak
2nk ak + ∑ n j a j − α k = 0, α k = 1 ,k ∈K , (8)
j∈K \ k b2
The solution of the system (8) by the Kramer's method can be written as follows:
nα k − ∑ α j
j∈K \ k
ak =
*
,k ∈ K , (9)
(n + 1)nk
where the symbol “*” denotes the equilibrium values, n is the number of agents in the system. For the
Cournot-Nash equilibrium (9), conditions (6) are satisfied considering (7).
1
In the case of these conditions, the system of equations b1 − b2 A0 = 0, b1 − b2 AD = pd leads to the solution (4a).
2
Proof of assertion 1. Function (3) under (4) has the form U k (ak ) = b1 − b2 ∑ n a a − p
k k k
1− δ ak
d ak . Therefore,
k ∈K
the extremum necessary condition U a/ k = 0 can be written in the form (5). The sufficient maximum condition
U a//k < 0 leads to (6).
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 45
Data Science
M I Geraskin
We simulate the equilibrium (11) using the example of the social groups of Russian volunteers
(Table 1). The actual propensity to the altruism is calculated by the formula δ ak (a ) =
ln ak
that
ln D
follows from formula (1), where the weekly time fund is taken equal to D=112 hours; the graph of the
function δ ak (a ) is shown in Fig. 1. The actual distribution function of the population in groups with
different propensities to the altruism is calculated by the formula f k (a ) =
nk
.
∑ nk
k ∈K
Table 1 Analysis of the volunteers in 2016.
Including time worked per week, hours
Indicator Total <9 9 – 15 16 – 20 21 – 30 31 – 40 41 – 50 >51
The average
"charitable" time a,
hours 8.6 2.35 15 20 30 40 50 60 70 80
The population nk,
thousand 1435 997 243 82 48 23 11 9.8 9 8
The total
"charitable" time А,
thousand hours 12398 2343 3645 1640 1440 920 550 588 630 640
The structure of
"charitable" time, % 100 18.90 29.40 13.23 11.62 7.42 4.44 4.74 5.08 5.16
δak (a ) 0.181 0.181 0.574 0.635 0.721 0.782 0.829 0.868 0.900 0.929
f k (a ) 0.070 0.018 0.005 0.000 0.000 0.000 0.000 0.000 0.000
The agent’s index 1 2 3 4 5 6 7 8 9
The model of the probability distribution of volunteers according to the "charitable" time has the
form of the normal law with certain values of the kurtosis and the asymmetry [7]:
w (a − a l )
2
−
f (a ) =
1
e 2σ a ,
2
(10)
σ a 2π
where a , σ a are the mathematical expectation and the standard deviation of the initial distribution of
the random variable of the "charitable" time; l is the coefficient taking into account the asymmetry
(l>1 is the left asymmetry, l<1 is the right asymmetry) in comparison with the normal law (l = 1); w is
the coefficient taking into account the kurtosis (w<1 is a more uniform distribution, w> 1 is a less
uniform distribution) in comparison with the normal law (w = 1).
According to a degree of the propensity to the altruism, the model of the distribution density of
volunteers has a similar form (10), but because the function δak (ak ) is calculated through a
logarithmic relationship, instead of formula (10), the following formula is used:
( )
2
l1
e w1 e δ a − e δ
−
f (δ a ) =
1
e ( )
2 e σδ
2
, (11)
σδ
e 2π
where δ, σ δ are the mathematical expectation and the standard deviation of the initial distribution of
the random value of the propensity to the altruism, the parameters w1 ,l1 are similar to the parameters
w, l for the distribution (10).
Using the least squares algorithm implemented in the MSExcel processor, the following values of
the coefficients of functions (10), (11) are obtained: a = 8,6, σ a = 0,57 , w = 0,007, l = 0,58 ,
δ = 0,18, σ δ = −0,65 , w1 = 0,43, l1 = 0,08 . The statistical estimates of regressions (for function (10)
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 46
Data Science
M I Geraskin
R 2 = 0,99, F = 257 , for function (11) R 2 = 0,99, F = 218 ) prove their adequacy. Therefore, the
distributions of volunteers according to the time and the propensity to the altruism correspond to the
right branch of the Gauss function, that is, the mathematical expectations of the time and the
propensity to the altruism are close to the minimum of these indicators. Functions (10), (11) with
regard to the indicated coefficients are shown in Fig. 1.2.
f (a ),δa (a )
1,0
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0,0
0 10 20 30 40 50 60 70 80
a
f(a) f~(a) δa
Figure 1. Density distribution function of volunteers by the time and the propensity to altruism.
f (δa )0,7
0,6
0,5
0,4
0,3
0,2
0,1
0,0
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
f(δa) f~(δa) δa
Figure 2. The density function of the distribution of volunteers by the degree of propensity to altruism.
Consider the Cournot-Nash equilibrium (9) for agents, which are indexed in accordance with
Table. 1. In Fig. 3 the distribution functions of the time depending on a degree of the propensity to the
altruism with different insensitives are shown. In Fig. 4 the dependence of the volunteers’ time on the
insensitive is demonstrated.
A (δa ) 200
180
160
140
120
100
80
60
40
20
0
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
A(Pa=0) A(Pa=200) A(Pa=400) A0 δa
Figure 3. The distribution of the volunteers’ time depending on the insensitive.
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 47
Data Science
M I Geraskin
AΣ(Pa)
10600
10400
10200
10000
9800
9600
9400
9200
9000
8800
8600
0 100 200 300 400 500
Pa
Figure 4. The dependence of the total volunteers’ time on the insensitive.
The analysis of the Cournot-Nash equilibria simulation leads to the following conclusions.
First, the distribution of “charitable” time is closest to the real distribution for Ра=0, but the
calculated equilibrium actions are lower than the actual values for the agents with low propensity to
the altruism, and higher than the actual values for the agents with high propensity to the altruism.
Moreover, the volunteers’ time at Ра=0 is lower than the actual value. However, the average deviation
of the total time for each agent from the total actual time is 4%. Consequently, in the developed
equilibrium model, the influence of the population’s propensity to the altruism on the volunteers’ time
is slightly exaggerated.
Second, in the case of the equilibrium distribution, the agents with the highest propensity to the
altruism make the greatest contribution to the volunteers’ time.
Third, with a growth of the insensitive, the distribution of “charitable” time becomes increasingly
uneven, that is, the “charitable” time increases slightly for the agents with low propensity to the
altruism and increases sharply for the agents with high propensity to the altruism. Therefore, the
agents with high propensity to the are the most sensitive to stimulation.
Fourth, with an increasing in the incentive, the total “charitable” time grows, i.e., the stimulation
influences on the involvement in volunteering for high and low altruistic agents.
4. Conclusion
The problem of the developing the information tools for the system of the volunteering stimulation is
considered. In the article, the following main results are obtained.
The game-theoretic model of the social groups’ behavior is developed, taking into account the
stimulation of volunteering, based on the differentiation of the population according to the altruism
and the individual rationality.
In the case of the linear diminishing incentive function and the linear agents’ cost functions, the
Cournot-Nash equilibrium mechanism in the corresponding game is proved.
The simulation of the volunteers’ behavior in Russia demonstrates the adequacy of the model, the
existence of the equilibrium actions vector and the effect of the stimulation on the volunteers’ time.
The analysis shows that the stimulation system has the greatest effect on the agents with high
propensity to altruism, but the growth of socially optimal actions is manifested in all social groups.
5. Literature
[1] Roland G 2000 Transition and Economics. Politics, Markets, and Firms (Cambridge: MIT
Press) p 840
[2] Braguinsky S, Yavlinsky G 2000 Incentives and Institutions. Transition to a Market Economy in
Russia (NJ.: Princeton University Press) p 420
[3] RF Government Decree of 30.12.2015 N 1493 "On State program" Patriotic Education of
Citizens of the Russian Federation for 2016-2020"
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 48
Data Science
M I Geraskin
[4] RF Government Decree of December 27, 2012 N 2567-r "On the state program of the Russian
Federation" Development of Culture and Tourism "2013-2020"
[5] RF Government Decree of 15.04.2014 N 313 (as amended on 10.21.2016.) "On approval of the
Russian Federation, the state program" Information Society (2011-2020)"
[6] Geraskin М I 2017 Algorithms of the information stimulation system of Russian citizens’ socio-
optimal actions CEUR Workshop Proceedings 1903 92-99
[7] Geraskin М I 2018 Analysis of the influence of citizens’ altruism on the effectiveness of the
socially-optimal actions stimulation system CEUR Workshop Proceedings 2212 431-439
[8] Burkov V N, Danev B, Enaleev A K, Nanev T B, Podvalny L D and Yusupov B S 1988
Competitive mechanisms in problems of distribution of scarce resources Avtomatika i
telemekhanika 11 142-153
[9] Burkov V N, Enaleev A K and Kalenchuk V F 1989 Coalition with the competitive mechanism
of resource distribution Avtomatika i telemekhanika 12 81-90
[10] Burkov V N, Enaleev A K and Lavrov Y G 1992 Synthesis of optimal planning and incentive
mechanisms in the active system Avtomatika i telemekhanika 10 113-120
[11] Burkov V N, Iskakov M B and Korgin N A 2010 Application of generalized median schemes
for the construction of non-manipulable mechanism multicriterion active expertise Automation
and Remote Control 71(8) 1681-1694
[12] Korgin N A 2009 Equivalence of non-manipulable and non-anonymous priority resource
distribution mechanisms Upravleniye bol'shimi sistemami 26.1 319-347
[13] Burkov V N, Gorgidze I I, Novikov D A and Yusupov B S 1997 Models and cost and revenue
distribution mechanisms in the market economy (Moskva: Institut problem upravleniya) p 356
[14] Korgin N A 2010 Use of intersection property for analysis of feasibility of multicriteria
expertise results Automation and Remote Control 71(6) 1169-1183
[15] Burkov V N, Korgin N A and Novikov D A 2016 Problems of aggregation and decomposition
mechanisms of management of organizational and technical systems Problemy upravleniya 5
14-23
[16] Khaimovich I N, Ramzaev V M and Chumak V G 2016 Use of big data technology in public
and municipal management CEUR Workshop Proceedings 1638 864-872
[17] Khaimovich I N, Ramzaev V M and Chumak V G 2015 Challenges of data access in economic
research based on Big Data technology CEUR Workshop Proceedings 1490 327-337
[18] Faizliev A R, Korotkovskaya E V, Sidorov S P, Smolov F M and Vlasov A A 2018 Utility
Maximization for an Investor with Asymmetric Attitude to Gains and Losses over the Mean-
Variance Efficient Frontier Journal of Physics: Conference Series 1141(1) 012017
[19] Sidorov S, Faizliev A and Balash V 2017 Measuring long-range correlations in news flow
intensity time series International Journal of Modern Physics C 28(8) 1750103
[20] Kulikovskikh I M 2017 Anomaly detection in an ecological feature space to improve the
accuracy of human activity identification in buildings Computer Optics 41(1) 126-133 DOI:
10.18287/2412-6179-2017-41-1-126-133
[21] Cournot A A 1960 Researches into the Mathematical Principles of the Theory of Wealth
(London: Hafner)
V International Conference on "Information Technology and Nanotechnology" (ITNT-2019) 49