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.
In Russia, in recent years, volunteering has been developing, because in 2016 a number of the volunteers was 1.435 million1, 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 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. ak = ζk (D) = Dδak , δak ∈ [0,1], D >> 1, k ∈ K , where D is the disposable time fund, i.e., the physical time fund with the exception of the rest time; δa 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 δak > 0,5 in function (
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.,
Uδ/a ( pd ) < 0 , (
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 ) = paak − p1d−δak ak , k ∈ K , (
The stimulation function determines the price of the insensitive as follows [7]: pa (A) = b1 − b2 ∑ nk ak , k ∈ K , b1, b2 > 0 ,
k∈K
(
b1 = pd A0 A−0AD ,b2 = A0 p−dAD , A0 = k∑∈K ak0, AD = D2 k∑∈K nk . (4а) The coefficients b1, b2 depend on the of agents’ actions vector in the previous period A0 = {ak0 , k ∈ K }. Formulas (4a) are obtained on the basis of the following conditions1: 1) in the absence of incentives (i.e., for pa (A) = 0 ), the total number of the socially optimal actions is equal to ∑ ak0 ; 2) when the price of the incentive is equal to the average wage pd , the disposable time fund is k∈K 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 (
∂pa
The equilibrium conditions are formulated as the following assertion.
Assertion 12. The actions vector А, satisfying the following conditions
b1 − b2 ∑j∈K n ja j − b2nk ak 1 + j∈∑K \k ∂∂aakj − p1d−δak = 0 ,
∑ ∂a j > −2, k ∈ K , (
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 , ∂ak which corresponds to the simultaneous and independent choice of actions.
Under condition (
b + p1d−δak
2nk ak + ∑ n ja j − αk = 0, αk = 1 , k ∈ K , (
nαk − ∑ α j
ak* = j∈K \k , k ∈ K , (
k
(
The population nk, thousand The total "charitable" time А, thousand hours The structure of "charitable" time, % δak (a) f k (a)
Total
8.6 1435 12398
100 0.181 2.35 997 2343 18.90 0.181 1 15 243 3645 29.40 0.574
2 0.070 0.018 0.005 0.000 0.000
0.000 20 82 1640 13.23 0.635 3 e 30 48 1440 11.62 0.721 4 40 23 920 7.42 0.782 5 50 11 550 4.44 0.829 6 60 9.8 588 4.74 0.868
We simulate the equilibrium (
Including time worked per week, hours <9 9 – 15 16 – 20 21 – 30 31 – 40 41 – 50 >51 70 9 630 5.08 0.900
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]: f (a) =
1 σa 2π − w(a−a l )2 2σa2 , where a , σa 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 (
1
eσδ 2π
e
ew1 eδa −(eδ )l1 2
−
2(eσδ )2
where δ, σδ 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 (
Using the least squares algorithm implemented in the MSExcel processor, the following values of
the coefficients of functions (
10 20 30 f(a) 40
50 f~(a) 60 δa 70 80 a
Consider the Cournot-Nash equilibrium (
AΣ(Pa) 10600 10400 10200 10000 9800 9600 9400 9200 9000 8800 8600 0 100 200 300 400 500
Pa
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.
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.