Corruption in the allocation of resources in a static model of a combination of general and private interests

Gorbaneva Ol'ga Ivanovna Doctor of Technical Science Associate Professor, Department of Applied Mathematics and Programming, Southern Federal University 344090, Russia, Rostovskaya oblast', g. Rostov-Na-Donu, ul. Mil'chakova, 8a, kab. 212 gorbaneva@mail.ru Other publications by this author

IntroductionThe topic of corruption in hierarchical management systems is raised in society quite often, especially when allocating resources.

The pioneering work on modeling corruption is considered ^[2].

Malsagov M.H., Usov A.B., Ugolnitsky G.A. in ^[3], having studied corruption in the allocation of resources in hierarchical systems, came to the conclusion that in order to combat it, it is necessary to increase the probability of catching a bribe taker or make bribery unprofitable for a supervisor.

In the author's works, corruption in the allocation of resources is considered in two-level hierarchical tree models ^[4]. The case of connivance and extortion in the allocation of resources is considered. It is proved that during extortion, the Center forces the agent to give half of the resources for a bribe, while the damage to society is 75%. With connivance, the losses of society and agents are not so significant.

The further structure of the work is as follows. In the first paragraph, the hierarchical superstructure built earlier in ^[1] over the SOCHI model is supplemented and complicated by the supervisor's target functions and limitations, a new agent-bribe strategy is introduced, a new system structure is described, corruption relations are introduced in the allocation of resources. The second paragraph applies a descriptive approach to the study of corruption in the allocation of resources in the SOCHI model, the third - an optimization approach, and the fourth paragraph describes the impact of corruption in the allocation of resources on the system consistency of the model. At the end of the article, the conclusions and general results of the study are presented, in particular recommendations on combating corruption in the allocation of resources.

1. Building a business model with corruption in resource allocationThere is a two–level fan control system in which the top-level governing body - the Principal - seeks to maximize the utilitarian function of public welfare.

 The models of the combination of public and private interests described earlier in ^[1] (SOCHI models) have the following form.

Agent target function:

;                          (1)

under restrictions:

The objective function of the principal:

                       (2)

under natural constraints:

Here r _i is the amount of resources of the i-th agent; u _i is a part of them directed to the creation of total income; c(u) is a function of total income; s _i is the participation share of the i-th agent in total income; p _i(r _i ? u _i) is a function of own income the i-th agent, M = {1,...,n} is the set of agents in the system forming the SOCHI model. The functions p _i, c are assumed to be continuously differentiable and concave over all arguments.

This article discusses the mechanism of the distribution of resources by the principal between agents r _i, 0 ? r _i ? 1, .

This article discusses the middle level introduced into the system by the principal – the supervisor, who should represent the interests of the agent.

We will assume that the upper level does not show corrupt behavior, but the middle level representing it can, in exchange for a bribe from agents, change the last amount of resources that the Principal allocated to them. Accordingly, corruption occurs in the allocation of resources.

A descriptive approach is applied to the study of corruption in the allocation of resources in the SOCHI model, in which the function of bribery is considered known and the solution of the problem is reduced to determining the optimal strategy of the agent, and an optimization approach, in which the optimal function of bribery for the supervisor is sought. At the same time, if the latter contained parameters that the supervisor can influence, then the optimal parameters for him were found.

Suppose that the principal has distributed his available resources among the agents in shares of r _i ^P, so that . The Supervisor reduces the amount of resources allocated by the Principal to the agent to the level of r _i ^S, 0?r _i ^S? ?r _i ^P, which the agent can increase in exchange for a bribe no more than to the level of r _i ^P:

                                      (3)

here ? i is the addition of resources to the i-th agent in exchange for a "rollback", b _i is the share of the "rollback" from ? _i sent to the agent from the supervisor. Then the model of corruption in the allocation of resources in the system under consideration has the form:

           (4)

(5)

where g _S, g _i are the target functions of the supervisor and the i-th agent, respectively.

As an example, we can cite a system in which the agents are hospital doctors, and the principal is the state. During the COVID-19 pandemic, the state allocates a salary supplement of 80,000 rubles to a hospital doctor who has worked with a patient who has been diagnosed with coronavirus. The supervisor in this system is the head of the hospital department, or even the entire hospital, who reduces the payments due to the doctor, assigning a proportional fee according to the hours worked, or even completely depriving the doctor of the payments due to him. The fact of corruption is that the head of the department still makes payments to some hospital employees due to the good relations of the latter with the head physician or in exchange for a share of these payments. This situation leads to the destabilization of society, as doctors lose interest in work, and representatives of society lose confidence in the state. In addition, this situation directly affects the security in the country, as a doctor may quit his job or work reluctantly. In the first case, the shortage of doctors in such a difficult period for the country can lead to increased mortality in this period. In the second case, trust in the state is already being lost among patients who are not sure that the state will do everything possible to cure the patient if the patient is infected with coronavirus. 

2. Descriptive approachConsider the model in general, where the agent function has the form g i(b i,? i,u) = p i(r i S+(1?b i)

_i?u _i)+_s i c(_u), the functions p _i⁽x), _c(x_{) are increasing concave, satisfying the obvious properties} p i_{(0) = 0,} c(0) = 0, and the functions _? i(x) are increasing, satisfying the property _i(0) = 0. This is an optimization problem in which the agent's controls are the values u i (the amount of resources directed by the agent to the realization of private interests) and b i (the share of the increase in resources allocated to the agent by the principal). In this case, we will assume that the principal has already assigned the amount of resources to each agent r _i and does not accept any further participation in the game of the supervisor and agent. The supervisor also reduced the amount of resources owed by the principal to the agent to the value of r i S _{and assigned a function depending on the relaxation of this restriction} ? i on the size of the bribe b _i, after which the outcome of the game now depends only on the agent's decision.

Theorem 1. Optimization problem (5) under conditions of increasing concave functions p _i(x), c(x) and ? _i(x) and ? _i(0)= =0 has a unique solution:

                                           (6)

where u _i** = (?p _i’(r _i ^S + (1 ? b _i^* )_i ? u_i) + s_ic’(u))^?1(0), it is not fulfilled under the conditions of increasing ? i(x) and ? i(0) = 0, since the condition i(b i) + (1 ? b i)?’(b i) < 0 must be fulfilled for this.

Taking into account _i(0) = 0, the inequality turns into (1 ? b _i) _i’(b _i) < 0, or, taking into account the positivity of the value 1 ? b _i, _i’(b _i) < 0, which contradicts the condition of increasing _i(x).

The conditions of the second convince of the uniqueness of this solution.

The statement is proven.

From (6) it can be concluded that 1) that the size of the bribe depends on the type of function of bribery, but in no way on the power of the functions of general and private income; 2) when introducing a corruption mechanism in the allocation of resources, it is always advantageous for an agent to give a bribe.

3. Optimization approachIn the optimization approach, the function ? i(b i) is defined as the optimal guaranteeing supervisor control mechanism in a game of type ? 2 with agents.

Let's define the objective function of the principal as a utilitarian function of public welfare:

          .               (7)

We will assume that the principal has resources in the amount of 1 (without limiting the generality of 100 percent). Recall that in this formulation, the principal affects the amount of resources available to the agent, i.e. assigns the values of r _i ^P to each agent, taking into account the restriction.

Let's apply an optimization approach to the study of economic corruption in the SOCHI model (4)–(5), (7). There are (n+2) participants in the model: a principal with an objective function g ^P(u) (7), a supervisor with an objective function g ^S(b,?,u) (4) and n agents with objective functions g _i(b _i,? _i,u _i) (5). The agent controls are still the quantities u _i and b _i with constraints u _i ? ^{[0,r i]} and b _i ? ^[0,1]. Moreover, the value u _i is the response to the control of the principal r _i ^P, which satisfies the constraints, . Then the supervisor intervenes in the game of the principal and the agent, who reduces the value to the level and offers the agent an increase of _i to the value of r _i ^S in exchange for a "rollback" of b _i from the increase. _i is a fraction, so 0 ? _i ? 1.

So, there is a hierarchical game of (n+2) persons defined by the following parameters:

1. The set of participants – players N = {P,S,1,2,...,n} is given. Subsets {P}, {S} and M = {1,2,...,n} define the upper (principal), middle (supervisor) and lower (agents) levels of the hierarchy. The principal has the right of the first move, i.e. he is the first to choose and inform the agents (players with numbers i) of his strategy. The supervisor has the right of the second move, choosing and informing the agents of his strategy.

2. The pair <u _i,b _i> defines the control parameters of the i-th player, i ? M, which can be selected from the sets, u _i ? U_i = r^[0;ri], b_i ? ^[0,1], i ? M. Let, therefore, <u _i,b _i>? U ? ^[0,1]n.

The vector defines the control parameters of the principal, which are selected from a compact set of the form.

The vector and the vector of functions = ( ₁,...,? _n) define the control parameters of the supervisor, where and are the functions of bribery, which are selected from the space ?_i of increasing functions satisfying the property _i(0) = 0, i.e. .

3. On the set UKP ?KS, the winning functions of agents g _i(5), supervisor g ^S (4) and principal g ^P (7) are determined.

4. For each player with number i, rules of behavior are defined that allow players P and S to evaluate the set of rational responses of players with numbers i:

- striving to maximize the winning function for your choices;

- striving to achieve a Nash-balanced situation.

5. The available game is a game with complete information, except that the principal may not know about the supervisor's objective function.

The principal may or may not use feedback from the management agent. The following rules are introduced:

Rule 5.1. The supervisor relies on information and will have it about the choice of b _i ? B _i, i ? M.

Rule 5.2. The supervisor makes the second move by choosing and informing the players i ? M of their strategies r _i ^S ? K ^S and _i ? ?_i.

Rule 5.3. Each agent, having received the information o r _i ^S and _i, tries to maximize its objective function by the appropriate choice <u _i,b _i>? U _i ? ^[0,1].

Rule 5.4. In the formulated conditions, the supervisor maximizes his guaranteed result.

That is, there are two games in the model: a meta–game (5), (7) and a hierarchical game (4)-(5).

The algorithm for investigating the model is as follows:

1. Find by means of a solution of the coordination of interests (perhaps, at least in a weak form) the values u^*, u ^max and the values r _i ^P (^[1]).

2. Find a balance in the game of ₂ supervisors and an agent.

Theorem 2. In the game ? ₂ (4) – (5), under the condition of increasing concave functions p _i(x), c(x), there is a single control mechanism that meets the interests of the supervisor:where

Proof.

The punishment strategy will be considered as a value (there is no premium to the share of total income in the punishment), (the optimal reaction of the agent to the punishment strategy is not to pay a bribe), where .

  – the optimal gain of the agent with the punishment strategy, - the supervisor's gain with zero agent's bribe.

 under restriction – the maximum income of the supervisor, provided that the gain of the agent's income is greater than with the punishment strategy.

From the constraint, we can find the dependence of b on u:

.

Substituting this value in , we get the maximization problem for 2 n variables u _i and _i:

subject . Note that in this case, the maximized function increases by _i, so the optimal value .

Therefore:

.

Denote

Let us prove that K₁? K₂. This will write out their difference

Given the arbitrariness of it , consider the sign of the following expression .

To do this, compare the following values: and .

In the power of non-decreasing p_i this comparison is equivalent to comparing expressions and , and it, in turn, is equivalent to comparing the expressions and . Ie actually need to compare the income of the agent in the case where it has r_i^P or r_i^S resources.In ^[1] proved that in SOCHI-a larger model r_i corresponds to a higher optimal u_i (and hence executed ) and a larger value of its objective function, which has the form (5). Therefore, the equivalent force of a chain of inequalities , and the equality occurs when r_i^P = r_i^S. Therefore, K₁ ? K₂.

Writing the terms of the second order, make sure that the found point — the only point of maximum. The assertion is proved.

Corollary 1. The agent is always advantageous application of the mechanism of corruption in the allocation of resources.

Corollary 2. The mechanism of corruption in the allocation of resources is manipulative on the part of the supervisor.

So, this mechanism is beneficial to the supervisor only because it is beneficial agents. As soon as at least one agent refuses the bribe, this mechanism becomes unprofitable supervisor.

4. The impact of corruption in the allocation of resources on the system consistency

Corollary 3. The mechanism of corruption in the allocation of resources will not increase the consistency of SOCHI-models (1) - (2).

The proof is based on the fact that system consistency is possible only at the ends of the segment r^{[0;r i]} (^[1]). But in the case of u _i ^max = 0 for the agent u _i ^NE = 0 due to the condition u _i ^NE ? u _i ^max = 0, i.e. in this situation, the consistency of the interests of the agent and the principal was without the introduction of a corruption mechanism. In the case of u _i ^max= r _i ^P, the supervisor limits the number of resources u _i ^NE? r _i ^S ? r _i ^P ? u _i ^max.

ConclusionsIn the model of the combination of general and private interests, the corruption mechanism in the allocation of resources is investigated using two approaches: descriptive and optimization.

The following conclusions were obtained.

1. When applying the descriptive approach, it is shown that the size of the bribe depends on the type of bribery function, but in no way on the power of the functions of general and private income. That is, both a rich and a poor agent, and an agent who is more eager to realize common interests, and an agent who is more eager to realize private interests, are ready to share part of the resource allowance with the supervisor, and the same fixed share. 2. When applying the optimization approach, it is proved that the surcharge should be the maximum possible, but the initial reduction of resources should be as much as possible - to zero.

If at the same time the agent's strategy for allocating resources for common purposes does not change, almost the entire surcharge of the agent's share of participation in the total income goes as a bribe to the supervisor.3. Unfortunately, when introducing this mechanism of corruption, it is always advantageous for an agent to give a bribe, which makes it difficult to fight this kind of corruption.

But for the supervisor, this mechanism is manipulative. That is, if the agent refuses to bribe the supervisor, the latter loses. If the agent refuses a bribe, it is advantageous for him to give the latter all the resources he is entitled to without a bribe. This fact can be used to fight corruption.4. In the case of the introduction of a mechanism of corruption in the allocation of resources, system consistency is not increased.

Even an initially agreed-upon SOCHI-model without a corruption mechanism will not remain consistent when it is introduced, except in cases when all agents with the consent of the principal are individualists (i.e. agents who allocate their available resources to the VSC only for private purposes). And the SOCHI model, which was not initially agreed upon, especially cannot become agreed upon. These results turned out to be comparable with the results in ^[2], namely: "The success of the fight against corruption within the framework of the proposed model depends entirely on whether it is profitable for the supervisor (in the sense of his target functionality) to take bribes from agents or not." Since in the model we are considering, the corruption mechanism in the allocation of resources is manipulative from the outside the supervisor, then it can be fought only by the lack of reaction of agents to manipulation.

This is not easy to do, since a bribe is always beneficial to the agent when allocating resources. Here, actions on the part of the state can only help, which call into question the whole meaning of the benefits of this kind of bribery, namely, "an increase in the amount of punishment for bribes and the likelihood of catching a bribe taker." That is, economic corruption in the allocation of resources can be fought only by administrative methods.As for the pioneering work ^[1] on modeling corruption, it also says that the use of political and administrative methods to combat it is effective for "resource corruption": the fight against monopoly, the protection of agents who are resistant to manipulation by bribe takers, strengthening the detection system and widespread control, which is consistent with the results obtained by us.

In both of these works, as well as in our case when applying the optimization approach to the study, it was revealed that only the interests of the supervisor are taken into account, but neither as an agent.

In our work, this has found expression in the maximum allowance, which almost all goes to the supervisor. The agent receives only a small part of this increase, which is approximately equal to his sensitivity threshold.