Problems of rationing become relevant in situations where the volume of claims by interested parties relative to a certain resource exceeds its volume.

The focus of this study is on the practical aspects of applying various methods (schemes) for solving problems of distribution-limited resources. In the context of Arctic region problems, it will primarily be mineral minerals, oil and gas.

#### Models (Schemes) of Rationing

Recall also the main provisions of the theory of rationing models (fair distribution), which are supposed to be used in the subsequent presentation. The simplest problem of rationing is defined by a set (set of three factors)

where

I = {1, …, n}—a set of agents (players) (i ∈ I);

t—the actual amount of the resource that agents demand;

y = (y_{1}, y_{2}, …, y_{i}, …, y_{n})—vector of the space R^{n}, defining the individual requirements of the agents (for the distributed resource).

It is assumed that

t ≥ 0 and

y_{i} ≥ 0 (∀

i ∈

I). The solution of the problem of rationing (method of rationing) is called a vector

x ∈

R^{n}, such that:

From a substantive point of view, research on the analysis and comparison of the properties possessed by certain possible methods of rationing is of natural interest. In terms of economic and socio-economic applications, this makes it possible to formalize the notion of requirements for “good”, “fair”, “mutually acceptable”, “objectively indisputable” distribution (satisfaction of competitive claims).

Formally, the requirements for good distribution are postulated in the form of axioms or rules. These rules determine the principles by which the distribution of a resource is formed. Obviously, for different types of resources, the rules can be different. The most famous of the axioms in which such requirements are formulated are presented below.

From a substantive point of view, studies on the analysis and comparison of the properties possessed by certain possible methods of rationing are of natural interest. Within the framework of the formulated tasks (1)–(3) it is necessary to find a total distribution of claims, which in economic and socio-economic terms would allow us to justify the best solution to satisfy competitive claims.

The basic requirements for such distribution can be formulated in the form of basic axioms [

8].

Equal Treatment of Equals (ETE)—agents who present equal demands obtain an equal share.

Symmetry (SYM). This axiom (requirement) assumes that the rationing method

x =

pr(

I,

t,

y) is a symmetric function. Recall that a function

f(

x_{1},

x_{2}, …,

x_{j}, …,

x_{n}) is called symmetric if, for any permutation of its variables, the value of the function does not change:

No Advantageous Reallocation (NAR)—the change in demand between agents within a group (coalition) S does not change the total share received by the coalition.

Irrelevance of Reallocations (IR)—shifts (changes) of claims within a certain group (coalition) S should not affect the shares received by other agents not affected by the redistribution.

Independence of Merging and Splitting (IMS)—independence from mergers and divisions of agents, i.e., as a result of the formation of a new agent when merging, its share should be equal to the sum of the shares of merged agents. When dividing one agent into several, its share without balance should be distributed among them.

Decomposition (DEC). The performance of this axiom implies the identity of the results of the method of rationing in the case of its application to all possible partitions of the set I.

Zero Consistency (ZC)—deleting a member with zero demand should not affect the shares received by other members.

As a rule, it is not possible to offer a distribution scheme in which all of the listed requirements would be fulfilled. However, by selecting a set of criteria that applies to a particular type of resource, you can create a model.

Next, we will consider several methods for solving the problem of allocation of limited resources or the problem of equitable distribution. Each of the methods satisfies a number of the proposed rules or axioms.

The simplest method for solving the problem (1) is a

proportional method that distributes the resource between the agents according to the demand:

or

where

${y}_{I}={{\displaystyle \sum}}_{i\in I}{y}_{i}$. It is assumed that if is

y_{I} = 0 then

x = 0.

It is easy to prove that the proportional method satisfies the properties NAR, IR, IMS, DEC [

31].

Pay attention to the following meaningful moment. NAR or IR axioms are quite reasonable and natural for many of the rationing problems. However, in the case of competition problems for Arctic resources, their feasibility raises serious doubts. This is primarily due to the need to take into account the complex of historical, political and geopolitical factors that are inevitably present in inter-country interaction.

Next in popularity after the proportional method are methods that focus on leveling winnings and losses of agents. There are methods of uniform gains and uniform losses. The

method of uniform gains (UG) specifies the shares of agents as:

where λ is found from the equation

In essence, the UG method implements the principle of leveling gains in relation to the initial requirements, which are obtained by the results of the distribution of the participants of the division.

Another method opposite to the method of uniform gains is

the uniform loss method (UL). The method (UL) defines the shares received by agents as:

where μ is found from the equation

The UG method implements the principle of leveling “net losses” of players (y_{i} − x_{i}).

Like the proportional method, the UG and UL methods satisfy axioms of relative ranking:

These methods differ significantly in terms of ranking relative wins. It is easy to show that the UG method satisfies the axiom of progressivity, according to which, if 0 <

y_{i} ≤

y_{j}, then

and the UL method satisfies the regressivity axiom:

At the same time, it should be noted that the UG does not satisfy the regressivity axiom and UG does not satisfy to progressivity axiom.

Important properties of the ration methods are postulated by axioms of the composition.

In accordance with the axiom top songs UC (

UC—upper composition)

In terms of content, the property of the upper composition is relevant in the case when the actual amount of allocated resources (t) was less than initially expected t′. UC means the opportunity to review the originally planned optimistic proportion (r(I, t′, y)) as the requirements of agents in the distribution of actually realized resources t.

According to the axiom of the

lower composition LC:

LC is related to situations when the initial distribution of pessimistic volume (t′) occurs, and in the case of additive (t − t′), it is distributed according to the same method, taking into account previously satisfied requirements (y − r(I, t′, y)).

E. Moulin [

31], in particular, proved that the method of uniform wins (UG) is characterized by axioms LC and ZC, and the method of uniform losses (UL)—axioms UC and ZC.

Let us introduce the notation. $\mathsf{\pi}=\left({\mathsf{\pi}}_{1},{\mathsf{\pi}}_{2},\dots ,{\mathsf{\pi}}_{i},\dots ,{\mathsf{\pi}}_{n}\right)$—is a permutation of the set of numbers of agents I.

The random priority method is defined as:

where Π—is the set of all possible permutations of the set

I.

The distribution rule ${{\displaystyle \sum}}_{\mathsf{\pi}\in \mathsf{\Pi}}prio\left(\mathsf{\pi}\right)\left(I,t,y\right)$ implies that agent π_{1} has the highest priority, agent π_{1}—next, etc. Resources are allocated in the descending order of priority (in accordance with the requirements of y_{i}). Thus, in the general case, the requirement of some π_{k}-th agent is satisfied in part, the requirements of the agents with the numbers π_{k+1}, π_{k+2}, etc. remain unmet.

In essence, the Talmud method halves the requirements of the agents and distributes the resource in accordance with the UG-method. In the case that after this the undistributed part of the resource remains, then it is would be distributed according to the UL-method on the basis of half requirements also.

Any of these principles can be implemented in the problem of allocation of scarce resources (oil, gas and other mineral resources) in the Arctic region. However, its application requires to take into account the specific conditions necessary to determine both resources in accordance with the requirements of agents [

64].

There is a possibility of implementation of such principles in the construction of models of cooperative games. Cooperative games with transferable utility and characteristic functions can be constructed on the basis of the rationing problem (2):

where

$S\subset I$ is a coalition of players—possible association of participants of the procedures for the allocation of the resource.

y_{s}—sum of demands of participants of the coalition

S.

Game (18) corresponds to optimistic expectations of participants, game (19), on the contrary–pessimistic.

In this game we will focus on inter-country cooperation in the distribution of rights to the Arctic resources. Perhaps such a distribution is associated with the uncertainty of the boundaries of resources located in the Arctic region.

The theorem [

2,

32] is valid, according to which the method of random priority distributes resources according to the Shapley-value of games (19), (20); the method of Talmud—in accordance with their—nucleolus.

In accordance with the statement of this theorem, we can express the shares obtained by agents using the random priority method as

where

s = |

S| is the number of members in coalition

S (compare with method (16)).

In turn, on the basis of these methods can be developed methods of resource allocation in the framework of stochastic normalization schemes, similar to the methods of the Talmud and random priority. Further development of the theory of resource allocation is associated with the construction of stochastic cooperative game interaction of countries in the allocation of rights to resources, in which the requirements of agents are defined as random variables $\tilde{{y}_{i}}$ (i ∈ I).

The baseline definition and parameters of a stochastic cooperative game (SCG) is a pair of sets $\mathsf{\Gamma}=\left(I,\tilde{v}\right)$ where I = {1 … n}—set of participants; $\tilde{v}\left(S\right)$—random variables with known density functions ${P}_{\tilde{v}\left(s\right)}\left(y\right)$, which are interpreted as income (utility, payoffs), and are received by the corresponding coalitions $S\subset I$.

The solution of this game will allow participants to realize the fair division of a limited resource subject based on the requirements of the participants. In this case, the problem of equitable distribution of Arctic resources among the main applicant countries can be solved. As part of solving the problem of distribution coalitions can be formed, which will receive certain advantages in the distribution of limited economic resources.

One of the most important problems in the practical use of models of rationing is identification of deterministic point values of the agent’s requirements

y_{i}. Overcoming this problem is possible due to the complexity of the models and the introduction of assumptions about which demands of the agents are random variables

$\left({\tilde{y}}_{i}\right)$ with known distribution functions. In fact, the introduction of this premise means the transition to stochastic schemes (models) of rationing [

65].

In this case, the characteristic functions of cooperative games, corresponding to stochastic normalization schemes (by analogy with the characteristic functions of games (18) and (19)), will have the form [

5]:

In turn, on the basis of these methods can be developed methods of resource allocation in the framework of stochastic normalization schemes, similar to the Talmud and random priority methods. This approach, involving a combination of stochastic cooperative games and rationing schemes, is relatively innovative and, from the point of view of the authors, can be quite fruitful.

The application of these approaches provides some insight into the possible distribution of resources among countries. You need to pay attention to the fact that resources of various kinds can be distributed based on different rules. For example, minerals are tied to the territory of occurrence and production of resources; fish stocks migrate, and to determine the volume of their catch in each country, it is advisable to apply quotas for fishing.