Value of Information in the Deterministic Optimal Resource Extraction Problem with Terminal Constraints

Abstract

In this paper, we consider a continuous-time differential game of non-renewable resource extraction with n players and assume that the terminal conditions of the game vary. To characterize the gain that the players obtain by using precise information, we use the notion of value of information in relation to the knowledge of terminal conditions. The stated problem is studied for two cases, a linear and a nonlinear one. The obtained results are illustrated for the real values of the parameters.

1. Introduction

In the problems of resource exploitation, information about resource availability plays a key role as it influences the future profit of the company [1,2]. The information about the boundary conditions influences the choice of the control strategy and the amount of the profit gained. To quantify the gain obtained from knowing more precise information, we use the characteristic dubbed value of information (VoI). For more information and examples of the use of the value of information in resource-extraction applications, we refer the interested reader to the papers [3,4]. Despite a substantial number of papers devoted to the concept of value of information, its role in the analysis of specific problems of resource extraction is still insufficiently studied. Thus, there is a clear need for a further investigation of the subject.

In this paper, we further develop the line of research initiated in the papers [5,6] as well as more recent research [7,8], and we consider the problem of determining the value of information in a continuous-time differential game of non-renewable resource extraction with n agents (players). In this particular case, we assume that the terminal conditions on the game vary and study the value of information in relation to the knowledge of terminal conditions. An alternative approach is presented in [9], which studies the same resource-extraction problem but assumes that the distribution of the time-until-failure is known in advance, while the concept of the value of information can be used to measure the inaccuracies in which such an approach incurs. Additionally, note that the problem of resource extraction considered in this paper should not be confused with a related research direction of resource mining in computer applications, such as, e.g., the one described in [10].

The rest of the paper is structured as follows. In Section 2, we describe the problem statement. In Section 3, two classes of model examples are introduced. Section 4 and Section 5 present a detailed analysis of the considered classes of problems. Section 6 presents a numerical illustration of the obtained results for the case of real life parameters. Finally, Section 7 concludes the paper.

2. Problem Formulation

Consider a differential game of n participants (players) $\Gamma (t_0,x_0,T_f)$ [11]. Let $N=\{1,\dots ,n\}$ be the set n of players. Let the game start at the initial moment $t_0$ from the initial state $x_0$, which has a prescribed duration $(T_f-t_0)$. The following assumptions are made:

• The dynamic constraints of the game are given by:

  $$\dot{x}=g(x\left(t\right),u_1\left(t\right),\dots ,u_n\left(t\right)),x\left(t_0\right)=x_0\in X\subset {\mathbb{R}}^m.$$

  (1)

  where the state Equations (1) are the ordinary differential equations, whose solutions satisfy the standard existence and uniqueness requirements, which holds if the function $g(x,u_1,\dots ,u_n)$ in (1) is a differentiable function of all its arguments;
• The controls $u_i\left(t\right)$ are open-loop strategies [12];
• The controls $u_i\left(t\right)$ belong to the sets of admissible controls ${\mathcal{U}}_i$, which consist of all measurable functions on the interval $[t_0,T_f]$, taking values in the set of admissible control values ${\mathbb{U}}_i$, which are in turn a convex compact subsets of ${\mathbb{R}}^k$;
• The integral payoff is

  $$K_i(x_0,t_0,T_f,u)=\underset{t_0}{\overset{T_f}{\int }}{f}_i(x\left(t\right),u\left(t\right))dt+F_i\left(x\left(T_f\right)\right),i=1,\dots ,n,$$

  (2)

  where $x\left(t\right)$ satisfies system (1), and the functions $f_i(·,·),F_i(·)$ are smooth. Denote $f(x\left(t\right),u\left(t\right))={\displaystyle \sum _{i=1}^n}{f}_i(x\left(t\right),u\left(t\right))$, $F\left(x\left(T_f\right)\right)={\displaystyle \sum _{i=1}^n}{F}_i\left(x\left(T_f\right)\right)$.

3. Model Example

Consider the model example within the framework of differential game formulation presented above. It is assumed that n players (firms or countries) simultaneously exploit a non-renewable natural resource (see, e.g., [11]). The strategy of each player i is to determine the extraction effort $u_i$ from some admissible set $U_i=[0,u_{max}],u_{max}>0$. Further, $x\left(t\right)$ denotes the state corresponding to the resource stock at the time t available to the extraction. Finally, it is assumed that the game is played over a finite time interval $[t_0,T]$.

Consider two theoretical model examples with differing dynamics. In the first case, the dynamics of the stock are given by the following linear differential equation with the initial and terminal conditions $x_0,x_f\ge 0$:

$$\left\{\begin{array}{c}\dot{x}\left(t\right)=-{\sum }_{i=1}^n{u}_i\left(t\right),t\in [t_0,T]\hfill \\ x\left(t_0\right)=x_0,\hfill \\ x\left(T\right)=x_f,\hfill \end{array}\right.$$

(3)

The second case model has different dynamic constraints and the same boundary conditions:

$$\left\{\begin{array}{c}\dot{x}\left(t\right)=-{\sum }_{i=1}^n{u}_i^2\left(t\right),t\in [t_0,T]\hfill \\ x\left(t_0\right)=x_0,\hfill \\ x\left(T\right)=x_f,\hfill \end{array}\right.$$

(4)

The objective function of the player i for both cases is defined as:

$$K_i(x_0,x_f,T-t_0,u)=\underset{t_0}{\overset{T}{\int }}{c}_i\sqrt{u_i}dt,$$

(5)

In the considered differential game, every i player’s goal is to maximize the total profit:

$$\begin{array}{c}{K}_i(x_0,x_f,T-t_0,u)\to \underset{u_i}{max}.\hfill \end{array}$$

(6)

We also consider the situation when the game is played in a cooperative mode and the players cooperate to achieve the maximum total payoff:

$$\begin{array}{c}K(x_0,x_f,T-t_0,u)={\displaystyle \sum _{i=1}^n}{K}_i(x_0,x_f,T-t_0,u)\to \underset{u_1,\dots ,u_n}{max}.\hfill \end{array}$$

(7)

The optimal open-loop cooperative strategies of players $u^{*}\left(t\right)=(u_1^{*}\left(t\right),\dots ,u_n^{*}\left(t\right))$, $t\in [t_0;T]$ are found from solving the optimization problem:

$$\begin{array}{c}{u}^{*}\left(t\right)=\underset{u_1\left(t\right),\dots ,u_n\left(t\right)}{arg\; max}{\displaystyle \sum _{i=1}^n}{K}_i(x_0,x_f,T-t_0,u).\hfill \end{array}$$

(8)

The trajectory that corresponds to the optimal cooperative strategies $u^{*}\left(t\right)$ is called the optimal cooperative trajectory and is denoted by $x^{*}\left(t\right)$.

The choice of optimal control influences the optimal trajectory and the payoffs of the players. On the other hand, the optimal control is selected in such a way that the boundary conditions are satisfied. Assume that we have imprecise information about the boundary conditions. To characterize the impact of the accuracy of the available information on the amount of the obtained payoff, we introduce the notion of value of information (V).

Definition 1.

The normalized value of information is defined as

$$V(x_0,x_f,T-t_0)=|{\displaystyle \frac{{\displaystyle \sum _{i=1}^n}{K}_i(x_0,x_f,T-t_0,u^{*})-{\displaystyle \sum _{i=1}^n}{K}_i(x_0,x_f,T-t_0,{\widehat{u}}^{*})}{{\displaystyle \sum _{i=1}^n}{K}_i(x_0,x_f,T-t_0,u^{*})}}|,$$

where ${\displaystyle \sum _{i=1}^n}{K}_i(x_0,x_f,T-t_0,u^{*})$ is the total payoff obtained with precise information, while ${\displaystyle \sum _{i=1}^n}{K}_i(x_0,x_f,T-t_0,{\widehat{u}}^{*})$ is the payoff obtained with imprecise information.

4. Computations for Model Example with Linear Dynamics

4.1. Cooperative Solution

Consider the model with dynamics (3) and the objective function (5). The optimization problem is solved using the Pontryagin maximum principle (PMP) [12], which is widely recognized as one of the central tools for determining optimal strategies. We mention the papers [13,14] to give an idea of just a couple of applications of this celebrated technique. We define the Hamiltonian function as

$$H(\psi ,u)=\psi \left(-\sum _{i=1}^n{u}_i\right)+\left(\sum _{i=1}^n{c}_i\sqrt{u_i}\right).$$

(9)

The optimal controls $u^{*}$ are computed from the first-order optimality conditions $\frac{\partial H}{\partial u_i}=0$ as

$$u_i^{*}={\left({\displaystyle \frac{c_i}{2\psi }}\right)}^2,i=\overline{1,n}.$$

(10)

Noting that the adjoint variable is constant according to optimality condition $\frac{\partial H}{\partial x}=0$, $\psi ={\psi }_0=const$ and, furthermore, has to satisfy $\psi >0$. Then, by substituting the expression of optimal control into system (3), we obtain

$${\psi }_0=\sqrt{\frac{(T-t_0)}{4(x_0-x_f)}{\displaystyle \sum _{j=1}^n}{c}_j^2}.$$

(11)

The optimal cooperative strategies and trajectory is thus

$$u_i^{*}={\displaystyle \frac{c_i^2(x_0-x_f)}{{\displaystyle \sum _{j=1}^n}{c}_j^2(T-t_0)}}.$$

(12)

$$x^{*}\left(t\right)=\left(\frac{x_f-x_0}{T-t_0}\right)(t-t_0)+x_0.$$

(13)

Further, we check the control constraints according to the of the problem, i.e., $u_i\in [0;u_{max}]$. For the admissible controls, the following condition should be satisfied:

$$x_f\ge x_0-\frac{{\displaystyle \sum _{j=1}^n}{c}_j^2}{c_i^2}{u}_{max}(T-t_0),i=\overline{1,n}.$$

(14)

The optimal value of the total payoff then equals

$$K(x_0,x_f,T-t_0,u^{*})=\sqrt{\sum _{j=1}^n{c}_j^2(x_0-x_f)(T-t_0)}.$$

(15)

4.2. Value of Information

Suppose that the information about the initial value of the resource stock is not available to the players and that they have information about the initial stock being ${\widehat{x}}_0$ instead. In this case, the controls chosen by the players would have the form

$${\widehat{u}}_i^{*}\left(t\right)={\displaystyle \frac{c_i^2({\widehat{x}}_0-x_f)}{{\displaystyle \sum _{j=1}^n}{c}_j^2(T-t_0)}},i=\overline{1,n},$$

(16)

and the trajectory corresponding to these controls is

$${\widehat{x}}^{*}\left(t\right)=\left({\displaystyle \frac{x_f-{\widehat{x}}_0}{T-t_0}}\right)(t-t_0)+x_0.$$

(17)

Next, we define an estimated initial value as ${\widehat{x}}_0=\alpha x_0$, where $\alpha >1$ corresponds to the case when the information is overestimated, and $\alpha <1$ to the case when the information is underestimated.

For convenience, we define the value at the final moment of time as $x_f=\beta x_0$, where, according to the problem statement, $\beta \in [0,1]$. In this case, $\beta =1$ corresponds to the case when the exact initial and final resource stocks coincide, i.e., the actual exploitation of the resource is not carried out with precise information. For this case, define the value of information as the maximum possible value, i.e., 1. Next, consider the cases where $\beta \ne 1$. Taking into account this notation, we obtain the value of information

$$V_1=\left\{\begin{array}{cc}1-\sqrt{{\displaystyle \frac{\alpha -\beta }{1-\beta }}},\hfill & \alpha ⩽1,\hfill \\ \sqrt{{\displaystyle \frac{\alpha -\beta }{1-\beta }}}-1,\hfill & \alpha >1.\hfill \end{array}\right.$$

(18)

Note that the value of $\alpha$ corresponds to the accuracy of resource reserves estimation, which also affects the terminal constraint, i.e., $x_f$. Thus, we can introduce ${\widehat{x}}_f=\alpha x_f$. We obtain controls for this case and the respective trajectory as

$${\tilde{u}}_i^{*}\left(t\right)={\displaystyle \frac{c_i^2({\widehat{x}}_0-{\widehat{x}}_f)}{{\displaystyle \sum _{j=1}^n}{c}_j^2(T-t_0)}},i=\overline{1,n},$$

(19)

$${\tilde{x}}^{*}\left(t\right)=\left({\displaystyle \frac{{\widehat{x}}_f-{\widehat{x}}_0}{T-t_0}}\right)(t-t_0)+x_0.$$

(20)

In these notations, the value of information will take the form

$$V_2=\left\{\begin{array}{cc}1-\sqrt{\alpha },\hfill & \alpha ⩽1,\hfill \\ \sqrt{\alpha }-1,\hfill & \alpha >1.\hfill \end{array}\right.$$

(21)

4.3. Nash Equilibrium

For the non-cooperative case, we define the Hamiltonian function as:

$$H_i({\psi }_i,u)={\psi }_i\left(-\sum _{i=1}^n{u}_i\right)+\left(c_i\sqrt{u_i}\right).$$

(22)

The optimal controls $u^{*}$ are obtained from the first-order optimality conditions $\frac{\partial H_i}{\partial u_i}=0$ as:

$$u_i^{*}={\left({\displaystyle \frac{c_i}{2{\psi }_i}}\right)}^2,i=\overline{1,n}.$$

(23)

Noting that the adjoint variable is constant according to optimality condition $\frac{\partial H_i}{\partial x}=0$, ${\psi }_i={\psi }_{0i}=const$ and, furthermore, has to satisfy ${\psi }_i>0$. Thus, we substitute this into (23), then substitute the expression of optimal control into system (3).

Solving this system, we come to the condition

$$\sum _{i=1}^3{\left({\displaystyle \frac{c_i}{2{\psi }_{0i}}}\right)}^2=\frac{x_0-x_f}{T-t_0}.$$

(24)

Let us assume that the coefficients are equal to each other, i.e., $c_i=c_j=c$. Then, the players are symmetrical, which means that their optimal controls should be the same, i.e., $u_i=u_j$. Consider the first equation of the system with respect to one player

$$\dot{x}=-n{u}_1=-n{\left({\displaystyle \frac{c}{2{\psi }_{01}}}\right)}^2.$$

(25)

Finally, we obtain the solution $x^{*}\left(t\right)$ and the optimal control of the first player:

$$x^{*}\left(t\right)=\left(\frac{x_f-x_0}{T-t_0}\right)(t-t_0)+x_0,$$

(26)

$$u_1^{*}=\frac{(x_0-x_f)}{n(T-t_0)}.$$

(27)

The same can be done for other players.

Next, we check the control constraints according to the conditions of the problem, i.e., $u_i\in [0,u_{max}]$. The controls are admissible if the following condition is satisfied:

$$x_f\ge x_0-n{u}_{max}(T-t_0).$$

Find the payoffs of each i player

$$K_i(x_0,x_f,T-t_0)=c\sqrt{\frac{(x_0-x_f)(T-t_0)}{n}},i=\overline{1,n}.$$

(28)

As a result, by substituting the optimal control into the objective function, we obtain

$$K(x_0,x_f,T-t_0)=c\sqrt{n(X_0-X_f)(T-t_0)}.$$

(29)

Similarly to the calculations from the Section 4.2, we can consider two cases of uncertainty in a non-cooperative scenario separately for each player. As a result, the values of the information coincide with the expressions (18) and (21).

5. Computations for Model Example with Non-Linear Dynamics

5.1. Cooperative Solution

In this section, we consider the model with dynamics (4) and the objective function (5). The optimization problem is solved using the Pontryagin maximum principle [12]. We define the Hamiltonian function as

$$H(\psi ,u)=\psi \left(-\sum _{i=1}^n{u}_i^2\right)+\left(\sum _{i=1}^n{c}_i\sqrt{u_i}\right).$$

(30)

The optimal controls $u^{*}$ are computed from the first-order optimality conditions $\frac{\partial H}{\partial u_i}=0$ as

$$u_i^{*}={\left({\displaystyle \frac{c_i}{4\psi }}\right)}^{\frac{2}{3}},i=\overline{1,n}.$$

(31)

Noting that the adjoint variable is constant according to the optimality condition $\frac{\partial H}{\partial x}=0$, $\psi ={\psi }_0=const$ and, furthermore, has to satisfy $\psi >0$. Then, by substituting the expression of optimal control into system (3), we obtain

$${\psi }_0=\frac{1}{4}{\left(\frac{(T-t_0)}{(x_0-x_f)}\sum _{j=1}^n{c}_j^{\frac{4}{3}}\right)}^{\frac{3}{4}}.$$

(32)

The optimal cooperative strategies and trajectory is thus

$$u_i^{*}=\frac{c_i^{\frac{2}{3}}}{\sqrt{{\displaystyle \sum _{j=1}^n}{c}_j^{\frac{4}{3}}}}\sqrt{{\displaystyle \frac{x_0-x_f}{T-t_0}}}.$$

(33)

$$x^{*}\left(t\right)=\left(\frac{x_f-x_0}{T-t_0}\right)(t-t_0)+x_0.$$

(34)

Further, we check the control constraints according to the problem, i.e., $u_i\in [0;u_{max}]$. To find admissible controls, we should satisfy the following condition:

$$x_f\ge x_0-\frac{{\displaystyle \sum _{j=1}^n}{c}_j^{\frac{4}{3}}}{c_i^{\frac{4}{3}}}{u}_{max}^2(T-t_0),i=\overline{1,n}.$$

(35)

The optimal value of the total payoff then equals

$$K(x_0,x_f,T-t_0,u^{*})={\displaystyle \frac{{\displaystyle \sum _{i=1}^n}{c}_i^{\frac{1}{3}}}{\sqrt[4]{{\displaystyle \sum _{j=1}^n}{c}_j^{\frac{4}{3}}}}}\sqrt[4]{(x_0-x_f)(T-t_0)}.$$

(36)

5.2. Value of Information

Now, suppose that the information about the initial value of the resource stock is not available to the players and that they have information about the initial stock being ${\widehat{x}}_0$ instead. In this case, the controls chosen by the players would have the form

$${\widehat{u}}_i^{*}\left(t\right)=\frac{c_i^{\frac{2}{3}}}{\sqrt{{\displaystyle \sum _{j=1}^n}{c}_j^{\frac{4}{3}}}}\sqrt{{\displaystyle \frac{{\widehat{x}}_0-x_f}{T-t_0}}},i=\overline{1,n},$$

(37)

and the trajectory corresponding to these controls is

$${\widehat{x}}^{*}\left(t\right)=\left({\displaystyle \frac{x_f-{\widehat{x}}_0}{T-t_0}}\right)(t-t_0)+x_0,$$

(38)

Next, we define an estimated initial value as ${\widehat{x}}_0=\alpha x_0$, where $\alpha >1$ corresponds to the case when the information is overestimated and $\alpha <1$ to the case when the information is underestimated.

For convenience, we define the value at the final moment of time as $x_f=\beta x_0$, where, according to the problem statement, $\beta \in [0,1]$. In this case, $\beta =1$ corresponds to the case when the exact initial and final resource stocks coincide, i.e., the actual exploitation of the resource is not carried out with precise information. For this case, define the value of information as the maximum possible value, i.e., 1. Next, consider the cases where $\beta \ne 1$. Taking into account these designations, we obtain the value of information

$$V_3=\left\{\begin{array}{cc}1-\sqrt[4]{{\displaystyle \frac{\alpha -\beta }{1-\beta }}},\hfill & \alpha ⩽1,\hfill \\ \sqrt[4]{{\displaystyle \frac{\alpha -\beta }{1-\beta }}}-1,\hfill & \alpha >1.\hfill \end{array}\right.$$

(39)

Note that the value of $\alpha$ corresponds to the accuracy of resource reserves estimation, which also affects the terminal constraint, i.e., $x_f$. Thus, we can introduce ${\widehat{x}}_f=\alpha x_f$. We obtain controls for this case and the corresponding trajectory as

$${\widehat{u}}_i^{*}\left(t\right)=\frac{c_i^{\frac{2}{3}}}{\sqrt{{\displaystyle \sum _{j=1}^n}{c}_j^{\frac{4}{3}}}}\sqrt{{\displaystyle \frac{{\widehat{x}}_0-{\widehat{x}}_f}{T-t_0}}},i=\overline{1,n},$$

(40)

$${\widehat{x}}^{*}\left(t\right)=\left({\displaystyle \frac{{\widehat{x}}_f-{\widehat{x}}_0}{T-t_0}}\right)(t-t_0)+x_0,$$

(41)

In these notations, the value of the information will take the form:

$$V_4=\left\{\begin{array}{cc}1-\sqrt[4]{\alpha },\hfill & \alpha ⩽1,\hfill \\ \sqrt[4]{\alpha }-1,\hfill & \alpha >1.\hfill \end{array}\right.$$

(42)

5.3. Nash Equilibrium

For the non-cooperative case, we define the Hamiltonian function as

$$H_i({\psi }_i,u)={\psi }_i\left(-\sum _{i=1}^n{u}_j^2\right)+\left(c_i\sqrt{u_i}\right).$$

(43)

The optimal controls $u^{*}$ are computed from the first-order optimality conditions $\frac{\partial H_i}{\partial u_i}=0$ as

$$u_i^{*}={\left({\displaystyle \frac{c_i}{4{\psi }_i}}\right)}^{\frac{2}{3}},i=\overline{1,n}.$$

(44)

Note that the adjoint variable is constant according to optimality condition $\frac{\partial H_i}{\partial x}=0$, ${\psi }_i={\psi }_{0i}=const$ and, furthermore, has to satisfy ${\psi }_i>0$. Thus, we substitute this into (44) and then substitute the expression of optimal control into system (4).

Solving this system, we come to the condition

$$\sum _{i=1}^3{\left({\displaystyle \frac{c_i}{4{\psi }_{0i}}}\right)}^{\frac{4}{3}}=\frac{x_0-x_f}{T-t_0}.$$

(45)

Let us assume that the coefficients are equal to each other, i.e., $c_i=c_j=c$. Then, the players are symmetrical, which means that their optimal controls should be the same, i.e., $u_i=u_j$. Consider the first equation of the system for one player and solve it with respect to the adjoint variable:

$$\dot{x}=-n{u}_1=-n{\left({\displaystyle \frac{c}{4{\psi }_{01}}}\right)}^{\frac{4}{3}}.$$

(46)

Finally, we obtain the solution $x^{*}\left(t\right)$ and the optimal control of the first player:

$$x^{*}\left(t\right)=\left(\frac{x_f-x_0}{T-t_0}\right)(t-t_0)+x_0,$$

(47)

$$u_1^{*}=\frac{(x_0-x_f)}{n(T-t_0)}.$$

(48)

The same can be done for other players.

Next, we check the control constraints according to the conditions of the problem, i.e., $u_i\in [0,u_{max}]$. The controls are admissible if the following condition is satisfied:

$$x_f\ge x_0-n{u}_{max}^2(T-t_0).$$

Find the payoffs of each i player

$$K_i(x_0,x_f,T-t_0,u^{*})=c\sqrt[4]{\frac{(x_0-x_f)(T-t_0)}{n}},i=\overline{1,n}.$$

(49)

As a result, substituting the optimal control into the objective function, we obtain

$$K(x_0,x_f,T-t_0,u^{*})=c\sqrt[4]{n^3(X_0-X_f)(T-t_0)}.$$

(50)

Similarly to the calculations from Section 5.2, we can consider two cases of uncertainty in a non-cooperative scenario separately for each player. As a result, the values of the information coincide with the expressions (39) and (42).

6. Numeric Example

This section is devoted to the specific numeric example of non-renewable resource extraction given by the system with linear dynamics. The same can be done for other system with non-linear dynamics.

Consider the coal mining problem [15,16]. The Kuznetsk Basin possesses some of the most extensive coal deposits anywhere in the world. There are a huge number of mining enterprises operating in the Russian coal industry. For example, coal assets in the Kuznetsk basin are owned by three companies: JSC SUEK, which specializes in the power and coal industry; and PJSC MECHEL and LLC EVRAZ, which are engaged in the metallurgical and coke industries. Based on the available data [17,18,19], we formalize the theoretical problem of coal mining in the form of a three-player differential cooperative game, where the players are the following companies: SUEK JSC, PJSC MECHEL, and LLC EVRAZ. The same can be done for non-cooperative games.

To determine the coefficients $c_i$ that determine the payoffs (5), we will use the data on the company’s revenue for 2020. The coefficients $c_i$ will be determined by the following formula:

$$c_i=\frac{R_i}{E_i},$$

(51)

where $R_i$ is i-company’s revenue and $E_i$ is i-company’s production volume for 2020.

Table 1. Enterprise data.

Coal Mining Company	${\mathit{R}}_{\mathit{i}}$ (Million USD)	${\mathit{E}}_{\mathit{i}}$ (Million Tons)
JSC SUEK	4817	101.2
PJSC MECHEL	1125	15.9
LLC EVRAZ	1490	20.7

contains information on the revenue of three companies for 2020 [17,18,19].

Resource extraction is regulated by Russian laws. Particularly, Russian laws divide reserves of minerals according to the degree of geological knowledge into the categories $A,B,C_1,C_2$. Moreover, there are categories of predictive resources ($P_1,P_2,P_3$), which are subdivided according to the degree of their validity [15].

Table 2. Distribution of coal reserves.

Category	Reserves of Categories, Billion Tons
$A+B+C_1$	55.2
$C_2$	13.9
$P_1+P_2+P_2$	222.5

contains information on coal reserves by category in the Kuznetsk Coal Basin [16].

Based on their assessment criteria, assume category $C_2$ to be less credible for the evaluation of the resource reserves level. Reserves of category $C_2$ determine about $20\%$ of all known resources, and predicted resources significantly exceed more accurately estimated resource reserves, which are redundant. Consider this information to set the parameters of the model, such as the initial level of resource reserves and the accuracy of resource reserves estimation. In addition, consider a game to be hold during one year.

Next, consider a case with estimated initial level of resource reserves, and indicate it as case 1. Based on the above-mentioned assumptions, consider the following values of the following parameters:

$$\begin{array}{cccc}\hfill & N=3\hfill & \hfill -& \mathrm{The}\mathrm{number}\mathrm{of}\mathrm{players};\hfill \\ \hfill & x_0=\mathrm{69,100}\hfill & \hfill -& \mathrm{The}\mathrm{initial}\mathrm{level}\mathrm{of}\mathrm{resource}\mathrm{reserves};\hfill \\ \hfill & t_0=0\hfill & \hfill -& \mathrm{The}\mathrm{initial}\mathrm{time};\hfill \\ \hfill & T=356\hfill & \hfill -& \mathrm{The}\mathrm{final}\mathrm{time};\hfill \\ \hfill & c_1=47.59,c_2=70.78,\hfill & \hfill -& \mathrm{The}\mathrm{values}\mathrm{of}\mathrm{the}\mathrm{coefficients}\mathrm{from}\mathrm{payoff};\hfill \\ \hfill & c_3=71.98\hfill & \hfill & \\ \hfill & \alpha \in [0.8,1.4]\hfill & \hfill -& \mathrm{The}\mathrm{accuracy}\mathrm{of}\mathrm{the}\mathrm{resource}\mathrm{reserves}\mathrm{estimation};\hfill \\ \hfill & \beta \in [0.4,0.7]\hfill & \hfill -& \mathrm{The}\mathrm{extraction}\text{-}\mathrm{level}\mathrm{parameter};\hfill \end{array}$$

Assume also that the set of admissible controls corresponds to the interval

$${\mathbb{U}}_i=[0;U],U<\infty .$$

For clarity, assume $U=100$. The obtained optimal controls and optimal trajectories for this case are presented in Figure 1.

Next, consider a case with an estimated initial and final level of resource reserves and designate it case 2.

The dependence of the normalized value of information on the accuracy parameter is presented for two cases with different extraction parameters in Figure 2. Since the logarithmic scale shows the relative change of a value, the logarithm of the normalized value of information is also presented in Figure 2.

7. Conclusions

We considered a continuous-time differential game of non-renewable resource extraction under the assumption that the terminal conditions of the problem are not precisely defined. To characterize the gain that the players obtain by using precise information, we used the notion of value of information. To illustrate the used approach, we considered two cases: a linear and a nonlinear one. Quite remarkably, the obtained results are rather similar: while the value of the payoff function in the Nash equilibrium case are equal, the value of information for both cases differs but has the same structure. The obtained results are illustrated for real values of parameters that characterize three coal mining companies from the Kuznetsk basin, Russia.