## 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:  
\[c_{i}=\frac{R_{i}}{E_{i}}, \tag{51}\]  
where \(R_{i}\) is \(i\)-company's revenue and \(E_{i}\) is \(i\)-company's production volume for 2020. Table 1 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 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{table}
\begin{tabular}{c c c} \hline
**Coal Mining Company** & \(R_{i}\) **(Million USD)** & \(E_{i}\) **(Million Tons)** \\ \hline JSC SUEK & 4817 & 101.2 \\ PJSC MECHEL & 1125 & 15.9 \\ LLC EVRAZ & 1490 & 20.7 \\ \hline \end{tabular}
\end{table} 
Table 1: Enterprise data. \begin{table}
\begin{tabular}{c c} \hline
**Category** & **Reserves of Categories, Billion Tons** \\ \hline \(A+B+C_{1}\) & 55.2 \\ \(C_{2}\) & 13.9 \\ \(P_{1}+P_{2}+P_{2}\) & 222.5 \\ \hline \end{tabular}
\end{table} 
Table 2: Distribution of coal reserves. \begin{tabular}{l l l} \(N=3\) & \(-\) & The number of players; \\ \(x_{0}=69,100\) & \(-\) & The initial level of resource reserves; \\ \(t_{0}=0\) & \(-\) & The initial time; \\ \(T=356\) & \(-\) & The final time; \\ \(c_{1}=47.59,\quad c_{2}=70.78,\quad-\) & The values of the coefficients from payoff; \\ \(c_{3}=71.98\) & & \\ \(\alpha\in[0.8,1.4]\) & \(-\) & The accuracy of the resource reserves estimation; \\ \(\beta\in[0.4,0.7]\) & \(-\) & The extraction-level parameter; \\ \end{tabular}   
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.