## 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, \(x(t)\) 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 \(x_{0},x_{f}\geq 0\):  
\[\left\{\begin{aligned} \dot{x}(t)&=-\sum\
olimits_{i=1}^{n }u_{i}(t),\quad t\in[t_{0},T]\\ x(t_{0})&=x_{0},\\ x(T)&=x_{f},\end{aligned}\right. \tag{3}\]  
The second case model has different dynamic constraints and the same boundary conditions:  
\[\left\{\begin{aligned} \dot{x}(t)&=-\sum\
olimits_{i=1}^{n }u_{i}^{2}(t),\quad t\in[t_{0},T]\\ x(t_{0})&=x_{0},\\ x(T)&=x_{f},\end{aligned}\right. \tag{4}\]  
The objective function of the player \(i\) for both cases is defined as:  
\[K_{i}(x_{0},x_{f},T-t_{0},u)=\int\limits_{t_{0}}^{T}c_{i}\sqrt{u_{i}}dt, \tag{5}\]  
In the considered differential game, every \(i\) player's goal is to maximize the total profit:\[K_{i}(x_{0},x_{f},T-t_{0},u)\rightarrow\max\limits_{u_{i}}. \tag{6}\]  
We also consider the situation when the game is played in a cooperative mode and the players cooperate to achieve the maximum total payoff:  
\[K(x_{0},x_{f},T-t_{0},u)=\sum\limits_{i=1}^{n}K_{i}(x_{0},x_{f},T-t_{0},u) \rightarrow\max\limits_{u_{1},\ldots,u_{n}}. \tag{7}\]  
The optimal open-loop cooperative strategies of players \(u^{*}(t)=(u_{1}^{*}(t),\ldots,u_{n}^{*}(t))\), \(t\in[t_{0};T]\) are found from solving the optimization problem:  
\[u^{*}(t)=\operatorname*{arg\,max}\limits_{u_{1}(t),\ldots,u_{n}(t)}\sum \limits_{i=1}^{n}K_{i}(x_{0},x_{f},T-t_{0},u). \tag{8}\]  
The trajectory that corresponds to the optimal cooperative strategies \(u^{*}(t)\) is called the optimal cooperative trajectory and is denoted by \(x^{*}(t)\).  
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  
\[V(x_{0},x_{f},T-t_{0})=|\frac{\sum\limits_{i=1}^{n}K_{i}(x_{0},x_{f},T-t_{0},u ^{*})-\sum\limits_{i=1}^{n}K_{i}(x_{0},x_{f},T-t_{0},\widehat{u}^{*})}{\sum \limits_{i=1}^{n}K_{i}(x_{0},x_{f},T-t_{0},u^{*})}|,\]  
where \(\sum\limits_{i=1}^{n}K_{i}(x_{0},x_{f},T-t_{0},u^{*})\) is the total payoff obtained with precise information, while \(\sum\limits_{i=1}^{n}K_{i}(x_{0},x_{f},T-t_{0},\widehat{u}^{*})\) is the payoff obtained with imprecise information.