Sufficiency.

Let and be the solution of (5), then, in accordance with the definition of and we have:

Let be now , then, repeating the previous considerations, we get . In the same way such type of statement is proved for the actor B. Therefore, the values and satisfy the conditions (4).

Thus, we have proved that the solution of system (5) describes the equilibrium. The equilibrium is stable. Indeed, if, for instance, the actor A chooses the emission , which differs from the equilibrium value , then the value of his utility function will be higher than the value of this function in the equilibrium. The stability is a very important argument in the process of selection among different virtual decision makings. This principle (stability) guarantees the ``fair play'' since none of the actors has an advantage to depart from the accepted agreement (equilibrium).

Nevertheless, one problem appears: is the suggested choice efficient? In other words, would there be another choice, which is more advantageous for both actors than the equilibrium one? It is proved that the equilibrium solution is efficient, i.e., it can not be improved for both actors simultaneously, and the solution belongs to the Pareto's set. If we would prove that decreasing, for instance, , we necessarily increase , then by the same token the main statement is proved.

Let and satisfy the conditions (4), hence, they are the solution of (5). Let us try to decrease the value changing only . Since in the equilibrium then by increasing we get , and increases. On the contrary, if we decrease then , and increases also. We can decrease only if we decrease , but this implies an increase of . Therefore, the equilibrium solution is efficient and the point belongs to the Pareto's set.

Note that all these results come from a partial application of the general theory mentioned in [4]. Speaking about an application of the game theory to the problem of CO emissions and their reductions we must say a few words about other approaches.