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Next: Marginal cases: either small Up: Equilibrium is a principle Previous: Necessity.

Sufficiency.

Let tex2html_wrap_inline915 and tex2html_wrap_inline917 be the solution of (5), then, in accordance with the definition of tex2html_wrap_inline887 and tex2html_wrap_inline947 we have:
eqnarray89
Let be now tex2html_wrap_inline949, then, repeating the previous considerations, we get tex2html_wrap_inline951. In the same way such type of statement is proved for the actor B. Therefore, the values tex2html_wrap_inline915 and tex2html_wrap_inline917 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 tex2html_wrap_inline845, which differs from the equilibrium value tex2html_wrap_inline901, then the value of his utility function tex2html_wrap_inline887 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, tex2html_wrap_inline887, we necessarily increase tex2html_wrap_inline947, then by the same token the main statement is proved.

Let tex2html_wrap_inline901 and tex2html_wrap_inline903 satisfy the conditions (4), hence, they are the solution of (5). Let us try to decrease the value tex2html_wrap_inline975 changing only tex2html_wrap_inline845. Since in the equilibrium tex2html_wrap_inline979 then by increasing tex2html_wrap_inline845 we get tex2html_wrap_inline983, and tex2html_wrap_inline887 increases. On the contrary, if we decrease tex2html_wrap_inline845 then tex2html_wrap_inline989, and tex2html_wrap_inline887 increases also. We can decrease tex2html_wrap_inline887 only if we decrease tex2html_wrap_inline849, but this implies an increase of tex2html_wrap_inline947. Therefore, the equilibrium solution is efficient and the point tex2html_wrap_inline999 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 COtex2html_wrap_inline817 emissions and their reductions we must say a few words about other approaches.


next up previous
Next: Marginal cases: either small Up: Equilibrium is a principle Previous: Necessity.

Werner von Bloh (Data & Computation)
Thu Jul 13 15:46:47 MEST 2000