The Optimal Principle of Stable Solutions in Lexicographic Cooperative Games

Neumann-Morgenstern’s solutions NM (v) as stable solution’s optimal principle is stated in a lexicographic v = (v1, v2 ,..., vm)T cooperative game. The conditions of NM (v) existence are proved for the cases, when: 1. v1 scalar cooperative game’s C-core C (v1) and NM (v1)solutions are equal; 2. Scalar cooperative v1 game’s C-core and NM (v1) solutions are different. In the first case the sufficient conditions are proved in order to say that a C-core C (v) of a lexicographic cooperative v game must not be empty and it should be coincided to NM (v) . In the second case the necessary condition of NM (v) existence is proved. In the case of the existence of NM (v) solutions their forms can be established. Some properties NM (v) of solutions are stated.