Participants' information awareness and existence of equilibrium in positional iteration games of many players

In positional games, dynamical decision-making models are studied for the situation when there is a conflict of interests and participants know the current position of the game. Each player is able to control the dynamical system partially. The control strategy chosen by a player is a function defined on the system's phase space. Players check the system's movement and obtain an implicit idea about strategies applied by their partners. The principle of players' rational behavior consists in trying to achieve the situation of Nash equilibrium. It is proved that an equilibrium can be reached as a result of collective efforts to choose the system's general program control. Stability of the solution is reached by using the threat of punishment to those who refuse to fulfill the program. Positions control and some additional information give players the possibility to identify the guilty player. Then, after a delay, he/she is punished by all other players. The theorem of existence of an equilibrium is applied to economic and mathematical model.