On a game of influence on system parameters

We study a game with finitely many players. The gain functions depend not only on the players' strategy profiles, but also on a certain vector state parameter of the system. The relation between a strategy profile and the value of the state parameter is defined quite generally as a correspondence. A player's decision on the best response to the system state and the other players' strategy profiles is made under the additional assumption that the variation of a player's strategy influences the variation of the system parameter. We consider both the version with constant assumptions and the version in which these assumptions by the players are in some correspondence with the strategy profiles and the value of the system parameter. We generalize the concept of equilibrium to this case and prove a theorem of its existence.