Here we begin to exploit another setting for major player behavior. We shall assume that the major player has some planning horizon with both running and (in case of a finite horizon) terminal costs. For instance, running costs can reflect real spending, while terminal costs can reflect some global objective, such as reducing the overall crime level by a specified amount. This setting will lead us to a class of problems that can be called Markov decision (or control) processes (for the principal) on the evolutionary background (of permanently varying profiles of small players). We shall obtain the corresponding LLN limit for both discrete and continuous time. For discrete time, the LLN limit turns into a deterministic multistep control problem in the case of one major player, and to a deterministic multistep game between major players in the case of several such players.