The task of managing general game playing in a multi-agent system is the problem addressed in this paper. It is considered to be done by an agent. There are many reasons for constructing such an agent, called general game management agent. This agent manages strategic interactions between other agents players, natural or also artificial. The agent records the interaction for further benchmarking and analysis. He can also be used for a kind of restricted communications. His behavior is defined by a game description written in a logic-based language. The language, we present for this application, is more expressive than the language GDL, which is already used for such purposes. Our language can represent imperfect information and time dependent elements of a game. Time dependent elements like delays and timeouts are of crucial importance for interactions between players with bounded processing power like humans. We provide examples to show the feasibility of our approach. A way for game theoretical solving of an interaction description in our language is considered as future work.