Notes on the Cross-Level Game ^†

Up till now, the research on game theory has concentrated on the game issues between agents, which are homogeneous and at the same level in the complex hierarchical system. There have already been many fruitful results that have been successfully used in many fields, such as economics, public management, sociology, and so on. However, in real life, the more frequently faced game problems are cross-level games, which are among agents from different levels. For example, in a supply chain, single factory and whole chain; in the public management, the single family and the whole community; in a tropical rain forest, single species and whole eco-system; in biology, an animal and the bacterial group in its body; and so on. Up till now, similar topics had not been discussed in detail. This situation seriously limited the research and applications for game theory. In this paper, the author introduces their own ideas for the difficulties and possible approaches on this issue. Some suggestions have been provided at the end. The main suggestion of this paper is to learn from the results of the complexity study, to focus on the payoff function, and to find an approach for the cross-level game model.