How cooperation arises from rational players?

In classical control systems,the plant to be controlled does not have intention to gain its payof or benefit,which is obviously not the case in various aspects of social and economic systems(or subsystems).In the latter case,competition and cooperation between players who will optimize their own payofs turn out to be an important feature,and a fundamental problem is how to achieve cooperation from these rational players.In this paper,we present a neat way to lead to cooperation in dynamical Prisoner’s Dilemma game.In our scenario,the two players are heterogenous with hierarchical roles as the‘leader’and the‘follower’respectively.It is shown that the system will co-evolve into and stay at the cooperation state if and only if the leader is restricted not to take the dominating strategies.For the special case of 1-step-memory,the optimal strategies for the leader and follower are‘Tit for Tat’and‘ALL C’respectively.In this framework,both the heterogeneity of the players’roles and the multiplicity of time-scales are crucial for cooperation,which are quite natural settings from the view point of control theory.Besides,the boundary for cooperation also turns out to depend on the relative payofs of the players.