What strategies can induce cooperation between heterogeneous players?

In many social systems, the emergence and maintenance of cooperation among rational agents is a central topic and a fundamental problem. Consider the case of the Prisoner´s Dilemma game which is played between heterogeneous players, i.e. a leader and a follower, cooperation can emerge when both players optimize their payoffs provided that the leader´s strategy is restricted in a certain set of k-step-memory strategies. However, it is difficult to analyze the condition for cooperation analytically since the number of leader´s strategies increases super-exponentially with the memory length k. So in this paper, a series of computer experiments are used to investigate the condition for cooperation by enumerating the leader´s strategies with 2-step-memory while the follower utilizes the Q-learning algorithm to update her strategy. The results illustrate that there are a small amount of strategies for the leader to induce mutual cooperation. Most of them can be featured as “reward of mutual cooperation” and “punishment of cheating kindness”. This is the first step to characterize the features of strategies which can induce mutual cooperation for the general memory length k.