Collusion, Randomization and Leadership in Groups I

We propose a theory of collusive groups in the context of finite non-cooperative games. We first consider a simple setting in which players are exogenously partitioned into groups within which players are symmetric. Given the play of the other groups there may be several symmetric equilibria for a particular group. We develop the idea that if group can collude they will agree to choose the equilibrium most favorable for its members. We then consider an alternative model of a strictly non-cooperative meta-game played between "leaders " of groups. We establish equivalence between equilibria of the collusive group game and the leadership game- this also establishes existence for the collusive group game. Finally, we extend the leadership model to games where players within groups are not necessarily symmetric and groups are endogenously formed. In this model leaders bid for followers by making credible offers of what the followers will receive if they follow the instructions of the leader. We illustrate the model with a simple application in which there are three potential groups. There are two symmetric groups that play a prisoners ' dilemma game with each other and there can also be a "group of the whole. " We show that the equilibrium necessarily involves randomization and that both cooperation and defection take place in equilibrium. We also show that the frequency of cooperation increases as the benefit to cooperation increases and the benefit of deviating decreases. JEL Classification Numbers: late