A prisoner's dilemma model of collusion deterrence

We examine a hierarchy formed by a principal, a supervisor and an agent, wherein the supervisor and the agent can collude. We consider a case where collusion-free supervisors are not available. We demonstrate first that it is easy for the principal to deter collusion by introducing a second supervisor and designing a mechanism similar to the prisoner's dilemma so that the two supervisors control each other. Since it could prove too costly for the principal to send two supervisors, a new question arises: whether it would be possible to deter collusion by sending the second supervisor with a probability less than one. We find that under reasonable assumptions on the size of rewards and punishments, the principal can prevent collusion only by 'creating' a new type of supervisor through sometimes informing the second supervisor of his position.