The Theory of Implementation When the Planner Is a Player: Incomplete Information

In the usual implementation problem agents are asked to play a game form-a mechanism which is designed by the planner and the outcome is determined by the outcome function of the mechanism. In the problem the planner can force agents to play the game, but she has to obey her commitment to the agents, thus she cannot change but have to accept the outcome. In this paper we study the situation where the planner can select an outcome arbitrarily after agents have send a message profile, based on the posterior belief she has on the state profile of the agents. We assume (i) agents have private information and play simultaneously and (ii) given the messages announced by the agents, the planner updates her belief about the probability distribution of the state profiles of the agents and chooses the best outcome for her according to the strategy profile of the agents. This transforms the implementation problem into a signaling game with agents as senders and the planner as receiver. We derive necessary and sufficient conditions for interactive implementation by using the refinement concept of FGP- equilibrium as the solution of the problem.