Incentive compatibility in sender-receiver stopping games

We introduce a model of sender-receiver stopping games, in which the sender observes the current state, and sends a message to the receiver to either stop the game, or to continue. The receiver, only seeing the message, then decides to stop the game, or to continue. The payoff to each player is a function of the state when the receiver quits, with higher states leading to better payoffs. We prove existence and uniqueness of responsive Perfect Bayesian Equilibrium (PBE) when players are sufficiently patient. The responsive PBE has a simple structure, with a threshold strategy for the sender, and the receiver obediently following the recommendations of the sender. Hence, the sender alone plays the decisive role, and therefore always obtains the best possible payoff for himself.