  We introduce a model of sender-receiver stopping games, where the state of
the world follows an iid--process throughout the game. At each period, the
sender observes the current state, and sends a message to the receiver,
suggesting either to stop or to continue. The receiver, only seeing the message
but not the state, decides either to stop the game, or to continue which takes
the game to the next period. The payoff to each player is a function of the
state when the receiver quits, with higher states leading to better payoffs.
The horizon of the game can be finite or infinite.
  We prove existence and uniqueness of responsive (i.e. non-babbling) Perfect
Bayesian Equilibrium (PBE) under mild conditions on the game primitives in the
case where the players are sufficiently patient. The responsive PBE has a
remarkably simple structure, which builds on the identification of an
easy-to-implement and compute class of threshold strategies for the sender.
With the help of these threshold strategies, we derive simple expressions
describing this PBE. It turns out that in this PBE the receiver obediently
follows the recommendations of the sender. Hence, surprisingly, the sender
alone plays the decisive role, and regardless of the payoff function of the
receiver the sender always obtains the best possible payoff for himself.
