Strategic information transmission with sender's approval: The single-crossing case

We consider games in which an informed sender first talks at no cost to a receiver; then, the latter proposes a decision and, finally, the sender accepts the proposal or “exits”. We make the following assumptions: the sender has finitely many types, the receiver's decision is real-valued, utility functions over decisions are concave, single-peaked and single-crossing, exit is damaging to the receiver. In this setup, it may happen that babbling equilibria necessarily involve exit. We nevertheless propose a constructive algorithm that achieves a pure perfect Bayesian equilibrium without exit in every game of the class considered.