Coordination, Efficiency and Two-Sided Cheap Talk

We consider the Battle of the Sexes game with incomplete information and allow two-sided cheap talk before the game is played. We characterise the set of fully revealing symmetric cheap talk equilibria. The best fully revealing symmetric cheap talk equilibrium, when exists, has a desirable characteristic. When the players' types are different, it fully coordinates on the ex-post efficient pure Nash equilibrium. We also analyse the mediated communication equilibria of the game. We find the range of the prior for which this desirable equilibrium exists under unmediated and mediated communication processes.