Multi-sender cheap talk with restricted state spaces∗

This paper analyzes multi-sender cheap talk when the state space might be restricted, either because the policy space is restricted, or the set of rationalizable policies of the receiver is not the whole space. We provide a necessary and sufficient condition for the existence of a fully revealing perfect Bayesian equilibrium for any state space. We show that if biases are large enough and are not of similar directions, where the notion of similarity depends on the shape of the state space, then there is no fully revealing perfect Bayesian equilibrium. The results suggest that boundedness, as opposed to dimensionality, of the state space plays an important role in determining the qualitative implications of a cheap talk model. We also investigate equilibria that satisfy a robustness property, diagonal continuity. ∗We would like to thank Drew Fudenberg for comments and suggestions, Jeffrey Ely and three anonymous referees for suggestions, seminar participants at Harvard-MIT, UPenn, UC Berkeley, UChicago, the 2006 North American Meeting of the Econometric Society, the 2006 European Meeting of the Econometric Society, and the 18th International Conference on Game Theory at Stony Brook for comments, and Niels Joaquin for valuable research assistance. †Department of Economics, Harvard University, Cambridge, MA 02138