Bargaining under almost complete information

We consider the canonical frequent-offer Rubinstein bargaining game between two leaders on a policy issue. The public sentiment on this issue leans toward one leader or the other. Public sentiment determines the maximum compromise a leader can make without risking getting primaried. We introduce a positive probability ε that the leaders are uninformed about the sentiment. We show that a unique equilibrium emerges that resembles a war of attrition, in which one of the leaders may demand too much even after learning that the sentiment leans the other way. For a given bargaining environment, as ε converges to zero, in every state, the leaders almost immediately agree to the same policy position as they do under complete information. However, this convergence does not hold in the uniform sense, nor does it necessarily hold for more than two states.