Posterior renegotiation-proofness in a two-person decision problem

When two agents with private information use a mechanism to determine an outcome, what happens if they are free to revise their messages and cannot commit to a mechanism? We study this problem by allowing agents to hold on to a proposed outcome in one mechanism while they play another mechanism and learn new information. A decision rule is posterior renegotiation-proof if it is posterior implementable and robust to a posterior proposal of any posterior implementable decision rule. We identify conditions under which such decision rules exist. We also show how the inability to commit to the mechanism constrains equilibrium: a posterior renegotiation-proof decision rule must be implemented with at most five messages for two agents.