Investment and Bargaining

We consider bargaining between two players who may invest ex ante in their agreement and disagreement payoffs. We characterize necessary conditions on equilibrium investment strategies in this environment, describe how investments and the probability of outcomes must vary across mechanisms, and specify what equilibrium conditions imply for constraints on various aspects of the design environment. In the case of private values any two trading rules that induce the same equilibrium lotteries over valuations induce the same probability of trade. We also show that equilibrium descriptions are fragile in the sense that descriptions that cannot be supported by any equilibrium investment decisions are dense in the space of problems. We exhibit an approach to recovering cost functions that support the primitives of a Bayesian Mechanism design problem. By way of an example, we use an unraveling argument to show that uniform distributions over valuations cannot emerge from equilibria investments to the Chatterjee-Samuelson bilateral trade mechanism for any strictly increasing cost functions.