Bargaining with nonanonymous disagreement: Decomposable rules

We analyze bargaining situations where the agents’ payoffs from disagreement depend on who among them breaks down the negotiations. We model such problems as a superset of the standard domain of Nash (1950). We first show that this domain extension creates a very large number of new rules. Particularly, decomposable rules (which are extensions of rules from the Nash domain) constitute a nowhere dense subset of all possible rules. We next enquire if the counterparts of some standard results on the Nash (1950) domain continue to hold for decomposable rules on our extended domain. We first show that an extension of the Kalai-Smorodinsky bargaining rule uniquely satisfies the Kalai-Smorodinsky (1975) properties. This uniqueness result, however, turns out to be an exception. We characterize the uncountably large classes of decomposable rules that survive the Nash (1950), Kalai (1977), and Thomson (1981) properties. We also show that extensions to our domain of a standard independence property (by Peters, 1986) imply decomposability. Finally, we analyze the process through which “good” properties of rules on the Nash domain extend to ours. JEL Classification numbers: C78, D74