Bargaining with non-anonymous 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. We first show that this domain extension creates a very large number of new rules. In particular, decomposable rules (which are extensions of rules from the Nash domain) constitute a nowhere dense subset of all possible rules. For them, we analyze the process through which "good" properties of rules on the Nash domain extend to ours. We then enquire whether the counterparts of some well-known results on the 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 properties. This uniqueness result, however, turns out to be an exception. We characterize the uncountably large classes of decomposable rules that survive the, and properties. [Copyright Elsevier B.V.]