Monotonicity in Bargaining Networks

We study bargaining networks, discussed in a recent impel of Kleinberg and Tardos [KT08], nom the perspective of cooperative game theory. In particular we examine three solution concepts, the nucleolus, the core center and the core median. All solution concepts define unique solutions, so they provide testable predictions. We define a new monotonicity property that is a natural axiom of any bargaining game solution, and we prove that all three of them satisfy this monotonicity property. This is actually in contrast to the conventional wisdom for general cooperative games that monotonicity and the core condition (which is a basic property that all three of them satisfy) ale incompatible with each other. Our proofs are based on a primal-dual arugument (for the nucleolus) and on the PKG inequality (for the core center and the core median). We further observe sonic qualitative differences between the solution concepts. In particular, there are cases where a strict version of our monotonicity property is a natural axiom, but only the core center and the core median satisfy it On the other hand, the nucleolus is easy to compute, whereas computing the core center or the core median is #P-hard (yet it can be approximated in polynomial time)