Random reduction consistency of the Weber set, the core and the anti-core

In this paper we introduce a new consistency condition and provide characterizations for several solution concepts in TU cooperative game theory. Our new consistency condition, which we call the random reduction consistency, requires the consistency of payoff vectors assigned by a solution concept when one of the players is removed with some probability. We show that the random reduction consistency and other standard properties characterize the Weber set, the convex hull of the marginal contribution vectors. Another salient feature of random reduction consistency is that, by slightly changing its definition, we can characterize the core and the anti-core in a parallel manner. Our result enables us to compare the difference between the three solution concepts from the viewpoint of consistency.