Considerations on the aggregate monotonicity of the nucleolus and the core-center

Even though aggregate monotonicity appears to be a reasonable requirement for solutions on the domain of convex games, there are well known allocations, the nucleolus for instance, that violate it. It is known that the nucleolus is aggregate monotonic on the domain of essential games with just three players. We provide a simple direct proof of this fact, obtaining an analytic formula for the nucleolus of a three-player essential game. We also show that the core-center, the center of gravity of the core, satisfies aggregate monotonicity for three-player balanced games. The core is aggregate monotonic as a set-valued solution, but this is a weak property. In fact, we show that the core-center is not aggregate monotonic on the domain of convex games with at least four players. Our analysis allows us to identify a subclass of bankruptcy games for which we can obtain analytic formulae for the nucleolus and the core-center. Moreover, on this particular subclass, aggregate monotonicity has a clear interpretation in terms of the associated bankruptcy problem and both the nucleolus and the core-center satisfy it.