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.

尽管总单调性似乎是凸博弈域上解的合理要求，但还是有一些众所周知的分配违反了它，例如核仁。众所周知，核仁在只有三个参与者的情况下在基本游戏领域是聚集单调的。我们提供了这一事实的简单直接证明，获得了三人基本游戏核仁的解析公式。我们还表明，核心中心（核心的重心）满足三人平衡游戏的总单调性。核心是聚合单调作为集值解决方案，但这是一个较弱的属性。实际上，我们证明了在至少有四个玩家的凸游戏领域，核心中心不是聚合的单调性。我们的分析使我们能够确定破产博弈的一个子类，可以为其获得核仁和核心中心的分析公式。而且，在这个特定的子类上，聚集的单调性对相关的破产问题有清晰的解释，核仁和核心中心都满足。