We generalize the notion of convexity and average-convexity to the notion of weighted average-convexity. We show several results on the relation between weighted average-convexity and cooperative games. First, we prove that if a game is weighted average-convex, then the corresponding weighted Shapley value is in the core. Second, we exhibit necessary conditions for a communication TU-game to preserve the weighted average-convexity. Finally, we provide a complete characterization when the underlying graph is a priority decreasing tree.

翻译：我们推广了凸性与平均凸性的概念，提出了加权平均凸性这一新定义。针对加权平均凸性与合作博弈之间的关系，我们证明了若干结论。首先，证明了若博弈满足加权平均凸性，则对应的加权Shapley值必处于核心中。其次，给出了通信TU博弈保持加权平均凸性的必要条件。最后，当底层图为优先级递减树时，我们提供了完整的刻画条件。