This paper provides conditions for a game with externalities to have a population monotonic allocation scheme (PMAS). We observe that the notion of convexity defined by Hafalir [Games Econ Behav 61:242–258, 2007 ] does not guarantee the existence of a PMAS in the presence of externalities. We introduce a new notion of convexity and show that while our convexity is not a stronger condition than Hafalir’s [Games Econ Behav 61:242–258, 2007 ] , it is a sufficient condition for a game to have a PMAS. Moreover, we show that the Aumann-Drèze value, which is defined for games with coalition structures, explicitly constructs a PMAS. In addition, we offer two necessary and sufficient conditions to guarantee a PMAS in the presence of externalities.

中文翻译：

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具有外部性的博弈的人口单调分配方案

本文为具有外部性的博弈提供了人口单调分配方案（PMAS）的条件。我们观察到 Hafalir [Games Econ Behav 61:242–258, 2007] 定义的凸性概念并不能保证在存在外部性的情况下存在 PMAS。我们引入了一个新的凸性概念，并表明虽然我们的凸性不是比 Hafalir 的 [Games Econ Behav 61:242–258, 2007] 更强的条件，但它是游戏具有 PMAS 的充分条件。此外，我们展示了为具有联盟结构的游戏定义的 Aumann-Drèze 值，明确地构建了一个 PMAS。此外，我们提供了两个充分必要条件来保证存在外部性的 PMAS。