Population Monotonic Allocation Schemes for Games with Externalities

This paper provides the conditions for a game with externalities to have a population monotonic allocation scheme (PMAS). We observe that the convexity defined by Hafalir (2007) does not guarantee the existence of a PMAS in the presence of externalities. We introduce a new convexity and show that while our convexity is not a stronger condition than Hafalir (2007)’s, 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 generalize Sprumont (1990)’s results and offer two necessary and sufficient conditions to guarantee a PMAS.