  A matching game is a cooperative profit game defined on an edge-weighted
graph, where the players are the vertices and the profit of a coalition is the
maximum weight of matchings in the subgraph induced by the coalition. A
population monotonic allocation scheme is a collection of rules defining how to
share the profit among players in each coalition such that every player is
better off when the coalition expands. In this paper, we study matching games
and provide a necessary and sufficient characterization for the existence of
population monotonic allocation schemes. Our characterization also implies that
whether a matching game admits population monotonic allocation schemes can be
determined efficiently.
