A note: characterizations of convex games by means of population monotonic allocation schemes

Convex cooperative games were first introduced by Shapley (Int J Game Theory 1:11–26, 1971) while population monotonic allocation schemes (PMAS) were subsequently proposed by Sprumont (Games Econ Behav 2:378–394, 1990). In this paper we provide several characterizations of convex games and introduce three new notions: PMAS-extendability, PMAS-exactness, and population monotonic set schemes, which imitate the classical concepts that they extend. We show that all of these notions provide new characterizations of the convexity of the game.