Cores of Cooperative Games, Superdifferentials of Functions, and the Minkowski Difference of Sets

Let ¨ be a cooperative TU.game and ¨ s¨1y¨2be a decomposition of ¨ as a difference of two convex games ¨1and ¨2. Then the core C ¨. of the game ¨ has a similar decomposition C ¨.sC ¨1.]C ¨2., where ] denotes the Minkowski difference. We prove such a decomposition as a consequence of two claims: the core of a game is equal to the superdifferential of its continuation, known as the Choquet integral, and the superdifferential of a difference of two concave functions equals the Minkowski difference of corresponding superdifferentials