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

Let v be a cooperative (TU) game and v = v1 − v2 be a decomposition of v as a difference of two convex games v1 and v2. Then the core C(v) of the game v has a similar decomposition C(v) = C(v1) ⊖ C(v2), 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.