Core, Shapley Value, and Weber Set

In Chap. 17 we have seen that the Shapley value of a game does not have to be in the core of the game, nor even an imputation (Problem 17.5). In this chapter we introduce a set-valued extension of the Shapley value, the Weber set, and show that it always contains the core (Sect. 18.1). Next, we study so-called convex games and show that these are exactly those games for which the core and the Weber set coincide. Hence, for such games the Shapley value is an attractive core selection (Sect. 18.2). Finally, we study random order values (Sect. 18.3), which fill out the Weber set, and the subset of weighted Shapley values, which still cover the core (Sect. 18.4).