Value theory without efficiency, Mathematics of Operations Research
A semivalue is a symmetric positive linear operator on a space of games, which leaves the additive games fixed. Such an operator satisfies all of the axioms defining the Shapley value, with the possible exception of the efficiency axiom. The class of semivalues is completely characterized for the space of finite-player games, and for the space pNA of nonatomic games.
Pradeep Dubey, Abraham Neyman, Robert Weber
Dubey, Pradeep, Abraham Neyman, and Robert Weber. 1981. Value theory without efficiency. Mathematics of Operations Research. 6(1): 122-128.