Weak values,the core,and new axioms for the Shapley value

An explicit formula for all linear symmetric values is established and is then used to show that, in the definition of the Shapley value, the efficiency and dummy axioms may be replaced by a projection axiom, a monotonicity axiom, and Roth's (1977) strategic risk neutrality axiom. It is also demonstrated that, for the set of super-additive games, there is no efficient linear symmetric value which is always in the core.