Nullified-game consistency and axiomatizations of the Core of TU-games with a fixed player set

We provide new axiomatic foundations for the Core of TU-games with a fixed player set. Our main axiom is referred to as nullified reduced game consistency, which is a natural counterpart of reduced game consistency in the setting with a universal player set. We consider three types of nullified reduced game consistency axioms, each of which corresponds to the max-, complement-, and projection-reduced game consistency in the literature. We show that each property, together with other standard axioms, characterizes the Core. As a result, our characterization results uncover implicit restrictions on solutions imposed by consistency in the universal player set.