An axiomatization of the proportional prenucleolus

The proportional prenucleolus is defined on the class of all positive TU games with finite sets of players. The set of axioms used by Sobolev (1975) for axiomatic justification of the prenucleolus is modified. It is proved that the proportional prenucleolus is a unique value that satisfies 4 axioms: efficiency, anonymity, proportionality, and proportional DM consistency. The proof is a modification of the proof of Sobolev's theorem.
For strictly increasing concave function $U$ defined on $(0,+\infty)$ with range equal to ${\rm R}^1$, a generalization of the proportional prenucleolus is called $U$–prenucleolus. The axioms proportionality and proportional DM consistency are generalized for its justification.