A dual description of the class of games with a population monotonic allocation scheme

A vector of balanced weights infers an inequality that games with a nonempty core obey. This paper gives a generalization of the notion ‘vector of balanced weights.’ Herewith, it provides necessary and sufficient conditions to determine whether a TU-game has a population monotonic allocation scheme or not. Furthermore, it shows that every four-person integer valued game with a population monotonic allocation scheme has an integer valued population monotonic allocation scheme and it gives an example of a seven-person integer valued game that has only noninteger valued population monotonic allocation schemes.