In this paper we focus on multinomial probabilistic values and we consider two special classes of players: necessary and nullifying players. By introducing new properties related to this kind of players, we provide new axiomatic characterizations of each multinomial probabilistic value, giving, in all cases, a set of independent properties that univocally determine them. Moreover, when the profile defining the value is positive, the multilinear extension is interpreted as a potential function and provides a computational tool.

This work focuses on the Owen value and the Owen-Banzhaf value, two classical concepts of solution defined on games with structure of coalition blocks. We provide a computation procedure for these solutions based on a method of double-level work obtained from the multilinear extension of the original game. Moreover, two applications to several possible political situations in the Madrid Assembly and the Andalusian Parliament (legislatures 2015-2019) are also given.

In this paper we focus on games with a coalition structure. Particularly, we deal with the Owen value, the coalitional value of the Shapley value, and we provide a computational procedure to calculate this coalitional value in terms of the multilinear extension of the original game.