The aim of this paper is to extend the classical Banzhaf index of power to voting games in which players have weights representing different cooperation or bargaining abilities. The obtained value does not satisfy the classical total power property, which is justified by the imperfect cooperation. Nevertheless, it is monotonous in the weights. We also obtain three different characterizations of the value. Then we relate it to the Owen multilinear extension.

Multicommunication is one of the methods at our disposal to address the significant challenges of reliability and latency in today’s communications. This paper presents a game-theoretical analysis of multicommunication, where we assume that players in a TU-game are restricted in their communication by a multigraph, whose links may or may not have limitations on capacity, frequency, or transmission speed. We generalize the classical position value defined for graph games to assess the importance of different communication providers. Extending to this context the classical properties of component efficiency and balanced link contributions we can characterize the new value. For a wide variety of multicommunications including but not limited to all those where the connections are in series or parallel, the value can be characterized by substituting balanced link contributions for the more restrictive balanced link pack contributions. In the case of limitations in capacity, parallel characterizations are obtained in which component efficiency must be substituted by a certain inefficiency that takes these limitations into account.

In this paper we generalize the allocation rule (point solution or value) known as the mixed value by introducing the weighted mixed value.The proposed solution assigns value in graph games where players, and/or links, have weights representing asymmetries of the players, and different flows, lengths, emotional intensities, trust in the transmission of the information, etc. in the links. We present several characterizations of this value using properties, such as mixed component efficiency, weighted mixed fairness, weighted balanced contributions and weighted balanced link contributions. These properties were inspired by the classical properties used to characterize the Myerson value and the position value.