Consonance, Symmetry and Extended Outputs

Abstract

In many real-world situations, performers always adopt different energy levels (strategies) to participate. Different from pre-existing results, this paper is devoted to proposing several generalized power outputs of the marginal index, the Banzhaf–Coleman index, and the Banzhaf–Owen index, respectively, by assigning different energy levels to all performers. Since these extended power outputs may not be efficacious, we further define the efficacious extensions of these power outputs, respectively. For each of these efficacious power outputs, we demonstrate that there exists a corresponding reduced game and related consonance property that can be used to characterize it. By focusing on the properties of symmetry and accordance, several axiomatic results are also introduced.