Shapley and Banzhaf Values as Probability Transformations

Abstract

We investigate the role of some game solutions, such the Shapley and the Banzhaf values, as probability transformations. The first one coincides with the pignistic transformation proposed in the Transferable Belief Model; the second one is not efficient in general, leading us to consider its normalized version. We study a number of particular models of lower probabilities: minitive measures, coherent lower probabilities, as well as the lower probabilities induced by comparative or distortion models. For them, we provide some alternative expressions of the Shapley and Banzhaf values and study under which conditions they belong to the core of the lower probability.

Outer approximating coherent lower probabilities with belief functions

“…Some early results were established in [27], where we studied how to inner approximate a coherent lower probability in C 2 , and in [20], where inner approximations in the family of k-additive measures were considered. Other relevant results may be found in [2,18,25,34].…”

“…, where the second inequality follows by the induction hypothesis and the third one follows from Eq. (25).…”

Outer Approximations of Coherent Lower Probabilities Using Belief Functions

“…Some early results were established in [26], where we studied how to inner approximate a coherent lower probability in C 2 , and in [20], where inner approximations in the family of k-additive measures were considered. Other relevant results may be found in [2,18,25,32].…”

Coalition Games for Performance Evaluation in 5G and Beyond Networks: A Survey

“…The dependence of the payoff of one coalition on the entire partition implies that the deviation of some sub-coalition may change the payoff of completely uninvolved, distinct coalitions. The deviators themselves can have different payoffs in different partitions [44]. A new solution concept over such a game is defined based on the concept of a Residual game.…”