Two Optimal Allocations under Management Systems: Game-theoretical Approaches

By applying the notion of the efficient Banzhaf index, any additional fixed utility should be distributed equally among the players who are concerned. In many applications, however, this notion seems unrealistic for the situation that is being modeled. Therefore, inspired by the notion of the weighted allocation of non-separable costs (WANSC), we adopt weight functions to introduce a modification of the efficient Banzhaf index, which we name the weighted Banzhaf index. In order to present the rationality of the weighted Banzhaf index, we adopt some reasonable properties to characterize the weighted Banzhaf index. Based on different viewpoints, we further define excess functions to propose alternative formulations and related dynamic processes for the weighted Banzhaf index and the WANSC respectively.