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In real-world management systems, agents might represent constituencies of different sizes; agents might have different bargaining abilities. It is reasonable that weights are assigned to agents and any utility should be divided according to these weights. On the other hand, agents might take different activity levels to participate; agents might represent administrative areas of different scales. Therefore, we introduce two indexes to investigate the distribution mechanism by applying the maximal-utilities among activity level vectors and the weights simultaneously. In order to present the rationality for these weighted indexes, we adopt extended reductions to provide several characterisations. Based on these extended reductions, we also introduce two dynamic processes to illustrate that these weighted indexes could be reached by agents who start from an arbitrary efficient payoff vector and make successive adjustments. By focusing on a different viewpoint, we apply excess functions to propose alternative formulations and related dynamic results for these weighted indexes.

Characterizations and dynamic results for consistent indexes under weighted consideration