Coincidence among potential and axiomatic approaches for a weighted resolution

By simultaneously considering the maximal utilities across operational step vectors among constituents and weighting structures, we introduced the weighted max-value as a novel resolution concept under game-theoretical frameworks. The proposed formulation generalized traditional resolution by incorporating individual heterogeneity via weight distributions. To justify its theoretical soundness, we established a set of coincident relations that characterized the class of all resolutions admitting a potential function defined with weights. Additionally, we proposed the dividend approach as a dual perspective to the potential formulation and demonstrated the equivalence between these two through rigorous structural analysis. These coincident relations formed the basis for two axiomatic characterizations, effectiveness for multi-choice games combined with either weighted balanced contributions under multi-choice games or weighted path independence under multi-choice games, that uniquely identified the weighted max-value. The findings not only extended the theory of potential functions under game-theoretical settings but also offered a unifying framework for modeling weighted participatory behavior. These theoretical insights are applicable to real-world contexts involving differentiated participatory stakes, such as public finance and collaborative ventures.