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Symmetry exists in a multitude of phenomena in varying forms. The main aim of this article is to analyze the plausibility of the equal allocation non-separable costs, the efficient Banzhaf–Owen index and the efficient Banzhaf–Coleman index from the perspective of symmetry. First, based on the difference between “participation processes” and “allocating results”, different forms of symmetry are proposed. Next, building on these forms of symmetry, axiomatic results are put forth for the three power indexes, whereby the plausibility of the three power indexes is analyzed. Finally, on the basis of these different forms of symmetry and related axiomatic results, this article introduces different dynamic processes to analyze how an initial allocation result approaches the results derived from the three power indexes through dynamically modification.

Axiomatic and Dynamic Results for Power Indexes under Symmetry