Shapley self-duality (${\varphi }_i(v^{\prime })={\varphi }_i(v)$)

Complementing predecessor sets within $N\setminus \{i\}$ preserves their Shapley value weights, since the factorials $|S|!$ and $(n-|S|-1)!$ interchange. For the dual coalitional game, the corresponding marginal contribution equals the original contribution at that complementary set. Hence the values coincide. The same bijection gives ${\varphi }_i(v)={\sum }_{S}w(S)(v(N\setminus S)-v(S))$.