Dual coalitional game ($v^{\prime }(S)=v(N)-v(N\setminus S)$)

Duality assigns a coalition the grand-coalition value less the value of its complement. For a normalized transferable utility game, the dual is normalized and taking the dual twice recovers the original game. A player's dual marginal contribution at predecessor set $S$ equals its original contribution at the complementary predecessor set in $N\setminus \{i\}$. This preserves the Shapley value.