Past exam of the MSc in advanced computer science course of the University of Oxford / 2022 / Computational game theory / 4 / c / iv

No. The denominator test from part ii is satisfied here, so a stronger obstruction is needed. Put $a_i=v(\{i\})$ and $b_{ij}=v(\{i,j\})$. In a three-player simple cooperative game, the Shapley value formula gives, for distinct $i,j,k$,Subtract the corresponding expression for ${\varphi }_j$:Each coalition value is zero or one, so this difference must be an integer multiple of $1/2$. This is the obstruction from pairwise Shapley differences in a three-player simple game. The proposed first two coordinates differ by $1/6$, which is not an integer multiple of $1/2$. Therefore the vector cannot occur, even though each coordinate separately passes the $1/6$ denominator test.