= Solution
\b[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 $i,j,k$,
$$
\phi_i=\frac13a_i+\frac16(b_{ij}-a_j)+\frac16(b_{ik}-a_k)+\frac13(1-b_{jk}).
$$
Subtract the corresponding expression for $\phi_j$:
$$
\phi_i-\phi_j=\frac12\bigl(a_i-a_j+b_{ik}-b_{jk}\bigr).
$$
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.
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