Pairwise Shapley differences in a three-player simple game (${\varphi }_i-{\varphi }_j=\frac{1}{2}(v(\{i\})-v(\{j\})+v(\{i,k\})-v(\{j,k\}))$)

For three distinct players $i,j,k$ in a normalized simple cooperative game, enumerate the predecessor coalitions in the Shapley value formula:Subtracting the formula with $i$ and $j$ exchanged proves the displayed identity. Every coalition value is zero or one, so $2({\varphi }_i-{\varphi }_j)$ is an integer. This rules out, for example, coordinates differing by $1/6$, although each may separately satisfy the Shapley value denominator in a simple game. The difference identity itself holds for arbitrary three-player transferable utility games; the integer restriction uses the zero-one values.