= Pairwise Shapley differences in a three-player simple game
{title2=$\phi_i-\phi_j=\tfrac12(v(\{i\})-v(\{j\})+v(\{i,k\})-v(\{j,k\}))$}
For three distinct players $i,j,k$ in a normalized <simple cooperative game>, enumerate the predecessor <coalitions> in the <Shapley value> formula:
$$
\phi_i=\tfrac13v(\{i\})+\tfrac16\bigl(v(\{i,j\})-v(\{j\})\bigr)+\tfrac16\bigl(v(\{i,k\})-v(\{k\})\bigr)+\tfrac13\bigl(1-v(\{j,k\})\bigr).
$$
Subtracting the formula with $i$ and $j$ exchanged proves the displayed identity. Every <coalition> value is zero or one, so $2(\phi_i-\phi_j)$ 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.
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