= Balanced contributions of Shapley values
{title2=$\phi_i(N,v)-\phi_i(N\setminus\{j\},v)=\phi_j(N,v)-\phi_j(N\setminus\{i\},v)$}
For a <coalitional game>, restriction to the remaining players preserves coalition values. Remove player $j$ from a uniformly random ordering when comparing player $i$'s <Shapley value>. The marginal contribution changes only when $j$ precedes $i$. If $S$ is the set of other predecessors, the change is $v(S\cup\{i,j\})-v(S\cup\{i\})-v(S\cup\{j\})+v(S)$, with coefficient $(|S|+1)!(n-|S|-2)!/n!$. The same expression and coefficient arise after interchanging $i,j$, proving balanced contributions.
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