Balanced contributions of Shapley values (${\varphi }_i(N,v)-{\varphi }_i(N\setminus \{j\},v)={\varphi }_j(N,v)-{\varphi }_j(N\setminus \{i\},v)$)

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.