 Solution

For an $n$-player transferable utility game with $v(\varnothing )=0$, the Shapley value isIt averages player $i$'s marginal contribution over uniformly random orders of all players. Exactly $|S|!(n-|S|-1)!$ of the $n!$ permutations put precisely $S$ before $i$. For a cooperative cost game, apply the same linear formula to the cost function; the result represents payments instead of distributed benefits.