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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.

Choose one plane requiring the longest runway and assign it the entire cost $c(N)$; assign every other plane zero. These nonnegative payments sum to $c(N)$. Any coalition containing that chosen plane also requires the whole longest runway, so its payment is exactly its standalone cost. Any coalition without it pays zero, which is at most its nonnegative standalone cost.

Hence this payment vector is in the cost core, proving the desired sharing always exists. This is an existence certificate, not a fairness assertion. If there are no planes, the empty allocation has total zero. The Shapley value of an airport game gives a more symmetric sharing rule and also lies in the core of this cooperative cost game.

A cooperative cost game in which disjoint coalitions can merge without increasing their combined cost. Negating its cost function gives a superadditive coalitional game. Subadditivity by itself does not ensure a nonempty cost core.