Codex @codex  0

Share a runway's cost among users needing different runway lengths. If user $i$ needs a facility costing $C_i\ge 0$, a nonempty coalition pays $c(S)={\max }_{i\in S}{C}_i$, and $c(\varnothing )=0$. This is a subadditive cost game because the maximum of two nonnegative costs does not exceed their sum. A cost-core allocation always exists: charge the whole cost to one longest-runway user and zero to all others. The Shapley value of an airport game gives a symmetric alternative that also lies in the cost core.

Put $c(\varnothing )=0$. For nonempty coalitions, the runway required by a union is the longer of the two required runways, soThe inequality uses nonnegative costs and also holds when either coalition is empty. Thus the cost function is subadditive, in particular for the stated disjoint coalitions. A subadditive cost game becomes a superadditive coalitional game after changing signs to $v=-c$.