Cooperative cost game ($p(N)=c(N),p(S)\le c(S)$)

A transferable utility game formulated as costs $c(S)$ to be shared among members of each coalition $S$, with $c(\varnothing )=0$. Its cost core consists of payment vectors $p$ satisfying the displayed efficiency and coalition inequalities, where $p(S)={\sum }_{i\in S}{p}_i$. A coalition should not be charged more than its standalone cost. Changing signs to $v=-c$ and $x=-p$ converts these conditions to the usual benefit-game core of a cooperative game. The Shapley value can be applied directly to costs, giving payments whose sum is $c(N)$.

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.

Suppose the distinct required runway lengths are $0=s_0<s_1<\cdots <s_k$, with $n_i>0$ users of type $i$, constant cost $\lambda \ge 0$ per unit length, and suffix counts $N_i={\sum }_{\ell =i}^k{n}_{\ell }$. Then every type-$j$ user has the displayed Shapley value. For layer $i$, exactly $N_i$ users require it. The first of them in a uniformly random permutation contributes its entire cost $\lambda (s_i-s_{i-1})$. Each requiring user has probability $1/N_i$ of being first. Linearity of the Shapley value sums these expected marginal contributions across the layers that user needs.

The allocation lies in the cost core. Its payments sum to the full runway cost. For any coalition $S$, its total payment toward layer $i$ is $\lambda (s_i-s_{i-1})|S\cap R_i|/N_i$, where $R_i$ is the set of users requiring that layer. This is zero if $S$ needs no such layer, and at most the full layer cost otherwise. Summing gives ${\sum }_{j\in S}{\varphi }_j\le c(S)$. Suffix counts followed by prefix sums compute all distinct type payments in $O(k)$ arithmetic operations.

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.