Cost sharing on prices for games on graphs

Let $$N=\{1,\dots ,n\}$$ N = { 1 , ⋯ , n } be a set of customers who want to buy a single homogenous goods in market. Let $$q_i>0$$ q i > 0 be the quantity that $$i\in N$$ i ∈ N demands, $$q=(q_1,\dots ,q_n)$$ q = ( q 1 , ⋯ , q n ) and $$q_S=\sum _{i\in S}q_i$$ q S = ∑ i ∈ S q i for $$S\subseteq N$$ S ⊆ N . If f(s) is a (increasing and concave) cost function, then it yields a cooperative game (N, f, q) by defining characteristic function $$v(S)=f(q_S)$$ v ( S ) = f ( q S ) for $$S\subseteq N$$ S ⊆ N . We now consider the way of taking packages of goods by customers and define a communication graph L on N, in which i and j are linked if they can take packages for each other. So if i and j are connected, then a package can be delivered from i to j by some intermediators. We thus admit any connected subset as a feasible coalition, and obtain a game (N, f, q, L) by defining characteristic function $$v_L(S)=\sum _{R\in S/L}f(q_R)$$ v L ( S ) = ∑ R ∈ S / L f ( q R ) for $$S\subseteq N$$ S ⊆ N , where S / L is the family of induced components (maximal connected subset) in S. It is shown that there is an allocation (cost shares) $$x=(x_1,\dots ,x_n)$$ x = ( x 1 , ⋯ , x n ) from the core for the game ( $$x_S\le v_L(S)$$ x S ≤ v L ( S ) for any $$S\subseteq N$$ S ⊆ N ) such that x satisfies Component Efficiency and Ranking for Unit Prices. If f(s) and q satisfy some further condition, then there is an allocation x from the core such that x satisfies Component Efficiency, and $$x_i \le x_j$$ x i ≤ x j and $$\frac{x_i}{q_i} \ge \frac{x_j}{q_j}$$ x i q i ≥ x j q j if $$q_i \le q_j$$ q i ≤ q j for i and j in the same component of N.