Cost sharing on prices for games on graphs

Let (N={1,dots ,n}) be a set of customers who want to buy a single homogenous goods in market. Let (q_i>0) be the quantity that (iin N) demands, (q=(q_1,dots ,q_n)) and (q_S=sum _{iin S}q_i) for (Ssubseteq 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)) for (Ssubseteq 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 _{Rin S/L}f(q_R)) for (Ssubseteq 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)) from the core for the game ((x_Sle v_L(S)) for any (Ssubseteq 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) and (frac{x_i}{q_i} ge frac{x_j}{q_j}) if (q_i le q_j) for i and j in the same component of N.