prove the following property of the

prove the following property of the AMEF value. Let (N, I) ∈ IT S. Given the inventory transportation systems (N,Ik), for each k ∈ E(N ,I) , the AMEF value of (N , I) is the average of the AMEF values of the systems (N , Ik ), i.e.
AMEF(N, I) = |E(N,I)|
AMEF(N, I ).
(2)
1k
k∈E(N ,I)
The next step is to define the properties we use in the characterization of the AMEF value. Let us denote by IT S1 the class of inventory transportation situations with only one extreme agent. Take (N, I) ∈ IT S1 and let us denote by e(N, I) the unique extreme agent of (N, I) and by (N, c) ∈ IT G its associated cost game. Let φ be an allocation rule on IT S.
One extreme agent balanced contributions (OBC). Take (N , I ) ∈ IT S1. For all pair of agents i, j ∈ N {e(N, I)} it holds
φi(N, I) − φi(N {j}, I−j) = φj(N, I) − φj(N {i}, I−i)
withI−j =I|N{j} andI−i =I|N{i}.
One extreme agent solidarity (EAS). Take (N, I) ∈ IT S1. It holds
φe(N,I)(N, I) = c(e(N, I)).
The OBC property is based on the Balanced Contribution property introduced by Myerson [7]. Roughly speaking, the OBC property requires that if i and j are not extreme agents, the effect on j of i leaving the system is the same as the effect on i of j leaving the system. The EAS property states that the extreme agent is cooperating unselfishly, because he is joining the grand coalition providing some savings in the total cost (since c(N {n})+c({n}) ≥ c(N)), but he is paying the same amount as if he is not cooperating with the others (we could say that he is resigning to savings). In some sense, this property implies that the extreme agent is altruistic. However, maybe this agent is altruistic in the present with the hope that other even more distant altruistic agents will join the game and pay in some time in the future.
The next theorem provides a characterization of the AMEF value on IT S1.