Averaged dual solution for linear production games and its characterization

In this paper, we deal with linear production planning problems in which multiple firms jointly produce some goods. Owen (Math Program 9:358–370, 1975) presents an allocation scheme for the joint profit of the firms through the cooperative game defined by formulating linear programming problems for obtaining optimal production planning. However, since the values of the resources are measured by the shadow prices which are the optimal dual solution to the linear programming problem for the grand coalition, the excess resources in the grand coalition have no value, and players receive no payoff for the excess resources possessed. Moreover, even when some coalitions cannot be formed, the Owen solution does not change and it is not affected by such situations because it is calculated using the optimal dual solution in the linear production planning problem only for the grand coalition. To cope with these difficulties, we revise the definition of the linear production game by introducing a characteristic function taking into account not only the maximized profit but also the value of the excess resources. To the revised linear production game, we introduce a solution concept with the following favorable aspects. (i) The shadow prices of the resources for all coalitions are utilized for calculating the payoffs of the players. (ii) When some coalitions cannot be formed, such situations are appropriately reflected in the payoffs. (iii) The proposed payoff vector is in the core of the revised linear production game. To demonstrate these properties, we give the numerical examples, and calculate the corresponding proposed payoff vectors. Finally, we give an axiomatic characterization of the proposed solution concept.