Fair Allocation of profits or costs arising from joint participation by multiple individuals or entities with different purposes is essential for their continuing involvement and for their dissatisfaction reduction. In this research, fair allocation of the profits of forming a grand coalition in Three-Echelon Supply Chain (TESC) game that is composed of manufacturer, distributor and retailer, is studied. In particular, the solutions of the proportional method of profit, the proportional method of marginal profit, and Shapley value based on cooperative game theory are proved to be in the desirable characteristics of the core. The proportional method of profit and the proportional method of marginal profit are often used because of their ease of application. These methods distribute total profit in proportion to profits or marginal profits of each game participant. In addition, Shapley value can be defined as the average marginal profit when one game player is added at a time. Even though the calculation of the average of all possible marginal profits is not simple, Shapley value are often used as a useful method. Experiments have shown that the solution of the incremental method, which calculates the marginal cost of adding game players in the order of manufacturers, distributors and retailers, does not exist in the core.

The cooperative game theory consists of a set of players and utility function that has positive values for a subset of players, called coalition, in the game. The purpose of cost allocation method is to allocate the relevant cost among game players in a fair and rational way. Therefore, cost allocation method based on cooperative game theory has been applied in many areas for fair and reasonable cost allocation. On the other hand, the desirable characteristics of the cost allocation method are Pareto optimality, rationality, and marginality. Pareto optimality means that costs are entirely paid by participating players. Rationality means that by joining the grand coalition, players do not pay more than they would if they chose to be part of any smaller coalition of players. Marginality means that players are charged at least enough to cover their marginal costs. If these characteristics are all met, the solution of cost allocation method exists in the core. In this study, proportional method is applied to EOQ inventory game and EPQ inventory game with shortage. Proportional method is a method that allocates costs proportionally to a certain allocator. This method has been applied to a variety of problems because of its convenience and simple calculations. However, depending on what the allocator is used for, the proportional method has a weakness that its solution may not exist in the core. Three allocators such as demand, marginal cost, and cost are considered. We prove that the solution of the proportional method to demand and the proportional method to marginal cost for EOQ game and EPQ game with shortage is in the core. The counterexample also shows that the solution of the proportional method to cost does not exist in the core.

Cost allocation studies on the rational allocation method for the common cost of the joint products or services that provide different benefits to each economic entity under the constraints of the efficiency and fairness. Cooperative game theory is often used for cost allocation and studies on a fair and efficient allocation of the utility if some feasible utility for a whole or subset of the players in a game is given. This study shows a variety of cooperative game theory approaches and discusses the pros and cons of each approach.