Abstract: Under the background of the "economic globalization" with both competition and cooperation, economic entities are increasingly reflecting the characteristics of competition and cooperation. They have not only the choice of strategies, but also the distribution of benefits or the allocation of costs. That is, competition and cooperation are interrelated. Therefore, Brandenburger and Stuart proposed a Biform game model to provide an effective tool for this game. At present, there are a few Biform game researches, and there are some shortcomings in the Biform game proposed by Brandenburger and Stuart:the core solution of cooperative games may be empty or not unique. Shapley value is an important single-valued solution of cooperative game, satisfying anonymity, validity, additivity, virtuality, and the expression form is simple and unique. It provides players with a fair and satisfactory allocation scheme for some cost allocation problems and benefit allocation problems. Therefore, when Shapley value is used as the solution of cooperative game, the conditions are studied for the existence of Biform game solutions. In order to analyze the new theoretical framework of the Biform game based on Shapley value proposed in this paper, the condition is first given that the characteristic function satisfies the no externalities of coalition (Shorthand for CNE, it means any player changes strategy will be not affect the return value of the coalition that it does not participate in). Under the satisfaction of this condition, the existence and nature of the Biform game solution are proven. The advantages and disadvantages of using core and Shapley value to solve Biform game solutions are compared and analyzed with numerical examples. The research shows that when the cooperative game solution is solved by the Shapley value, the existence conditions of the Biform game solution are reduced. Therefore, the research in this paper not only makes up for the Biform game proposed by Brandenburger and Stuart, the core of the cooperative game is empty or not unique, and provides a new theoretical framework for the solution of the Biform game. Therefore, it provides a new solution method for the game problem of both competition and cooperation. Therefore, the research of this paper has certain theoretical value and application value.

Key words: non-cooperative game, cooperative game, Biform game, Shapley value