The Kernel of the Stochastic Cooperative Game

Based on the concept of the stochastic cooperative game, the concepts such as the kernel and the nucleolus of the countermeasure are extended in this article, and many concepts such as the maximum excess value, the kernel and the nucleolus of the stochastic cooperative countermeasure are defined, and some characters and properties, and the relationships among the kernel, the nucleolus and the minimum core of the stochastic cooperative game are discussed.

The stochastic cooperative game introduced by Suijs et al (1995)   ( ) is the optimal one in ( ) . Define the set of all un-optimal payments of the stochastic cooperative game Γ as the core of Γ , and it is denoted by

Kernel of the stochastic cooperative game
Definition 2.1: Suppose ( ,{ } ,( ) ) with n players, ( , ) ( ) is the excess value of the alliance S about ( , ) This value reflects the attitude of the alliance S to the distribution ( , ) is not welcomed by S. For the fixed ( , ) ( ) N N d r .And rank them from big to small, a 2 n dimensional vector can be obtained.
Where, ( , ) [ ,( , )] ( 1,2, ,2 ) e. the core ( ) C Γ of Γ is the 0 nucleolus, and when ε is small enough, ( ) is the maximum excess value of the player i exceeding j at the position of ( , ) is the stochastic cooperative game with n players, and the stochastic payment ( , ) ( ) , so at the position of ( , ) Definition 2.5: At the position of ( , ) N N d r , if i doesn't outweight j, and j doesn't outweight i, so i and j are equilibrium at the position of ( , ) Definition 2.6: The kernel ( ) K Γ of the stochastic cooperative game is the collectivity of all these distributions, and for the distribution ( , ) N N d r , any two players are in equilibrium, i.e.
Definition 2.7: The nucleolus ( ) Nu Γ of the stochastic cooperative game is the collectivity of those distributions which minimize ( , ) Q d r according to the dictionary sequence, i.e. .

Main conclusions
Theorem 3.1: Suppose Γ is the stochastic cooperative game, so ( ) Q d r according to the dictionary sequence, and because S N ∀ ⊆ , so the set of player, and S Χ is the payment function of the alliance S, and each stochastic payment has limited anticipation, and ( ) N i i∈ f is the optimal relation about the player i .The stochastic payment of the alliance S, S Χ , is denoted by a binary pair ( ) variable, and the set of all payments of the alliance S is denoted by ( ) Z S .The set of all individual reasonable payments of the alliance S is denoted by The nucleolus of the stochastic cooperative game is the subset of the intersection set of the kernel and the minimum core, i.e.