Abstract:

In the present paper, we define a basis of [see pdf for notation] relative to a Semivalue, we compute the potentials of the subgames of a given game, to show that the basis is a potential basis, from which we get the Semivalues of the basic vectors. In this way we discover a basis of the null space of a Semivalue and derive, as in the previous work, a solution of the inverse problem, this time for a Semivalue. As the Shapley value was considered in detail in the previous work, we give a complete description for the Banzhaf value. As a byproduct of the results on the potential basis relative to a Semivalue, we give an algorithm for the computation of a Semivalue, called a dynamic algorithm, because the algorithm is building
a finite sequence of games, with the same Semivalue as the given one, where the last game is providing the Semivalue by an easy computation. A similar algorithm for computing the Shapley value has been developed by M.Maschler (1982).
We show also an accelerated algorithm which solves the problem in n steps. The case of a general three person cooperative TU game is shown for illustrating the concepts introduced, and a particular game is chosen to exhibit the application of the dynamic algorithm. 