Equivalent representations of set functions.

*(English)*Zbl 0982.91009Characteristic functions in cooperative game theory, belief functions in evidence theory and weight functions in multicriteria decision making share a common structure. Alternative representations of such a set function, equivalent to it in the sense of being linked by means of an invertible transformation, are studied here. Thus, in addition to the well-known dividend function or Möbius transform, the dual representation, the co-Möbius representation and the Banzhaf and Shapley interaction indices are considered, the latter two having been already introduced by (some of) the authors in previous work.

It is shown that the co-Möbius and the interaction indices are linear representations of the characteristic function, and a complete set of conversion formulas is provided. To this end, use is made of pseudo-Boolean functions and their multilinear and Lovász extensions. As to the transformations (fractal and upper and lower-cardinality transformations) remarkable properties of the underlying matrices are stated.

Finally, an application is done to solve the problem of approximating a pseudo-Boolean function by means of a multilinear polynomial by using the Banzhaf interaction index. The eventual readers should be warned that the paper, but interesting, is highly technical.

It is shown that the co-Möbius and the interaction indices are linear representations of the characteristic function, and a complete set of conversion formulas is provided. To this end, use is made of pseudo-Boolean functions and their multilinear and Lovász extensions. As to the transformations (fractal and upper and lower-cardinality transformations) remarkable properties of the underlying matrices are stated.

Finally, an application is done to solve the problem of approximating a pseudo-Boolean function by means of a multilinear polynomial by using the Banzhaf interaction index. The eventual readers should be warned that the paper, but interesting, is highly technical.

Reviewer: F.Carreras (Terrassa)

##### MSC:

91A12 | Cooperative games |