Approximations of Lovász extensions and their induced interaction index

The Lovasz extension of a pseudo-Boolean function f : (0, 1)(n) -> R is defined on each simplex of the standard triangulation of [0, 1](n) as the unique affine function (f) over cap : [0, 1](n) -> R that interpolates f at the n + 1 vertices of the simplex. Its degree is that of the unique multilinear polynomial that expresses f. In this paper we investigate the least squares approximation problem of an arbitrary Lovasz extension (f) over cap by Lovasz extensions of (at most) a specified degree. We derive explicit expressions of these approximations. The corresponding approximation problem for pseudo-Boolean functions was investigated by Hammer and Holzman [Approximations of pseudo-Boolean functions; applications to game theory, Z. Oper. Res. 36(1) (1992) 3-21] and then solved explicitly by Grabisch et al. [Equivalent representations of set functions, Math. Oper. Res. 25(2) (2000) 157-178], giving rise to an alternative definition of Banzhaf interaction index. Similarly we introduce a new interaction index from approximations of (f) over cap and we present some of its properties. It turns out that its corresponding power index identifies with the power index introduced by Grabisch and Labreuche [How to improve acts: an alternative representation of the importance of criteria in MCDM, Internat. J. Uncertain. Fuzziness Knowledge-Based Syst. 9(2) (2001) 145-157]. (c) 2007 Elsevier B.V. All rights reserved.