Approximations of Lovasz extensions and their induced interaction index

The Lovász 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 f̂ : [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 Lovász extension f̂ by Lovász extensions of (at most) a specified degree. We derive explicit expressions of these approximations. The corresponding approximation problem for pseudoBoolean functions was investigated by Hammer and Holzman [14] and then solved explicitly by Grabisch, Marichal, and Roubens [11], giving rise to an alternative definition of Banzhaf interaction index. Similarly we introduce a new interaction index from approximations of f̂ 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 [10].