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The Lovasz extension of a pseudo-Boolean function $f : \{0,1\}^n \to R$ isdefined on each simplex of the standard triangulation of $[0,1]^n$ as theunique affine function $\hat f : [0,1]^n \to R$ that interpolates $f$ at the$n+1$ vertices of the simplex. Its degree is that of the unique multilinearpolynomial that expresses $f$. In this paper we investigate the least squaresapproximation problem of an arbitrary Lovasz extension $\hat f$ by Lovaszextensions of (at most) a specified degree. We derive explicit expressions ofthese approximations. The corresponding approximation problem forpseudo-Boolean functions was investigated by Hammer and Holzman (1992) and thensolved explicitly by Grabisch, Marichal, and Roubens (2000), giving rise to analternative definition of Banzhaf interaction index. Similarly we introduce anew interaction index from approximations of $\hat f$ and we present some ofits properties. It turns out that its corresponding power index identifies withthe power index introduced by Grabisch and Labreuche (2001).

Approximations of Lovasz extensions and their induced interaction index