Weighted Banzhaf power and interaction indexes through weighted approximations of games

The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players. It was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo-Boolean functions. Considering certain weighted versions of this approximation problem, we define a class of weighted interaction indexes that generalize the Banzhaf interaction index. We show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties. Finally, we give an interpretation of the Banzhaf and Shapley interaction indexes as centers of mass of this subclass of interaction indexes.

💡 Research Summary

The paper revisits two cornerstone concepts in cooperative game theory – the Banzhaf power index and the Banzhaf interaction index – and places them within a unified approximation framework that can be weighted arbitrarily. The authors start by representing a cooperative game as a pseudo‑Boolean function f : {0,1}ⁿ → ℝ, where each binary vector indicates a coalition. In the classical setting, the Banzhaf power of player i coincides with the first‑order Fourier coefficient of f, while the Banzhaf interaction index for a coalition S corresponds to the higher‑order coefficient associated with the monomial χ_S. This observation motivates the view that both indices arise as solutions of a least‑squares approximation problem: find a low‑degree polynomial p_k that best fits f in the ordinary (unweighted) L² norm.

The novelty of the work lies in replacing the uniform weight by an arbitrary probability distribution w on the set of coalitions. The weighted least‑squares problem is defined as
 min_{a_S} ∑_{x∈{0,1}ⁿ} w(x)