Weighted Banzhaf power and interaction indexes through weighted approximations of games

In cooperative game theory, various kinds of power indexes are used to measure the influence that a given player has on the outcome of the game or to define a way of sharing the benefits of the game among the players. The best known power indexes are due to Shapley [15,16] and Banzhaf [1,5] and there are many other examples of such indexes in the literature.
When one is concerned by the analysis of the behavior of players in a game, the information provided by power indexes might be far insufficient, for instance due to the lack of information on how the players interact within the game. The notion of interaction index was then introduced to measure an interaction degree among players in coalitions; see [13,12,7,8,14,10,6] for the definitions and axiomatic characterizations of the Shapley and Banzhaf interaction indexes as well as many others.
In addition to the axiomatic characterizations the Shapley power index and the Banzhaf power and interaction indexes were shown to be solutions of simple least squares approximation problems (see [2] for the Shapley index, [11] for the Banzhaf power index and [9] for the Banzhaf interaction index).
We generalize the non-weighted approach of [11,9] by adding a weighted, probabilistic viewpoint: A weight w(S) is assigned to every coalition S of players that represents the probability that coalition S forms. The solution of the weighted least squares problem associated with the probability distribution w was given in [3,4] in the special case when the players behave independently of each other to form coalitions.
In this particular setting we introduce a weighted Banzhaf interaction index associated with w by considering, as in [11,9], the leading coefficients of the approximations of the game by polynomials of specified degrees.We then study the most important properties of these weighted indexes and their relations with the classical Banzhaf and Shapley indexes.