Weighted Banzhaf power and interaction indexes through weighted approximations of games

Abstract. 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 A cooperative game on a finite set of players N = {1, . . . , n} is a set function v : 2 N → R which assigns to each coalition S of players a real number v(S) representing the worth of S. 1 Identifying the subsets of N with the elements of {0, 1} n , we see that a game v : 2 N → R corresponds to a pseudo-Boolean function f : {0, 1} n → R (the correspondence is given by v(S) = f (1 S ), where 1 S denotes the characteristic vector of 1 Usually, the condition v(∅) = 0 is required for v to define a game. However, we do not need this restriction in the present work