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Some Bounds for the Banzhaf Index and Other Semivalues

The normalized Banzhaf index of a player in a monotone simple game thought of as a voting model is the probability for that player to swing the outcome of the vote. We bound the Euclidean norm of the vector of Banzhaf indices of simple games in terms of the number winning coalitions. The Banzhaf index is a semivalue, so we proceed to estimate norms of general semivalues.