NPPP-Completeness of Control by Adding Players to Change the Penrose-Banzhaf Power Index in Weighted Voting Games

Weighted voting games are an important class of simple games that can be compactly represented and have many real-world applications. Rey and Rothe [14] introduced the notion of structural control by adding players to or deleting them from weighted voting games, with the goal to either change or maintain a given player's power in a given game with respect to the (probabilistic) Penrose-Banzhaf power index [4] or the Shapley-Shubik power index [17]. For control by adding players, they showed PP-hardness as the best known lower bound and an upper bound of NP PP , where PP is "probabilistic polynomial time. " We optimally improve their results by establishing NP PP-hardness (and thus NP PP-completeness) of all problems related to the Penrose-Banzhaf index and for the problem of maintaining the Shapley-Shubik index when players are added.