Weighted voting games are a well-known and useful class of succinctly representable simple games that have many real-world applications, e.g., to model collective decision-making in legislative bodies or shareholder voting. Among the structural control types being analyzing, one is control by adding players to weighted voting games, so as to either change or to maintain a player's power in the sense of the (probabilistic) Penrose-Banzhaf power index or the Shapley-Shubik power index. For the problems related to this control, the best known lower bound is PP-hardness, where PP is "probabilistic polynomial time," and the best known upper bound is the class NP^PP, i.e., the class NP with a PP oracle. We optimally raise this lower bound by showing NP^PP-hardness of all these problems for the Penrose-Banzhaf and the Shapley-Shubik indices, thus establishing completeness for them in that class. Our proof technique may turn out to be useful for solving other open problems related to weighted voting games with such a complexity gap as well.

翻译：加权投票博弈是一类具有简洁表示形式的经典简单博弈，在实际应用中具有广泛价值，例如可用于立法机构集体决策或股东投票建模。在已分析的结构控制类型中，有一类是通过向加权投票博弈添加参与者来实现控制，其目的在于改变或维持某个参与者在（概率性）Penrose-Banzhaf权力指数或Shapley-Shubik权力指数意义上的权力。针对这类控制问题，已知最佳下界是PP-困难性（其中PP指“概率多项式时间”），而最佳上界是NP^PP类（即配备PP预言机的NP类）。我们通过证明所有涉及Penrose-Banzhaf指数和Shapley-Shubik指数的此类问题均具有NP^PP-困难性，从而将下界提升至最优水平，并确立其在该复杂性类中的完备性。我们的证明技术可能对解决其他具有类似复杂性间隙的加权投票博弈开放问题也具有参考价值。