False-name manipulation refers to the question of whether a player in a weighted voting game can increase her power by splitting into several players and distributing her weight among these false identities. Analogously to this splitting problem, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. Aziz et al. [ABEP11] analyze the problem of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley-Shubik and the normalized Banzhaf index, and so do Rey and Rothe [RR10] for the probabilistic Banzhaf index. All these results provide merely NP-hardness lower bounds for these problems, leaving the question about their exact complexity open. For the Shapley--Shubik and the probabilistic Banzhaf index, we raise these lower bounds to hardness for PP, "probabilistic polynomial time", and provide matching upper bounds for beneficial merging and, whenever the number of false identities is fixed, also for beneficial splitting, thus resolving previous conjectures in the affirmative. It follows from our results that beneficial merging and splitting for these two power indices cannot be solved in NP, unless the polynomial hierarchy collapses, which is considered highly unlikely.
加权投票博弈（WVG）是一种联盟博弈，其中代理人对联盟的贡献由他的权重给出，如果联盟的总权重达到或超过给定的配额，则联盟获胜。这些游戏模型的决策，在政治机构以及合作和盈余分割多智能体域。近年来，加权投票博弈的各种解概念的计算复杂性受到了广泛的关注。特别是，Elkind et al.（2007）研究了WVG中与稳定性相关的解决方案概念的复杂性，即核心，最小核心和核仁。虽然他们已经完全描述了核心和最小核心的算法复杂性，但对于核仁，他们只提供了NP-硬度结果。在本文中，我们解决了Elkind等人提出的一个公开问题。通过证明WVG的核仁，以及更一般地，具有固定k的k-向量加权投票游戏，可以在伪多项式时间内计算，即，存在一种算法，其正确地计算核仁并且在玩家数量和最大权重的时间多项式中运行。在这样做的时候，我们提出了一个计算核仁，这可能是适用于更广泛的一类游戏的一般框架。
