Control by Adding Players to Change or Maintain the Shapley-Shubik or the Penrose-Banzhaf Power Index in Weighted Voting Games Is Complete for NP^PP
Joanna Kaczmarek, Jörg Rothe
TL;DR
The paper proves that control-by-adding-players in weighted voting games, aimed at changing or maintaining the Shapley-Shubik or Penrose-Banzhaf power indices, is complete for $\mathrm{NP}^{\mathrm{PP}}$. It introduces SAT-based reduction tools, including two weight-vector constructions that bijectively encode CNF assignments into WVG coalitions with carefully chosen quotas, and leverages seed problems from the counting and majority/minority SAT literature to establish both hardness and membership in $\mathrm{NP}^{\mathrm{PP}}$ for all variants. The results close the known complexity gap by showing $\mathrm{NP}^{\mathrm{PP}}$-hardness and containment for five control problems (increase/decrease/nondecrease/maintain) across both $\beta$ and $\varphi$. The techniques, including intricate weight gadgets and exact-coalition accounting, may be useful for adjacent open problems in weighted voting games and beyond.
Abstract
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.