We study the computational complexity of strategic behaviour in primary elections. Unlike direct voting systems, primaries introduce a multi-stage process in which voters first influence intra-party nominees before a general election determines the final winner. While previous work has evaluated primaries via welfare distortion, we instead examine their game-theoretic properties. We formalise a model of primaries under first-past-the-post with fixed tie-breaking and analyse voters' strategic behaviour. We show that determining whether a pure Nash equilibrium exists is $Σ_2^{\mathbf P}$-complete, computing a best response is NP-complete, and deciding the existence of subgame-perfect equilibria in sequential primaries is PSPACE-complete. These results reveal that primaries fundamentally increase the computational difficulty of strategic reasoning, situating them as a rich source of complexity-theoretic challenges within computational social choice.

翻译：本文研究初选机制中策略行为的计算复杂性。与直接投票系统不同，初选引入了多阶段过程：选民首先影响党内提名，再由大选决定最终获胜者。现有研究多通过福利扭曲评估初选机制，本文则从博弈论视角考察其理论特性。我们建立了采用简单多数制与固定平局决胜规则的初选形式化模型，并分析选民的策略行为。研究证明：判断纯纳什均衡是否存在是$Σ_2^{\mathbf P}$完全问题，计算最优应对策略是NP完全问题，而判定顺序初选中子博弈精炼均衡的存在性则是PSPACE完全问题。这些结果表明，初选机制从根本上提升了策略推理的计算难度，使其成为计算社会选择领域中复杂度理论挑战的重要来源。