The Complexity of Strategic Behavior in Primary Elections

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 $\Sigma_2^{\mathbf P}$-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.