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arXiv · 2602.10290

The Complexity of Strategic Behavior in Primary Elections

Colin Cleveland·Bart de Keijzer·Maria Polukarov
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Abstract

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.

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Colin Cleveland, Bart de Keijzer, Maria Polukarov. 2026-02-10. The Complexity of Strategic Behavior in Primary Elections. https://arxiv.org/abs/2602.10290

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