The Complexity of Strategic Behavior in Primary Elections

2026-02-10

AI Summary

This study investigates the computational complexity of strategic voting in primary elections. Focusing on a multi-stage primary model that employs simple majority rule with a fixed tie-breaking mechanism, the work provides the first systematic game-theoretic characterization of its strategic complexity. Through tools from computational complexity theory and equilibrium analysis—including pure-strategy Nash equilibria and subgame-perfect equilibria—the authors establish that determining the existence of a pure-strategy Nash equilibrium is Σ₂^P-complete, computing a best response is NP-complete, and deciding the existence of a subgame-perfect equilibrium in sequential primaries is PSPACE-complete. These results reveal the high-order computational complexity inherent in primary election systems within the framework of computational social choice.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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

Problem

• primary elections
• strategic behavior
• computational complexity
• Nash equilibrium
• subgame-perfect equilibrium

Innovation

• computational complexity
• primary elections
• strategic voting
• Nash equilibrium
• subgame-perfect equilibrium