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The Complexity of Strategic Behavior in Primary Elections

Colin Cleveland, Bart de Keijzer, Maria Polukarov

TL;DR

This work provides a formal complexity-theoretic analysis of strategic behavior in primary elections, framing primaries as a two-stage (or multi-stage) decision process with first-past-the-post voting and fixed tie-breaking. It proves that computing a best response is NP-hard, deciding the existence of a pure Nash equilibrium is $Σ_2^{\mathbf P}$-complete, and that subgame-perfect equilibria in sequential primaries are PSPACE-complete, with a refined $\Sigma_{O(p)}^{\mathbf P}$ bound for fixed horizons. The results show that introducing sequential primaries amplifies the logical depth of equilibrium reasoning, bridging the polynomial hierarchy and PSPACE, and highlighting substantial computational barriers to strategic manipulation in complex electoral settings. These findings have implications for understanding the design and robustness of primary systems and motivate further exploration of richer models, information dynamics, and simulation-based analyses. The work thus establishes a foundational link between electoral institution design and the complexity of multi-agent strategic reasoning.

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.

The Complexity of Strategic Behavior in Primary Elections

TL;DR

This work provides a formal complexity-theoretic analysis of strategic behavior in primary elections, framing primaries as a two-stage (or multi-stage) decision process with first-past-the-post voting and fixed tie-breaking. It proves that computing a best response is NP-hard, deciding the existence of a pure Nash equilibrium is -complete, and that subgame-perfect equilibria in sequential primaries are PSPACE-complete, with a refined bound for fixed horizons. The results show that introducing sequential primaries amplifies the logical depth of equilibrium reasoning, bridging the polynomial hierarchy and PSPACE, and highlighting substantial computational barriers to strategic manipulation in complex electoral settings. These findings have implications for understanding the design and robustness of primary systems and motivate further exploration of richer models, information dynamics, and simulation-based analyses. The work thus establishes a foundational link between electoral institution design and the complexity of multi-agent strategic reasoning.

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 -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.
Paper Structure (33 sections, 7 theorems, 22 equations, 3 figures)

This paper contains 33 sections, 7 theorems, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Complexity overview.
  3. Related Work
  4. Preliminaries
  5. Voting Framework
  6. Voter strategies.
  7. Registration.
  8. Participation regimes.
  9. Access model for $g_i$.
  10. Voter Cost and Utility
  11. Computational Questions Over Primaries
  12. Best-Response Search Space and Running Time
  13. Baseline procedure.
  14. Explicit vs. succinct GE behaviour.
  15. Special regimes.
  16. ...and 18 more sections

Key Result

Proposition 1

Let $\tau$ be the time to evaluate a single voter’s GE decision $g_i(c)$ for a given finalist profile $c$. For a fixed voter $i$, the enumeration algorithm over $b_i \in \prod_{k=1}^p (A_k \cup \{0\})$ runs in For fixed $p$ and constant $\tau$, so best response is computable in time polynomial in $n$ and $m$.

Figures (3)

  • Figure 1: Politician and voter positions in the toy example.
  • Figure 2: A partial game tree of two voters in a sequential primary. Nodes labelled "1’s" and "2’s $\mid a_j$" represent party 1’s primary and party 2’s primary conditional on candidate $a_j$ winning party 1’s primary, respectively. At each node, both voters simultaneously cast ballots $(a_i,a_{i'})$. Leaf nodes $c=\{c_1,c_2\}$ denote the general-election finalists.
  • Figure 3: Illustration of Example \ref{['EXM:NO-NE']}. Solid arrows: if $c=(x,y,a_7)$, then $x$ wins the GE. Dashed arrows: cyclic tie-breaking in primaries ($w\!\to\! z$ means $w$ defeats $z$ in a tie).

Theorems & Definitions (14)

  • Proposition 1: Best-response running time
  • Remark 1: Polynomial time versus input size
  • Definition 1: Best-Response-at-Least-$U$ (BR-$\ge U$)
  • Theorem 2: NP-completeness of BR-$\ge U$
  • Definition 2: Equilibrium Verification (NE-VERIFY)
  • Definition 3: Equilibrium Existence (NE-EXIST)
  • Theorem 3: coNP-completeness of NE-VERIFY
  • Example 4.1
  • Proposition 4
  • Theorem 5: $\Sigma_2^{\mathbf P}$-completeness of NE existence
  • ...and 4 more