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Control for Coalitions in Parliamentary Elections

Hodaya Barr, Eden Hartman, Yonatan Aumann, Sarit Kraus

TL;DR

This paper studies control in parliamentary elections by focusing on coalitions of parties rather than individual candidates, introducing the problems CC and CCFP. It analyzes these controls under general and symmetric single-peaked preferences, leveraging compact input representations to obtain polynomial-time and pseudo-polynomial algorithms in SSP settings, while proving W[1]-hardness and immune results in the general case. The authors provide a detailed model with plurality allocation, formal objective definitions, and separation of control actions into adding/deleting coalition/opposition parties, along with extensive lemmas that underpin interval-based analyses. They also outline a roadmap for future work, including alternative voting rules, new control types, and broader notions of manipulation and governance influence, highlighting the practical relevance of coalition dynamics in real-world elections and policy decisions.

Abstract

The traditional election control problem focuses on the use of control to promote a single candidate. In parliamentary elections, however, the focus shifts: voters care no less about the overall governing coalition than the individual parties' seat count. This paper introduces a new problem: controlling parliamentary elections, where the goal extends beyond promoting a single party to influencing the collective seat count of coalitions of parties. We focus on plurality rule and control through the addition or deletion of parties. Our analysis reveals that, without restrictions on voters' preferences, these control problems are W[1]-hard. In some cases, the problems are immune to control, making such efforts ineffective. We then study the special case where preferences are symmetric single-peaked. We show that in the single-peaked setting, aggregation of voters into types allows for a compact representation of the problem. Our findings show that for the single-peaked setting, some cases are solvable in polynomial time, while others are NP-hard for the compact representation - but admit a polynomial algorithm for the extensive representation.

Control for Coalitions in Parliamentary Elections

TL;DR

This paper studies control in parliamentary elections by focusing on coalitions of parties rather than individual candidates, introducing the problems CC and CCFP. It analyzes these controls under general and symmetric single-peaked preferences, leveraging compact input representations to obtain polynomial-time and pseudo-polynomial algorithms in SSP settings, while proving W[1]-hardness and immune results in the general case. The authors provide a detailed model with plurality allocation, formal objective definitions, and separation of control actions into adding/deleting coalition/opposition parties, along with extensive lemmas that underpin interval-based analyses. They also outline a roadmap for future work, including alternative voting rules, new control types, and broader notions of manipulation and governance influence, highlighting the practical relevance of coalition dynamics in real-world elections and policy decisions.

Abstract

The traditional election control problem focuses on the use of control to promote a single candidate. In parliamentary elections, however, the focus shifts: voters care no less about the overall governing coalition than the individual parties' seat count. This paper introduces a new problem: controlling parliamentary elections, where the goal extends beyond promoting a single party to influencing the collective seat count of coalitions of parties. We focus on plurality rule and control through the addition or deletion of parties. Our analysis reveals that, without restrictions on voters' preferences, these control problems are W[1]-hard. In some cases, the problems are immune to control, making such efforts ineffective. We then study the special case where preferences are symmetric single-peaked. We show that in the single-peaked setting, aggregation of voters into types allows for a compact representation of the problem. Our findings show that for the single-peaked setting, some cases are solvable in polynomial time, while others are NP-hard for the compact representation - but admit a polynomial algorithm for the extensive representation.

Paper Structure

This paper contains 94 sections, 33 theorems, 28 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Organization.
  4. Related Work
  5. Time Complexity.
  6. Symmetric-Single-Peaked Preferences.
  7. Strategic Voting and Coalition Preferences.
  8. Model and Definitions
  9. Allocation Rule.
  10. Plurality.
  11. Objectives.
  12. Control by Deleting Parties.
  13. Control by Adding Parties.
  14. Control Variants.
  15. Symmetric Single-Peaked Preferences
  16. ...and 79 more sections

Key Result

Lemma 1

Let $p_1, p_2 \in P$ and $R \subseteq (P\setminus\{p_2\})$ such that $p_1 \in R$. Then, $\pi(R, p_1) \geq \pi(R \cup \{p_2\}, p_1)$.

Figures (1)

  • Figure 1: Dividers

Theorems & Definitions (69)

  • Definition 1: Coalition Objective
  • Definition 2: Favored-Party Objective
  • Lemma 1
  • Corollary 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Corollary 7
  • Lemma 8
  • ...and 59 more