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Bribery for Coalitions in Parliamentary Elections

Hodaya Barr, Yonatan Aumann, Sarit Kraus

TL;DR

This work examines bribery in parliamentary elections where a briber seeks to optimize a coalition of parties rather than a single party. It formalizes two problems, CB and CBP, and analyzes them under Plurality and Borda scoring with four bribery types across threshold and non-threshold seat allocation. The authors provide polynomial-time algorithms for several 1-bribery and $\$$-bribery cases under Plurality (and various Borda settings without thresholds), and establish NP-hardness for swap-bribery and coalition-shift-bribery under thresholds, as well as for all Borda variants with thresholds, using reductions from 3-4-Exact-Cover and Min-Bisection, and a polynomial-time minimum-cost-flow construction for some non-threshold Plurality cases. The results map the computational boundaries of coalition bribery, informing both theory and defense against coalitional manipulation in parliamentary settings. Overall, the paper advances understanding of how coalition considerations alter the complexity landscape of election bribery and highlights avenues for future work on other multi-winner rules and strategic interactions.

Abstract

We study the computational complexity of bribery in parliamentary voting, in settings where the briber is (also) interested in the success of an entire set of political parties - a ``coalition'' - rather than an individual party. We introduce two variants of the problem: the Coalition-Bribery Problem (CB) and the Coalition-Bribery-with-Preferred-party Problem (CBP). In CB, the goal is to maximize the total number of seats held by a coalition, while in CBP, there are two objectives: to maximize the votes for the preferred party, while also ensuring that the total number of seats held by the coalition is above the target support (e.g. majority). We study the complexity of these bribery problems under two positional scoring functions - Plurality and Borda - and for multiple bribery types - $1$-bribery, $\$$-bribery, swap-bribery, and coalition-shift-bribery. We also consider both the case where seats are only allotted to parties whose number of votes passes some minimum support level and the case with no such minimum. We provide polynomial-time algorithms to solve some of these problems and prove that the others are NP-hard.

Bribery for Coalitions in Parliamentary Elections

TL;DR

This work examines bribery in parliamentary elections where a briber seeks to optimize a coalition of parties rather than a single party. It formalizes two problems, CB and CBP, and analyzes them under Plurality and Borda scoring with four bribery types across threshold and non-threshold seat allocation. The authors provide polynomial-time algorithms for several 1-bribery and $-bribery cases under Plurality (and various Borda settings without thresholds), and establish NP-hardness for swap-bribery and coalition-shift-bribery under thresholds, as well as for all Borda variants with thresholds, using reductions from 3-4-Exact-Cover and Min-Bisection, and a polynomial-time minimum-cost-flow construction for some non-threshold Plurality cases. The results map the computational boundaries of coalition bribery, informing both theory and defense against coalitional manipulation in parliamentary settings. Overall, the paper advances understanding of how coalition considerations alter the complexity landscape of election bribery and highlights avenues for future work on other multi-winner rules and strategic interactions.

Abstract

We study the computational complexity of bribery in parliamentary voting, in settings where the briber is (also) interested in the success of an entire set of political parties - a ``coalition'' - rather than an individual party. We introduce two variants of the problem: the Coalition-Bribery Problem (CB) and the Coalition-Bribery-with-Preferred-party Problem (CBP). In CB, the goal is to maximize the total number of seats held by a coalition, while in CBP, there are two objectives: to maximize the votes for the preferred party, while also ensuring that the total number of seats held by the coalition is above the target support (e.g. majority). We study the complexity of these bribery problems under two positional scoring functions - Plurality and Borda - and for multiple bribery types - -bribery, $-bribery, swap-bribery, and coalition-shift-bribery. We also consider both the case where seats are only allotted to parties whose number of votes passes some minimum support level and the case with no such minimum. We provide polynomial-time algorithms to solve some of these problems and prove that the others are NP-hard.

Paper Structure

This paper contains 27 sections, 16 theorems, 8 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Proofs.
  3. Related Work
  4. Model
  5. Problem Definition
  6. Notations.
  7. Plurality
  8. 1-Bribery and $-Bribery
  9. Computing $\hat{V}_{\bar{A}}$.
  10. Computing $g$.
  11. Swap-Bribery and Coalition-Shift-Bribery without Threshold
  12. Swap-Bribery and Coalition-Shift-Bribery with Threshold
  13. Borda
  14. Borda$_{0}$
  15. Computing $h$ for coalition-shift-bribery.
  16. ...and 12 more sections

Key Result

Lemma 1

It is possible to compute $g(\ell, a_{\bar{A}},d, a_{A_{-1}})$, for all $0\leq \ell, a_{A_{-1}}, d, a_{\bar{A}}\leq n$, and $mincost(c_j,k)$, for $1\leq j\leq m, 1\leq \ell \leq n$ in polynomial time.

Figures (1)

  • Figure 1: Network for proof of Theorem \ref{['thm:Plurality-ccbp']}. Unless otherwise stated, the default values are $cost = 0$ and $cap = 1$

Theorems & Definitions (29)

  • Definition 1: Coalition-Bribery-Problem (CB)
  • Definition 2: Coalition-Bribery-with-Preferred-party-Problem (CBP)
  • Lemma 1
  • Theorem 2
  • Corollary 3
  • Definition 3: Minimum-Cost Flow (MCF)
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • ...and 19 more