How to Tamper with a Parliament: Strategic Campaigns in Apportionment Elections
Robert Bredereck, Piotr Faliszewski, Michał Furdyna, Andrzej Kaczmarczyk, Joanna Kaczmarek, Martin Lackner, Christian Laußmann, Jörg Rothe, Tessa Seeger
TL;DR
The paper analyzes the computational feasibility of strategic campaigns (bribery-like vote changes) in parliamentary apportionment elections under common procedures including $D\'Hondt$ (Jefferson), Sainte-Laguë (Webster), Largest-Remainer, and $FPTP$, accounting for thresholds and multi-district structures. It develops polynomial-time algorithms for single-district bribery problems across divisor-sequence methods and LRMs, and establishes NP-hardness and $W[1]$-hardness results for multi-district settings, showing that maximizing the distinguished party’s seats is intractable in general. It introduces a second-chance mode that allows below-threshold voters a second vote opportunity, proving that strategic campaigns become NP-hard (even NP-complete when $\mathcal{R}$ is poly-time) under this mode for majority-consistent methods, both in single and multi-district settings. The work couples rigorous complexity results with extensive experiments on real election data, demonstrating that optimal campaigns typically outperform heuristics, that electoral thresholds can create sharp gains, and that the number of districts materially affects campaign costs. The authors also provide code and data for reproducibility and outline avenues for future work, including imperfect information, distance-based bribery models, and analysis of additional apportionment rules.
Abstract
In parliamentary elections, parties compete for a limited, typically fixed number of seats. Most parliaments are assembled using apportionment methods that distribute the seats based on the parties' vote counts. Common apportionment methods include divisor sequence methods (like D'Hondt or Sainte-Laguë), the largest-remainder method, and first-past-the-post. In many countries, an electoral threshold is implemented to prevent very small parties from entering the parliament. Further, several countries have apportionment systems that incorporate multiple districts. We study how computationally hard it is to change the election outcome (i.e., to increase or limit the influence of a distinguished party) by convincing a limited number of voters to change their vote. We refer to these bribery-style attacks as \emph{strategic campaigns} and study the corresponding problems in terms of their computational (both classical and parameterized) complexity. We also run extensive experiments on real-world election data and study the effectiveness of optimal campaigns, in particular as opposed to using heuristic bribing strategies and with respect to the influence of the threshold and the influence of the number of districts. For apportionment elections with threshold, finally, we propose -- as an alternative to the standard top-choice mode -- the second-chance mode where voters of parties below the threshold receive a second chance to vote for another party, and we establish computational complexity results also in this setting.