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Main Authors: Bredereck, Robert, Faliszewski, Piotr, Furdyna, Michał, Kaczmarczyk, Andrzej, Kaczmarek, Joanna, Lackner, Martin, Laußmann, Christian, Rothe, Jörg, Seeger, Tessa
Format: Preprint
Published: 2026
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Online Access:https://arxiv.org/abs/2601.15855
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author Bredereck, Robert
Faliszewski, Piotr
Furdyna, Michał
Kaczmarczyk, Andrzej
Kaczmarek, Joanna
Lackner, Martin
Laußmann, Christian
Rothe, Jörg
Seeger, Tessa
author_facet Bredereck, Robert
Faliszewski, Piotr
Furdyna, Michał
Kaczmarczyk, Andrzej
Kaczmarek, Joanna
Lackner, Martin
Laußmann, Christian
Rothe, Jörg
Seeger, Tessa
contents 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.
format Preprint
id arxiv_https___arxiv_org_abs_2601_15855
institution arXiv
publishDate 2026
record_format arxiv
spellingShingle How to Tamper with a Parliament: Strategic Campaigns in Apportionment Elections
Bredereck, Robert
Faliszewski, Piotr
Furdyna, Michał
Kaczmarczyk, Andrzej
Kaczmarek, Joanna
Lackner, Martin
Laußmann, Christian
Rothe, Jörg
Seeger, Tessa
Computer Science and Game Theory
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.
title How to Tamper with a Parliament: Strategic Campaigns in Apportionment Elections
topic Computer Science and Game Theory
url https://arxiv.org/abs/2601.15855