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.

翻译：在议会选举中，各政党竞争数量有限且通常固定的席位。大多数议会采用席位分配方法，根据各政党得票数分配席位。常见的席位分配方法包括除数序列法（如D'Hondt法或Sainte-Laguë法）、最大余数法以及简单多数制。许多国家设置了选举门槛以防止极小型政党进入议会。此外，若干国家采用包含多个选区的席位分配体系。我们研究了通过说服有限数量的选民改变投票，从而改变选举结果（即增加或限制特定政党的影响力）的计算难度。我们将这类贿赂式攻击称为\emph{策略性竞选活动}，并从计算复杂性（包括经典复杂性与参数化复杂性）角度研究相应问题。我们还在真实选举数据上进行了大量实验，研究了最优竞选活动的有效性，特别是与启发式贿赂策略相比的效果，以及选举门槛和选区数量对结果的影响。最后，针对设有门槛的席位分配选举，我们提出——作为标准首选模式的替代方案——"二次机会"模式，即得票低于门槛的政党支持者可获得为其他政党投票的第二次机会，并在此设定下建立了相应的计算复杂性结果。