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.

Apportionment methods allocate a fixed number of seats in a parliament to parties based on their vote counts. In many countries, parliamentary elections are organized by first holding separate elections in several districts and then putting the single results together. We call such elections multidistrict apportionment elections. Moreover, many countries have an additional general electoral threshold, i.e., a minimum number of votes a party must receive to win any seats in a parliament. For such methods, we study the complexity of bribery problems where an external agent seeks to increase the number of seats for a distinguished party by bribing voters within a given budget. Specifically, we investigate how adding either a general electoral threshold, or district limits for the budget, or both influences the complexity of bribery. District limits—which the external agent must adhere to while still respecting the overall budget—denote a maximum budget for each district, each to be spent for changing the votes only in that district. We show that adding a general electoral threshold, with or without district limits, makes the problems NP-complete for the largest-remainder method and all divisor sequence methods (including the prominent D’Hondt and Sainte-Laguë apportionment methods). We also study parameterized complexity and domain restrictions of these problems.

We survey the notion of control in various areas of computational social choice (COMSOC) such as voting, fair allocation, cooperative game theory, matching under preferences, and group identification. In all these scenarios, control can be exerted, for instance, by adding or deleting agents with the goal of influencing the outcome. We conclude by briefly covering control in some other COMSOC areas including participatory budgeting, judgment aggregation, and opinion diffusion.