Control for Coalitions in Parliamentary Elections

The traditional election control problem focuses on the use of control to promote a single candidate. In parliamentary elections, however, the focus shifts: voters care no less about the overall governing coalition than the individual parties' seat count. This paper introduces a new problem: controlling parliamentary elections, where the goal extends beyond promoting a single party to influencing the collective seat count of coalitions of parties. We focus on plurality rule and control through the addition or deletion of parties. Our analysis reveals that, without restrictions on voters' preferences, these control problems are W[1]-hard. In some cases, the problems are immune to control, making such efforts ineffective. We then study the special case where preferences are symmetric single-peaked. We show that in the single-peaked setting, aggregation of voters into types allows for a compact representation of the problem. Our findings show that for the single-peaked setting, some cases are solvable in polynomial time, while others are NP-hard for the compact representation - but admit a polynomial algorithm for the extensive representation.