Apportionment with Thresholds: Strategic Campaigns are Easy in the Top-Choice but Hard in the Second-Chance Mode

In apportionment elections, a fixed number of seats in a parliament are distributed to parties according to their vote counts. Common procedures are divisor sequence methods like D’Hondt or Sainte-Laguë. In many countries, an electoral threshold is used to prevent very small parties from entering the parliament. Parties with fewer than a given number of votes are simply removed. We (experimentally) show that by exploiting this threshold, the effectiveness of strategic campaigns (where an external agent seeks to change the outcome by bribing voters) can be increased significantly, and prove that it is computationally easy to determine the required actions. To resolve this, we propose an alternative second-chance mode where voters of parties below the threshold receive a second chance to vote for another party. We establish complexity results showing that this makes elections more resistant to strategic campaigns.