Control Complexity in Bucklin and Fallback Voting

Electoral control models ways of changing the outcome of an election via such actions as adding/deleting/partitioning either candidates or voters. To protect elections from such control attempts, computational complexity has been investigated and the corresponding NP-hardness results are termed “resistance.” It has been a long-running project of research in this area to classify the major voting systems in terms of their resistance properties. We show that fallback voting, an election system proposed by Brams and Sanver (2009) to combine Bucklin with approval voting, is resistant to each of the common types of control except to destructive control by either adding or deleting voters. Thus fallback voting displays the broadest control resistance currently known to hold among natural election systems with a polynomial-time winner problem. We also study the control complexity of Bucklin voting and show that it performs at least almost as well as fallback voting in terms of control resistance. As Bucklin voting is a special case of fallback voting, each resistance shown for Bucklin voting strengthens the corresponding resistance for fallback voting. Such worst-case complexity analysis is at best an indication of security against control attempts, rather than a proof. In practice, the difficulty of control will depend on the structure of typical instances. We investigate the parameterized control complexity of Bucklin and fallback voting, according to several parameters that are often likely to be small for typical instances. Our results, though still in the worst-case complexity model, can be interpreted as significant strengthenings of the resistance demonstrations based on NP-hardness.

💡 Research Summary

The paper conducts a comprehensive study of the computational complexity of electoral control for two voting systems: Bucklin voting and its hybrid variant, fallback voting. Electoral control models the actions of an external chair who may add, delete, or partition either candidates or voters with the goal of influencing the election outcome. Control problems are classified as constructive (making a distinguished candidate win) or destructive (preventing a distinguished candidate from winning), and each can be applied to candidates or voters, yielding a total of 22 standard control types.

Fallback voting, introduced by Brams and Sanver (2009), combines Bucklin’s level‑score mechanism with approval voting: each voter submits a set of approved candidates together with a strict ranking of all candidates. The authors prove that fallback voting is resistant (i.e., the corresponding decision problem is NP‑hard) to 18 of the 22 control types. Specifically, it is resistant to all constructive control actions (candidate addition, deletion, partition; voter addition, deletion, partition) and to all destructive candidate control actions. The only control types for which fallback voting is vulnerable (i.e., solvable in polynomial time) are destructive control by adding voters and destructive control by deleting voters. Consequently, fallback voting exhibits the broadest known resistance among natural voting systems that have a polynomial‑time winner determination algorithm.

Bucklin voting, which can be seen as a special case of fallback voting without the approval component, inherits almost the same resistance profile. The paper shows that Bucklin voting is also NP‑hard for all constructive control actions and for all destructive candidate control actions. For destructive voter control, the same two exceptions (adding and deleting voters) remain polynomial‑time solvable, mirroring the fallback case. Thus Bucklin voting is “almost as resistant” as fallback voting, and every resistance result for Bucklin automatically strengthens the corresponding result for fallback voting.

Beyond classical NP‑hardness, the authors explore parameterized complexity. They treat the number of added or deleted candidates/voters (denoted k) as a parameter and prove that the corresponding control problems are W