Complexity of Manipulative Actions When Voting with Ties

Most of the computational study of election problems has assumed that each voter’s preferences are, or should be extended to, a total order. However in practice voters may have preferences with ties. We study the complexity of manipulative actions on elections where voters can have ties, extending the definitions of the election systems (when necessary) to handle voters with ties. We show that for natural election systems allowing ties can both increase and decrease the complexity of manipulation and bribery, and we state a general result on the effect of voters with ties on the complexity of control.

💡 Research Summary

The paper investigates how allowing ties (indifferences) in voters’ preference orders affects the computational complexity of three classic strategic election problems: manipulation, bribery, and control. While most prior work assumes that each voter’s ballot is a total order, the authors consider weak orders, i.e., transitive, reflexive, antisymmetric relations that may contain ties at any rank. They first formalize four natural extensions of scoring rules to handle ties: Min, Max, Round‑down, and Average. These extensions determine how a candidate in a tied group receives points based on the underlying scoring vector (e.g., Borda, Plurality, t‑Approval). The definitions are consistent with earlier work on top‑orders (where only the lowest‑ranked candidates may tie) and generalize them to arbitrary tie patterns.

The authors then analyze a variety of election systems under these extensions. For scoring rules, they show that allowing ties can both raise and lower the difficulty of manipulation and bribery. For example, 3‑candidate Borda with single‑peaked preferences is polynomial‑time solvable when ballots are total orders, but becomes NP‑complete when ballots are top‑orders and the Max extension is used. The hardness proof reduces from a variant of the Partition problem (named Partition′) and exploits the ability of manipulators to split their votes among different tie‑breaking orders to achieve a precise score balance. Conversely, for Plurality and t‑Approval, manipulation and bribery remain in P regardless of the tie‑handling extension, because the optimal strategic vote is still to give the maximum possible points to the preferred candidate.

For pairwise‑majority based systems, the paper focuses on Copeland α (α∈