The Complexity of Controlling Candidate-Sequential Elections

Candidate control of elections is the study of how adding or removing candidates can affect the outcome. However, the traditional study of the complexity of candidate control is in the model in which all candidates and votes are known up front. This paper develops a model for studying online control for elections where the structure is sequential with respect to the candidates, and in which the decision regarding adding and deleting must be irrevocably made at the moment the candidate is presented. We show that great complexity—PSPACE-completeness—can occur in this setting, but we also provide within this setting polynomial-time algorithms for the most important of election systems, plurality.

💡 Research Summary

The paper introduces a novel framework for studying “online” candidate control in elections where candidates appear sequentially and the election chair must make irrevocable add‑or‑delete decisions at the exact moment each candidate is presented. This setting contrasts with the traditional model of candidate control, which assumes full knowledge of all candidates and votes before any manipulation takes place. The authors formalize four fundamental decision problems: constructive control by deleting candidates (online‑E‑CCDC), constructive control by adding candidates (online‑E‑CCAC), destructive control by deleting candidates (online‑E‑DCDC), and destructive control by adding candidates (online‑E‑DCAC). Each problem is defined by an input consisting of the set of candidates, the set of voters, a predetermined order in which candidates will be presented, a flag indicating the current candidate, a bound k on the total number of deletions or additions allowed, the chair’s preference ranking σ over all candidates, a distinguished target candidate d, and a history of past decisions and voter preferences up to the current moment. The central question is whether the chair can make a decision about the current candidate (delete or keep, add or not) such that, regardless of how future preferences are revealed, there exists a strategy for the remaining steps that guarantees the desired outcome (either that a candidate at least as good as d wins, or that no candidate worse than d wins).

From a complexity‑theoretic perspective, the authors first observe that if the underlying election system’s winner‑determination problem lies in polynomial time (P), then each of the four online control problems belongs to PSPACE. This follows from the classic characterization of PSPACE as the class of problems solvable by an alternating polynomial‑time Turing machine (Chandra, Kozen, Stockmeyer 1981). The upper bound holds even when the winner problem is itself PSPACE‑hard.

The more striking contribution is the matching lower bound: for each of the four control problems there exist artificial election systems E (and E′) whose winner problems are in P, yet the corresponding online control problem is PSPACE‑complete. The proof proceeds via a polynomial‑time many‑one reduction from the PSPACE‑complete Quantified Boolean Formula (QBF) problem. The reduction encodes a QBF instance into a specially crafted election system where each candidate is a pair (F,i): F is a Boolean formula and i is a non‑negative integer. All candidates must share the same formula ˆF; otherwise the election is declared a loss for everyone. The integer i serves two purposes: for odd i it determines the truth value of the i‑th variable in the formula, and for even i it encodes the voter’s preference between candidate (ˆF,i) and a baseline candidate (ˆF,0). The chair’s preference order σ ranks candidates in decreasing order of i, making higher‑indexed candidates more desirable. The target candidate d is (ˆF,0). The reduction sets the initial candidate set to all even‑indexed candidates up to 2j, the “spoil” (potentially addable) set to the odd‑indexed candidates, and the addition/deletion bound k to j, where j is the number of quantifier blocks in the QBF formula. Under this construction, the chair can achieve the desired outcome if and only if the original QBF formula is true. Consequently, each online control problem is PSPACE‑hard, and together with the upper bound they are PSPACE‑complete.

In contrast, the authors show that for the widely used plurality rule (the candidate with the most first‑place votes wins) the online control problems become tractable. Because plurality’s winner determination depends only on the number of first‑place votes, the chair’s sequential decisions can be guided by a simple greedy strategy: at each step keep a candidate if doing so does not jeopardize the possibility of a preferred candidate winning later, otherwise delete it. This yields a polynomial‑time algorithm for both constructive and destructive online control under plurality, demonstrating that high complexity is not inherent to all voting systems.

The paper also distinguishes two models of chair restrictions: the “hand‑tied” model, where the chair is forbidden to delete any candidate that is already at or below the target’s rank, and the “non‑hand‑tied” model, where the chair may delete some but not all such candidates. Both models retain the PSPACE‑completeness results for the artificial systems.

Overall, the work makes several key contributions:

1. It defines a realistic online candidate‑control framework reflecting sequential candidate entry and irrevocable decisions.
2. It establishes that, for general election systems with polynomial‑time winner problems, online candidate control lies in PSPACE.
3. It proves PSPACE‑completeness for carefully constructed election systems, showing that the upper bound is tight.
4. It identifies plurality as an important exception where online control is efficiently solvable.
5. It provides a foundation for future research on online manipulation, including extensions to other voting rules, probabilistic models of voter behavior, and mechanisms to mitigate online control attacks.

The results have practical implications for the design of election processes in settings such as talent shows, academic hiring, or any scenario where candidates are evaluated one after another. Understanding that certain voting rules can be vulnerable to computationally hard online control while others admit efficient safeguards helps policymakers and system designers choose appropriate mechanisms to preserve fairness and resistance to manipulation.