How Hard Is It to Control an Election by Breaking Ties? (1304.6174v2)

Abstract: We study the computational complexity of controlling the result of an election by breaking ties strategically. This problem is equivalent to the problem of deciding the winner of an election under parallel universes tie-breaking. When the chair of the election is only asked to break ties to choose between one of the co-winners, the problem is trivially easy. However, in multi-round elections, we prove that it can be NP-hard for the chair to compute how to break ties to ensure a given result. Additionally, we show that the form of the tie-breaking function can increase the opportunities for control. Indeed, we prove that it can be NP-hard to control an election by breaking ties even with a two-stage voting rule.

Analyzing the Computational Complexity of Tie-Breaking as a Control Mechanism in Elections

The paper conducted by Mattei, Narodytska, and Walsh explores the intricate issues surrounding electoral control, specifically through the manipulation of tie-breaking mechanisms. The research primarily concerns itself with the computational complexity involved in deciding the winner of an election when two or more candidates tie, a situation akin to the parallel universes tie-breaking (PUT) problem.

Context and Importance

In the domain of computational social choice, understanding how electoral outcomes can be influenced or controlled is paramount. The problem of controlling election outcomes through various manipulation techniques, such as strategic voting or candidate removal, has been extensively explored and characterized as NP-hard in many instances. However, an intriguing aspect that lacks comprehensive paper is electoral control through tie-breaking—a common real-world scenario where decision-makers, often positioned as impartial "chairs," may possess opportunities to influence outcomes by strategically breaking ties.

Key Contributions

The paper rigorously establishes that controlling the outcome of an election by breaking ties can range from being trivially computable to NP-complete, depending on the electoral system and the specifics of the tie-breaking mechanism. The main findings include:

• Complexity Results for Single and Multi-Round Voting Rules: The authors prove that when tie-breaking happens once, typically at the end of the election process, control by tie-breaking is polynomially solvable. This result applies to a majority of traditional scoring rules, Bucklin, Black's rule, Maximin, and Copeland$^\alpha$. Conversely, in multi-round voting systems where the chair has to break ties at successive stages, such as in STV, Baldwin, and Coombs rules, the problem becomes NP-complete.
• Extension to Two-Stage Voting Rules: Interestingly, even some two-stage rules, which ostensibly require fewer decisions, can present NP-complete complexity. The authors craft a two-stage voting rule that initially eliminates half of the candidates using veto and then selects a winner with plurality. They demonstrate that control by strategic tie-breaking in this hybrid system is NP-complete.
• Nuanced Control in Specific Tournament Settings: Further, for systems like Cup schedules or certain iterations of Copeland that demand non-traditional tie-breaking (e.g., non-transitive orders), the opportunities for control increase significantly, further complicating the computational landscape.

Implications and Future Directions

The implications of these results reach far into both theoretical understanding and practical applications. From a theoretical perspective, these findings contribute to the ongoing discourse on the limits of electoral control and manipulation, adding depth to the nuanced role that tie-breaking rules play. Practically, the research highlights the potential vulnerabilities in real-world elections, particularly in systems with ambiguous or poorly defined tie-breaking methods.

For future research, the authors suggest empirical evaluations extending beyond worst-case scenarios to assess real-world applicability. The integration of datasets, like those from PrefLib, may offer insights into how often such computationally theoretical challenges present in actual electoral contexts.

In conclusion, this paper not only sharpens the understanding of election control mechanisms through the lens of computational complexity but also provides a robust analytical framework for future explorations into the strategic dimensions of electoral processes.