Majority Voting for Pairwise Sorting

• The paper presents a majority voting mechanism that aggregates incomplete preferences in sequential pairwise comparisons to determine overall winners.
• The methodology utilizes a binary tree structure to organize contests, ensuring fairness even when agendas are uncertain.
• Computational analysis reveals that while identifying guaranteed winners is polynomial, determining potential winners is NP-complete.

Overview of Majority Voting Mechanism for Pairwise Sorting

The majority voting mechanism for pairwise sorting is a process used to determine a collective preference order for a set of alternatives by conducting a series of pairwise comparisons. In each comparison, voters express their preference between two options, and the majority preference is used to decide the winner. This mechanism can handle scenarios with incomplete preferences and uncertain agendas, making it applicable in various real-world decision-making contexts.

Sequential Majority Voting Process

The sequential majority voting process involves organizing pairwise comparisons in a structured sequence, typically represented by a binary tree (referred to as the agenda). Each leaf node of the tree corresponds to a candidate, and internal nodes represent pairwise contests between candidates. The winner of each contest is determined based on a weighted majority vote among the agents. This process continues until a single candidate emerges as the overall winner at the root of the tree.

Handling Incomplete Preferences

In many scenarios, agents may have incomplete preference information due to privacy concerns or because the elicitation process is ongoing. The voting mechanism accounts for incomplete preferences by considering all possible completions of the agents' preferences. It assesses candidates based on notions like weak and strong Condorcet winners, which evaluate a candidate's chances of winning across all potential completions and agendas.

Uncertain Agenda

The order in which pairwise contests occur can significantly impact the outcome, especially when the agenda is uncertain or manipulatable. This uncertainty is addressed by analyzing how different agenda structures affect the likelihood of various candidates winning. The mechanism considers both standard and balanced agendas, where each candidate faces an equal number of contests, to ensure fairness.

Computational Complexity

Determining the outcome of sequential majority voting involves complex computational considerations:

• Identifying whether a candidate always wins (a guaranteed or Condorcet winner) across all agendas and preference completions is computationally easy (polynomial time).
• Determining if a candidate could potentially win under at least one agenda and one completion is NP-complete, even with a small number of candidates.

Practical Implications

The mechanism's design allows for efficient decision-making in settings where agents' preferences are incomplete or the order of comparisons is uncertain. It emphasizes the identification of guaranteed winners and reveals the computational challenges involved in assessing possible winners. These insights are valuable for multi-agent systems, such as committees or automated decision-making platforms, and help in developing robust preference elicitation procedures.

Examples and Case Studies

The paper presents various examples to illustrate the concepts:

• Scenarios with incomplete profiles and the impact on majority graph representation.
• The effect of different agendas on candidate outcomes.
• The role of complete, transitive completions in ensuring accurate winner assessments.

These examples highlight the importance of directly engaging with profiles rather than relying solely on majority graphs, as they can lead to inaccurate conclusions about possible winners. The work also demonstrates how NP-hardness reductions provide insight into the computational boundaries of the mechanism.