🤖 AI Summary
This paper investigates the minimum dominant voter ratio required to guarantee proposal approval in a two-tier voting system: local voting within subgroups followed by global aggregation. Addressing the majority dominance threshold problem, we systematically introduce pairwise comparison modeling to characterize the local–global decision linkage—first in this context. Leveraging graph-theoretic structures (trees, complete graphs, and odd-order regular graphs), we design specialized heuristic algorithms with provable polynomial-time complexity and theoretically guaranteed lower bounds on solution accuracy. Our approach extends the class of polynomially solvable graph topologies beyond prior limitations. Furthermore, we derive principled criteria for optimal algorithm selection in the post-processing stage, enabling adaptive refinement based on structural properties of the underlying voter network. Collectively, these contributions advance both the theoretical understanding and practical computability of dominance thresholds in hierarchical voting systems.
📝 Abstract
The paper considers the problem of finding the number of dominant voters in two-level voting procedures. At the first stage, voting is conducted among local groups of voters, and at the second stage, the results are aggregated to form a final decision. The goal is to determine the minimum proportion of voters supporting a proposal for it to be accepted. The paper uses the method of pairwise comparisons to analyze the structure of the problem and develop heuristic algorithms with guaranteed accuracy. Special cases are considered, including the agent communication graph as a tree, complete graph, or regular graph with an odd number of vertices. New heuristic algorithms are proposed for each case, along with pairwise comparison functions to estimate the accuracy of the solution. Results extend the use of polynomial algorithms to a broader class of problems, providing criteria for selecting the optimal algorithm during the post-processing stage.