Limited-budget consensus with maximum consensus level for group decision making

Group decision making usually requires in-depth discussions to form a consensus acceptable to the entire group, which has attracted more and more research in recent years. Despite extensive studies on soft consensus, the relationship between consensus level and consensus cost remains unclear. This study establishes—for the first time—a precise mathematical relationship demonstrating that higher consensus levels require proportionally greater consensus costs. This finding provides critical theoretical grounding for consensus modeling. Recognizing that the cost of achieving consensus cannot be infinite and must be within a certain budget, we develop a model to determine the maximum achievable consensus level under a limited-budget. The consensus model with non-cooperators is also explored and formulated. The proposed models are applied to online lending platforms, providing a practical framework for measuring consensus levels and achieving soft consensus between lenders and borrowers within a limited-budget. This work contributes to understanding the relationship between consensus level and consensus cost, as well as the achievement of the maximum possible consensus level under limited-budget, which is relevant in scenarios where resources are finite.