📊 statistics

A two-stage minimum compromise consensus model with dynamic compromise learning

This paper proposes a two-stage minimum-adjustment consensus framework that dynamically learns and extends individuals' compromise limits to achieve required consensus levels when initial adjustments are insufficient, as demonstrated in an anticancer drug screening application.

Original authors: Qun Su, Dong Cheng, Yong Wu

Published 2026-10-28

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Qun Su, Dong Cheng, Yong Wu

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the complex world of making big decisions, from choosing the next breakthrough medicine to approving a new scientific project, a group of experts must often agree on a single path forward. This process, known as group decision-making, is rarely simple. People bring different backgrounds, experiences, and fears to the table, leading to widely varying opinions on the same options. The goal is not to force everyone to think exactly alike, but to find a common ground that respects individual judgment while moving the group toward a unified conclusion. This journey toward agreement is called the consensus-reaching process. A critical part of this journey is compromise: the willingness of an individual to adjust their view slightly to get closer to the group. Traditionally, researchers have treated this willingness as a fixed setting, like a dial that is turned to a specific number before the discussion even begins. They assume that if a person is willing to move their opinion by a certain amount, that limit never changes, regardless of how the conversation evolves.

However, a new study challenges this static view, suggesting that the willingness to compromise is not a pre-set rule but a dynamic behavior that can be learned and adjusted in real time. The researchers, focusing on a scenario where experts screen potential drugs to fight cancer, found that assuming a fixed limit on how much someone is willing to change their mind often leads to a dead end. If the group needs a very high level of agreement, and everyone is stuck within their initial, unchangeable limits, they may never reach a consensus. The study proposes a two-stage approach that first learns what a person is naturally willing to accept based on their starting opinions, and then, only if necessary, gently asks them to stretch that limit just enough to reach the goal. This method treats compromise not as a rigid constraint, but as a flexible response that grows as the group gets closer to agreement.

The researchers applied this new framework to a realistic problem: selecting the best candidate for an anticancer drug. Imagine a panel of six experts, each with a different specialty, such as pharmacology, clinical medicine, or safety testing. They are asked to rate six different drug molecules. Initially, their opinions are scattered; one expert might rate a drug very highly while another rates it poorly. In a traditional model, the researchers would have to guess beforehand how much each expert is willing to move their score toward the group average. They might arbitrarily decide that everyone is willing to shift their opinion by 20 percent. The problem with this approach is that it ignores the reality of the situation. Some experts might be naturally more flexible, while others are more stubborn, and these differences are often invisible until the discussion actually begins. Furthermore, if the group needs a 98.5 percent level of agreement, a fixed 20 percent shift might not be enough to bridge the gap, leaving the group stuck in disagreement.

To solve this, the authors developed a method that works in two distinct phases. In the first stage, the computer analyzes the experts' initial opinions to figure out their natural "compromise limit." Instead of guessing, the system looks at how far apart an expert's opinion is from the group's average and calculates the maximum amount that expert could reasonably move without violating their own internal logic. It is a process of discovery, identifying the boundary of what each person can accept based on the data they provided. In the case of the drug screening, this learning phase revealed that the experts had very different natural limits. Some were willing to move their opinions significantly, while others were much more resistant. The system then checked if these natural limits were enough to reach the required 98.5 percent agreement.

In the specific case study, the initial learning phase showed that the experts could not reach the high consensus target just by staying within their naturally learned limits. The group was close, but not quite there. This is where the second stage of the model kicks in. Rather than forcing everyone to change their minds by the same amount, the system identifies which experts need to stretch their limits just a little bit more to close the gap. It does this by calculating the smallest possible extra adjustment needed to push the group over the finish line. The goal is to achieve the consensus with the absolute minimum amount of extra effort. In the drug screening simulation, this meant that some experts had to extend their willingness to compromise slightly beyond what they had initially shown, while others stayed within their original range. The result was a successful consensus that respected the individual differences of the group while meeting the strict requirement for agreement.

The study also tested how sensitive this process is to different settings. When the researchers demanded an even higher level of agreement, the system required more iterations and slightly larger adjustments, showing that the path to consensus is not a straight line but a curve that gets steeper as the goal becomes harder to reach. They also looked at how fast the system could find a solution by changing the size of the steps it took to adjust opinions. They found that taking larger steps made the process faster, but the final result remained the same regardless of the step size. This suggests the method is robust and reliable, capable of finding the optimal solution without getting stuck or wasting time.

Perhaps most importantly, the researchers compared their new dynamic learning method against older models that used fixed, pre-set compromise limits. The results were clear: the dynamic approach achieved a higher level of consensus. In the simulation, the fixed models managed to reach agreement levels of roughly 87 percent and 91 percent, depending on how high they set the initial limits. The new dynamic method, which learned and then extended the limits as needed, reached 95.7 percent. This represents a significant improvement, proving that allowing the compromise behavior to evolve during the process is more effective than trying to predict it in advance. The study concludes that by learning from the data and adapting to the specific needs of the group, decision-makers can achieve better outcomes without forcing unnatural or uniform changes on the participants.

This work offers a practical way forward for any situation where diverse experts must agree on a complex issue. Whether it is selecting a new drug, evaluating scientific projects, or planning urban development, the ability to learn how much people are willing to compromise and then gently extend that willingness only when necessary provides a more realistic and effective path to agreement. The study does not claim to have solved every problem in group decision-making, but it demonstrates that treating compromise as a dynamic, learnable behavior rather than a fixed rule leads to more stable and higher-quality consensus. By focusing on the actual behavior of the group rather than theoretical assumptions, the researchers have provided a tool that respects individual judgment while ensuring the group can move forward together.

Technical Summary: A Two-Stage Minimum Compromise Consensus Model with Dynamic Compromise Learning

Problem Statement
In Group Decision Making (GDM), particularly in complex contexts like drug screening or scientific evaluation, achieving consensus requires balancing the coordination of divergent opinions with the preservation of individual judgments. Existing Consensus Reaching Process (CRP) models typically rely on pre-set, fixed compromise limits (parameters representing an individual's willingness to adjust). This approach faces three critical limitations:

Static Assumptions: Individuals often cannot specify acceptable compromise ranges before the process begins, and pre-assigned parameters fail to capture the dynamic nature of opinion adjustment.
Infeasibility: When initial compromise limits are insufficient to overlap or reach a required consensus threshold, existing models often fail to provide a mechanism for further adjustment.
Lack of Differentiation: Most studies do not distinguish between initial compromise learning and the additional adjustments required when consensus is not met, nor do they optimize for minimum adjustment when extending these limits.

Methodology
The paper proposes a Two-Stage Minimum Compromise Consensus (MCC) framework that integrates dynamic compromise learning. The framework operates under the premise that compromise limits should be learned from opinion data rather than pre-defined.

Stage 1: Initial Compromise Learning
- Objective: Identify each individual's initial compromise limit ( $\beta_i$ ) directly from the interaction between individual opinions ( $o_{ij}$ ) and the group opinion ( $o_{cj}$ ) under deviation constraints.
- Mechanism: The model introduces a deviation constraint where the adjustment of an individual's opinion must not exceed their persistence degree ( $1 - \beta_i$ ). The update rule is defined as $\bar{o}_{ij} = (1 - \beta_i)o_{ij} + \beta_i o_{cj}$ .
- Optimization: The model (M-I) seeks to maximize the initial compromise degree $\beta_i$ such that the deviation between the updated individual opinion and the initial opinion remains within the acceptable bound ( $|\bar{o}_{ij} - o_{ij}| \leq 1 - \beta_i$ ). This is solved iteratively until the group opinion converges.
- Outcome: If the consensus level ( $CL$ ) meets the predefined threshold ( $\varepsilon$ ), the process ends. If not, the system proceeds to Stage 2.
Stage 2: Extended Compromise Learning
- Objective: Achieve the required consensus level with the minimum total compromise adjustment when the initial state is insufficient.
- Mechanism: The model activates an extended compromise process where individuals can expand their compromise limits beyond the initial learned state ( $\beta^*_i$ ) to a new state ( $\bar{\beta}_i$ ).
- Optimization: The model (M-II) minimizes a quadratic cost function representing the total adjustment: $F = \sum (\bar{\beta}_i - \beta^*_i)^2$ .
- Strategy: The algorithm iteratively increases the compromise degree of selected individuals, prioritizing those who can leverage the maximum consensus improvement with the minimum compromise expansion. This continues until $CL \geq \varepsilon$ .

Key Contributions

Two-Stage MCC Model: The study separates initial compromise learning from extended compromise learning. Unlike single-process optimization models, this framework clarifies the role of learning in CRP and achieves target consensus through a dynamic optimization form.
Data-Driven Initial Compromise Learning: The paper proposes a method to identify initial compromise limits from opinion data using deviation constraints, overcoming the limitation of pre-set parameters that ignore individual heterogeneity.
Extended Compromise Mechanism: The model addresses the infeasibility of consensus when initial limits are insufficient. It introduces an adaptive mechanism to extend compromise degrees while minimizing the additional deviation from the initially learned state, ensuring the MCC is achieved efficiently.

Experimental Results
The proposed method was applied to an anticancer drug screening problem involving six experts evaluating six drug candidates.

Convergence: The initial compromise learning converged rapidly (within 9 iterations), identifying stable and heterogeneous compromise limits for each expert.
Consensus Evolution: When the initial compromise state failed to meet the strict consensus threshold ( $\varepsilon = 0.985$ ), the extended compromise stage successfully improved the consensus level to the target.
Adjustment Analysis: The results demonstrated a nonlinear relationship between consensus improvement and adjustment; as the threshold was approached, further improvements required steeper adjustments.
Comparative Performance: Compared to static low and static high compromise settings, the proposed dynamic learning model achieved a higher consensus level ( $CL = 0.957$ vs. $0.868$ and $0.910$). It achieved this by allowing selective adjustments beyond the initial boundary rather than uniform changes, effectively balancing individual opinion preservation with group consensus.

Significance and Claims
The paper claims that the proposed framework offers a more realistic and effective approach to CRP by:

Eliminating Parameter Dependence: Shifting from fixed, pre-defined compromise parameters to a data-driven learning approach that captures individual differences.
Handling Infeasibility: Providing a structured mechanism to resolve consensus failures when initial compromise ranges do not overlap, converting infeasible problems into optimizable ones.
Optimizing Adjustment: Ensuring that when consensus requires extending limits, the total adjustment is minimized, thereby respecting individual persistence while achieving group goals.
Behavioral Interpretability: The model reflects the behavioral pattern of individual concession and consensus evolution more accurately than static models, as it learns compromise limits from observed opinion data rather than assuming them.

The authors conclude that this approach balances consensus improvement, individual acceptability, and adjustment control, offering a robust tool for complex GDM scenarios where expert opinions are heterogeneous and consensus requirements are high.

Technical Summary: A Two-Stage Minimum Compromise Consensus Model with Dynamic Compromise Learning

More like this