The framework operates within a Pythagorean fuzzy multi-scale information system, a setting designed for situations in which different decision makers grade their assessments at different levels of granularity. One expert might rate a transportation option simply as poor, fair, or excellent, while another provides a finely tuned linguistic scale with seven or nine gradations. The researchers first deploy a recursive scale-expansion mechanism that converts these heterogeneous Pythagorean fuzzy evaluations into representations at different levels of granularity. Pythagorean fuzzy sets extend ordinary fuzzy logic by assigning each element both a membership degree and a non-membership degree whose squares must sum to no more than one, giving evaluators more room to express hesitation. The recursive mechanism ensures that opinions expressed at coarse scales can be consistently compared and merged with those expressed at fine scales without discarding information in the process.

What distinguishes the new framework from earlier consensus models is its refusal to treat decision makers as identical rational agents. Decades of behavioral economics research, most famously the work of Daniel Kahneman and Amos Tversky, have shown that real people distort probabilities, fear losses more than they value equivalent gains, and weigh unlikely events in systematically biased ways. The team captures this heterogeneity through a hybrid behavioral model that combines cumulative prospect theory with rank-dependent utility. Cumulative prospect theory describes how individuals evaluate gains and losses relative to a reference point, weighting outcomes by a distorted probability function, while rank-dependent utility transforms cumulative probabilities in a rank-ordered fashion. Merging the two yields five psychological parameters for each decision maker: sensitivity to gains, sensitivity to losses, a loss-aversion coefficient, and two curvature parameters describing probability distortion in the gain and loss domains.

Estimating these parameters is far from trivial, and the paper devotes a detailed appendix to the problem. The authors designed a structured questionnaire of nineteen binary choices, each pitting a sure outcome against a probabilistic lottery, following the standard elicitation paradigm in behavioral decision theory. Participants switch preferences at points that reveal their certainty equivalents for various lotteries, and those switching points allow the researchers to back out each parameter. Gain sensitivity, for example, is estimated from the ratio of the probability weighting at even odds to the logarithm of the normalized certainty equivalent, while the loss-aversion coefficient follows from the indifference gain and loss identified in a mixed gamble. Probability-weighting curvatures are fitted by least squares across probability levels ranging from one percent to eighty percent. The team notes that this choice-based approach reduces the cognitive burden on respondents and improves the reliability of the resulting estimates compared with direct questioning.

Once every decision maker carries a psychological profile vector, the framework clusters participants hierarchically according to the similarity of their estimated parameters. This step recognizes that a risk-seeking optimist and a loss-averse pessimist will interpret the same fuzzy evaluation in profoundly different ways, and that consensus mechanics should account for those differences rather than averaging them away. The clustering partitions the large group into subgroups of psychologically like-minded members, each of which can then be managed with its own consensus strategy. Within each subgroup, social network analysis takes over: by mapping who trusts, consults, or influences whom, the method computes degree-centrality weights that identify the most connected and persuasive individuals. Feedback and opinion adjustment are targeted at those influential members, amplifying the effect of each intervention without forcing every participant to revise their views repeatedly.

The second layer of the framework coordinates the subgroups themselves, and this is where machine learning enters the picture. A regularized graph attention network learns adaptive weights for the different clusters, attending more strongly to subgroups whose current positions matter most for closing the consensus gap. Graph attention networks compute weights dynamically from the structure and content of the network rather than fixing them in advance, which makes them well suited to the shifting landscape of a consensus process as opinions evolve across iterations. Crucially, the authors add a regularization term that penalizes excessive weight concentration, preventing the model from pouring all its attention into one dominant cluster and neglecting the tail of smaller groups whose agreement is still needed. The result is an inter-group negotiation that adapts its weighting scheme as the discussion progresses while remaining numerically stable.

To demonstrate the machinery in action, the researchers applied the framework to a real-world class of problem: selecting a low-carbon urban transportation scheme. Such decisions are ideal testbeds because they involve large panels of stakeholders, from traffic engineers to environmental planners, who naturally differ in both the precision of their assessments and their tolerance for risk about future costs and emissions. When the authors benchmarked their method against five representative group decision-making approaches, the double-layer framework reached the prescribed consensus threshold in fewer inter-group iterations and with lower total adjustment cost, meaning that experts had to modify their opinions less aggressively to reach collective agreement. The method also preserved more of the original evaluation information, an important consideration since forcing coarse or over-smoothed judgments erodes the very expertise that large panels are convened to harvest.

The team went beyond a single demonstration, subjecting the framework to both ablation studies and sensitivity analyses. In the ablation experiments, removing or disabling individual components—such as the behavioral clustering, the social-network-based influence weighting, or the regularization of the attention network—degraded performance, confirming that each element contributes measurably to the overall result. The sensitivity analyses varied key parameters across plausible ranges and found that the outcomes remained stable, suggesting that the method is not a fragile artifact of carefully tuned settings but a robust procedure that practitioners could realistically deploy. That robustness matters, because consensus models that only work under idealized conditions rarely survive contact with the noisy, deadline-driven environments of municipal planning and corporate strategy.

The implications extend well beyond transportation policy. Any setting in which many stakeholders with vague, multi-granularity information and divergent risk attitudes must converge—health technology assessment, disaster response planning, infrastructure investment, environmental regulation—could benefit from a consensus mechanism that respects both the uncertainty of the data and the psychology of the people. By uniting Pythagorean fuzzy mathematics, behavioral economics, social network analysis, and graph neural networks in a single coherent architecture, the Shanxi University team has sketched a template for what next-generation group decision support might look like: models that do not pretend humans are perfectly rational, information is perfectly comparable, or influence is perfectly flat, but instead build those imperfections into the mathematics of agreement itself. The work was supported by the Humanities and Social Sciences Research Project of the Ministry of Education of China, and the article is available open access.

Subject of Research: Group decision-making with multi-scale fuzzy information and behavioral preferences

Article Title: A double-layer consensus framework for group decision-making with multi-scale fuzzy information and behavioral preferences

Article References: Chai, M., Fan, J., Lu, J., Wu, M., Cheng, R., & Chen, R. (2026). A double-layer consensus framework for group decision-making with multi-scale fuzzy information and behavioral preferences. Complex & Intelligent Systems. https://doi.org/10.1007/s40747-026-02492-0

Image Credits: AI Generated

DOI: 10.1007/s40747-026-02492-0

Keywords: group decision-making, Pythagorean fuzzy sets, consensus framework, cumulative prospect theory, rank-dependent utility, social network analysis, graph attention network, behavioral preferences, multi-scale information system, loss aversion, machine learning, low-carbon transportation