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We consider a supply chain supernetwork constrained by stringent carbon policy of mandatory cap, the purpose of which is to examine how stringent carbon regulations and operational decision modes jointly influnce the profitability and emission control of the system. We show that two key aspects of caps sharing schemes and decision modes eventually constitute four basic configurations of operations environment. Drawing on these configurations, this paper formulates novel carbon-capped mathematical gaming models to portray the supply chain supernetwork operations, which is then transformed into equivalent ones based on variational inequality approach and solved by devising augmented algorithms. The algorithms empower us to conduct typical computational experiments to compare supply chain performances resulting from diverse scenarios. The results reveal that only when the mandatory cap is set in intermediate level rather than excessively mild or tight can it be effective to balance system profitability and emission control well. Furthermore, regardless of decision modes, caps sharing scheme can always incur more system profit than the nonsharing scheme, while this rule does not apply to total carbon emission. Specifically, it is illustrated that properly changing the combination of decision mode and cap scheme can achieve Pareto improvement on low-carbon supernetwork performances (i.e. increasing profit and reducing emission), which is however dependent on the emission intensities contrast of all emitting firms. This study provides the advanced analytical framework and new insights for better understanding the joint impact of stringent (mandatory) cap policies and decision modes on supply chain performances. We shed light on that properly adjusting the cap tightness and the combination of decision mode and cap-sharing scheme may achieve Paretoly improvement for individual profits and particularly attain win-win result for system profit and emission abatement. [ABSTRACT FROM AUTHOR]