With the rise of climate disasters, consumers have growing interest in low-carbon products. Considering the information updating of low-carbon preferences, this paper analyses incentive strategies for low-carbon supply chains. Based on the public's increase in environmental awareness during the lead time of supply chains, this paper first describes information updating in low-carbon supply chains. Second, we analyze the response decisions of this supply chain under the given incentive strategies. Based on these decisions, an optimal model of carbon reduction is then designed for the government. Finally, these best incentive strategies (including promotion allowance and carbon reduction) are optimized using a heuristic algorithm. The numerical results reveal that cooperation of profit-driven supply chain members improves not only their profits but also carbon reduction efficiency. Governments should promote the coordination of low-carbon supply chains to realize win-win outcomes. In addition, a reasonably higher carbon reduction level and sale price can both help to weaken the bullwhip effect of the supply chain. Effective information updating improves the carbon reduction efficiency better than a promotion allowance.